# Linear algebra

ivy.cholesky(x, /, *, upper=False, out=None)[source]

Computes the cholesky decomposition of the x matrix.

Parameters
• x (`Union`[`Array`, `NativeArray`]) – input array having shape (…, M, M) and whose innermost two dimensions form square symmetric positive-definite matrices. Should have a floating-point data type.

• upper (`bool`) – If True, the result must be the upper-triangular Cholesky factor U. If False, (default: `False`) the result must be the lower-triangular Cholesky factor L. Default: `False`.

• out (`Optional`[`Array`]) – optional output array, for writing the result to. It must have a shape that the (default: `None`) inputs broadcast to.

Return type

`Array`

Returns

ret – an array containing the Cholesky factors for each square matrix. If upper is False, the returned array must contain lower-triangular matrices; otherwise, the returned array must contain upper-triangular matrices. The returned array must have a floating-point data type determined by Type Promotion Rules and must have the same shape as x.

This function conforms to the Array API Standard. This docstring is an extension of the docstring in the standard.

Both the description and the type hints above assumes an array input for simplicity, but this function is nestable, and therefore also accepts `ivy.Container` instances in place of any of the arguments.

Examples

With `ivy.Array` input:

```>>> x = ivy.array([[4.0, 1.0, 2.0, 0.5, 2.0],
...                [1.0, 0.5, 0.0, 0.0, 0.0],
...                [2.0, 0.0, 3.0, 0.0, 0.0],
...                [0.5, 0.0, 0.0, 0.625, 0.0],
...                [2.0, 0.0, 0.0, 0.0, 16.0]])
>>> l = ivy.cholesky(x, upper='false')
>>> print(l)
ivy.array([[ 2.  ,  0.5 ,  1.  ,  0.25,  1.  ],
[ 0.  ,  0.5 , -1.  , -0.25, -1.  ],
[ 0.  ,  0.  ,  1.  , -0.5 , -2.  ],
[ 0.  ,  0.  ,  0.  ,  0.5 , -3.  ],
[ 0.  ,  0.  ,  0.  ,  0.  ,  1.  ]])
```
```>>> x = ivy.array([[4.0, 1.0, 2.0, 0.5, 2.0],
...                [1.0, 0.5, 0.0, 0.0, 0.0],
...                [2.0, 0.0, 3.0, 0.0, 0.0],
...                [0.5, 0.0, 0.0, 0.625, 0.0],
...                [2.0, 0.0, 0.0, 0.0, 16.0]])
>>> y = ivy.zeros([5,5])
>>> ivy.cholesky(x, upper='false', out=y)
>>> print(y)
ivy.array([[ 2.  ,  0.5 ,  1.  ,  0.25,  1.  ],
[ 0.  ,  0.5 , -1.  , -0.25, -1.  ],
[ 0.  ,  0.  ,  1.  , -0.5 , -2.  ],
[ 0.  ,  0.  ,  0.  ,  0.5 , -3.  ],
[ 0.  ,  0.  ,  0.  ,  0.  ,  1.  ]])
```
```>>> x = ivy.array([[4.0, 1.0, 2.0, 0.5, 2.0],
...                [1.0, 0.5, 0.0, 0.0, 0.0],
...                [2.0, 0.0, 3.0, 0.0, 0.0],
...                [0.5, 0.0, 0.0, 0.625, 0.0],
...                [2.0, 0.0, 0.0, 0.0, 16.0]])
>>> ivy.cholesky(x, upper='false', out=x)
>>> print(x)
ivy.array([[ 2.  ,  0.5 ,  1.  ,  0.25,  1.  ],
[ 0.  ,  0.5 , -1.  , -0.25, -1.  ],
[ 0.  ,  0.  ,  1.  , -0.5 , -2.  ],
[ 0.  ,  0.  ,  0.  ,  0.5 , -3.  ],
[ 0.  ,  0.  ,  0.  ,  0.  ,  1.  ]])
```

With `ivy.NativeArray` input:

```>>> x = ivy.array([[1., -2.], [2., 5.]])
>>> u = ivy.cholesky(x, upper='false')
>>> print(u)
ivy.array([[ 1., -2.],
[ 0.,  1.]])
```

With `ivy.Container` input:

```>>> x = ivy.Container(a=ivy.array([[3., -1],[-1., 3.]]),
...                   b=ivy.array([[2., 1.],[1., 1.]]))
>>> y = ivy.cholesky(x, upper='false')
>>> print(y)
{
a: ivy.array([[1.73, -0.577],
[0., 1.63]]),
b: ivy.array([[1.41, 0.707],
[0., 0.707]])
}
```

With multiple `ivy.Container` inputs:

```>>> x = ivy.Container(a=ivy.array([[3., -1],[-1., 3.]]),
...                   b=ivy.array([[2., 1.],[1., 1.]]))
>>> upper = ivy.Container(a=1, b=-1)
>>> y = ivy.cholesky(x, upper='false')
>>> print(y)
{
a: ivy.array([[1.73, -0.577],
[0., 1.63]]),
b: ivy.array([[1.41, 0.707],
[0., 0.707]])
}
```

With a mix of `ivy.Array` and `ivy.Container` inputs:

```>>> x = ivy.array([[1., -2.], [2., 5.]])
>>> upper = ivy.Container(a=1, b=-1)
>>> y = ivy.cholesky(x, upper='false')
>>> print(y)
ivy.array([[ 1., -2.],
[ 0.,  1.]])
```
ivy.cross(x1, x2, /, *, axisa=-1, axisb=-1, axisc=-1, axis=None, out=None)[source]

The cross product of 3-element vectors. If x1 and x2 are multi-dimensional arrays (i.e., both have a rank greater than 1), then the cross- product of each pair of corresponding 3-element vectors is independently computed.

Parameters
• x1 (`Union`[`Array`, `NativeArray`]) – first input array. Should have a numeric data type.

• x2 (`Union`[`Array`, `NativeArray`]) – second input array. Must have the same shape as x1. Should have a numeric data type.

• axis (`Optional`[`int`]) – the axis (dimension) of x1 and x2 containing the vectors for which to compute (default: `None`) the cross product.vIf set to -1, the function computes the cross product for vectors defined by the last axis (dimension). Default: `-1`.

• out (`Optional`[`Array`]) – optional output array, for writing the result to. It must have a shape that the (default: `None`) inputs broadcast to.

Return type

`Array`

Returns

ret – an array containing the cross products. The returned array must have a data type determined by Type Promotion Rules.

This function conforms to the Array API Standard. This docstring is an extension of the docstring in the standard.

Both the description and the type hints above assumes an array input for simplicity, but this function is nestable, and therefore also accepts `ivy.Container` instances in place of any of the arguments.

Examples

With `ivy.Array` inputs:

```>>> x = ivy.array([1., 0., 0.])
>>> y = ivy.array([0., 1., 0.])
>>> z = ivy.cross(x, y)
>>> print(z)
ivy.array([0., 0., 1.])
```

With `ivy.Container` inputs:

```>>> x = ivy.Container(a=ivy.array([5., 0., 0.]),
...                   b=ivy.array([0., 0., 2.]))
>>> y = ivy.Container(a=ivy.array([0., 7., 0.]),
...                   b=ivy.array([3., 0., 0.]))
>>> z = ivy.cross(x,y)
>>> print(z)
{
a: ivy.array([0., 0., 35.]),
b: ivy.array([0., 6., 0.])
}
```

With a combination of `ivy.Array` and `ivy.Container` inputs:

```>>> x = ivy.array([9., 0., 3.])
>>> y = ivy.Container(a=ivy.array([1., 1., 0.]),
...                   b=ivy.array([1., 0., 1.]))
>>> z = ivy.cross(x,y)
>>> print(z)
{
a: ivy.array([-3., 3., 9.]),
b: ivy.array([0., -6., 0.])
}
```
ivy.det(x, /, *, out=None)[source]

Returns the determinant of a square matrix (or a stack of square matrices)``x``.

Parameters
• x (`Union`[`Array`, `NativeArray`]) – input array having shape `(..., M, M)` and whose innermost two dimensions form square matrices. Should have a floating-point data type.

• out (`Optional`[`Array`]) – optional output array, for writing the result to. It must have a shape that the (default: `None`) inputs broadcast to.

Return type

`Array`

Returns

ret – if `x` is a two-dimensional array, a zero-dimensional array containing the determinant; otherwise,a non-zero dimensional array containing the determinant for each square matrix. The returned array must have the same data type as `x`.

This function conforms to the Array API Standard. This docstring is an extension of the docstring in the standard.

Both the description and the type hints above assumes an array input for simplicity, but this function is nestable, and therefore also accepts `ivy.Container` instances in place of any of the arguments.

Functional Examples

With `ivy.Array` input:

```>>> x = ivy.array([[2.,4.],[6.,7.]])
>>> y = ivy.det(x)
>>> print(y)
ivy.array(-10.)
```
```>>> x = ivy.array([[3.4,-0.7,0.9],[6.,-7.4,0.],[-8.5,92,7.]])
>>> y = ivy.det(x)
>>> print(y)
ivy.array(293.46997)
```

With `ivy.NativeArray` input:

```>>> x = ivy.native_array([[3.4,-0.7,0.9],[6.,-7.4,0.],[-8.5,92,7.]])
>>> y = ivy.det(x)
>>> print(y)
ivy.array(293.46997)
```

With `ivy.Container` input:

```>>> x = ivy.Container(a = ivy.array([[3., -1.], [-1., 3.]]) ,
...                   b = ivy.array([[2., 1.], [1., 1.]]))
>>> y = ivy.det(x)
>>> print(y)
{a:ivy.array(8.),b:ivy.array(1.)}
```
ivy.diag(x, /, *, k=0, out=None)[source]

Returns the specified diagonals of the input array, or an array with the input array’s elements as diagonals.

Parameters
• x (`Union`[`Array`, `NativeArray`]) – An array with rank >= 1.

• k (`int`) – An integer that controls which diagonal to consider. (default: `0`) Positive value means superdiagonal, 0 refers to the main diagonal, and negative value means subdiagonal.

• out (`Optional`[`Array`]) – optional output array, for writing the result to. It must have a shape that the (default: `None`) inputs broadcast to.

Return type

`Array`

Returns

ret – If x is a 1-D array, the function returns a 2-D square array with the elements of input as diagonals. If x is a 2-D array, the function returns a 1-D array with the diagonal elements of x.

This function conforms to the Array API Standard. This docstring is an extension of the docstring in the standard.

Both the description and the type hints above assumes an array input for simplicity, but this function is nestable, and therefore also accepts `ivy.Container` instances in place of any of the arguments.

Functional Examples

With `ivy.Array` inputs:

```>>> x = ivy.array([[0, 1, 2],
>>>                [3, 4, 5],
>>>                [6, 7, 8]])
>>> ivy.diag(x)
ivy.array([0, 4, 8])
```
```>>> x = ivy.array([[0, 1, 2],
>>>                [3, 4, 5],
>>>                [6, 7, 8]])
>>> ivy.diag(x, k=1)
ivy.array([1, 5])
```
```>>> x = ivy.array([[0, 1, 2],
>>>                [3, 4, 5],
>>>                [6, 7, 8]])
>>> ivy.diag(x, k=-1)
ivy.array([3, 7])
```
```>>> x = ivy.array([[0, 1, 2],
>>>                [3, 4, 5],
>>>                [6, 7, 8]])
>>> ivy.diag(ivy.diag(x))
ivy.array([[0, 0, 0],
[0, 4, 0],
[0, 0, 8]])
```
ivy.diagonal(x, /, *, offset=0, axis1=-2, axis2=-1, out=None)[source]

Returns the specified diagonals of a matrix (or a stack of matrices) `x`.

Parameters
• x (`Union`[`Array`, `NativeArray`]) – input array having shape `(..., M, N)` and whose innermost two dimensions form `MxN` matrices.

• offset (`int`) – offset specifying the off-diagonal relative to the main diagonal. (default: `0`) - `offset = 0`: the main diagonal. - `offset > 0`: off-diagonal above the main diagonal. - `offset < 0`: off-diagonal below the main diagonal. Default: 0.

• axis1 (`int`) – axis to be used as the first axis of the 2-D sub-arrays from which the diagonals (default: `-2`) should be taken. Defaults to first axis (-2).

• axis2 (`int`) – axis to be used as the second axis of the 2-D sub-arrays from which the (default: `-1`) diagonals should be taken. Defaults to second axis (-1).

• out (`Optional`[`Array`]) – optional output array, for writing the result to. It must have a shape that the (default: `None`) inputs broadcast to.

• out – optional output array to write the result in. Must have the same number of dimensions as the function output.

Return type

`Array`

Returns

ret – an array containing the diagonals and whose shape is determined by removing the last two dimensions and appending a dimension equal to the size of the resulting diagonals. The returned array must have the same data type as `x`.

This function conforms to the Array API Standard. This docstring is an extension of the docstring in the standard.

Both the description and the type hints above assumes an array input for simplicity, but this function is nestable, and therefore also accepts `ivy.Container` instances in place of any of the arguments.

Functional Examples

With `ivy.Array` inputs:

```>>> x = ivy.array([[1., 2.],
...                [3., 4.]])
```
```>>> d = ivy.diagonal(x)
>>> print(d)
ivy.array([1., 4.])
```
```>>> x = ivy.array([[[1., 2.],
...                 [3., 4.]],
...                [[5., 6.],
...                 [7., 8.]]])
>>> d = ivy.diagonal(x)
>>> print(d)
ivy.array([[1., 4.],
[5., 8.]])
```
```>>> x = ivy.array([[1., 2.],
...                [3., 4.]])
```
```>>> d = ivy.diagonal(x, offset=1)
>>> print(d)
ivy.array([2.])
```
```>>> x = ivy.array([[0, 1, 2],
...                   [3, 4, 5],
...                   [6, 7, 8]])
>>> d = ivy.diagonal(x, offset=-1, axis1=0)
>>> print(d)
ivy.array([3, 7])
```
```>>> x = ivy.array([[[ 0,  1,  2],
...                 [ 3,  4,  5],
...                 [ 6,  7,  8]],
...                [[ 9, 10, 11],
...                 [12, 13, 14],
...                 [15, 16, 17]],
...                [[18, 19, 20],
...                 [21, 22, 23],
...                 [24, 25, 26]]])
>>> d = ivy.diagonal(x, offset=1, axis1=-3)
>>> print(d)
ivy.array([[1, 11],
[4, 14],
[7, 17]])
```
```>>> x = ivy.array([[[0, 1],
...                 [2, 3]],
...                [[4, 5],
...                 [6, 7]]])
>>> d = ivy.diagonal(x, offset=0, axis1=0, axis2=1)
>>> print(d)
ivy.array([[0, 6],
[1, 7]])
```
```>>> x = ivy.array([[[1., 2.],
...                 [3., 4.]],
...                [[5., 6.],
...                 [7., 8.]]])
>>> d = ivy.diagonal(x, offset=1, axis1=0, axis2=1)
>>> print(d)
ivy.array([[3.],
[4.]])
```

With `ivy.NativeArray` inputs:

```>>> x = ivy.native_array([[1., 2.],
...                       [3., 4.]])
>>> d = ivy.diagonal(x)
>>> print(d)
ivy.array([1., 4.])
```
```>>> x = ivy.native_array([[[ 0,  1,  2],
...                        [ 3,  4,  5],
...                        [ 6,  7,  8]],
...                       [[ 9, 10, 11],
...                        [12, 13, 14],
...                        [15, 16, 17]],
...                       [[18, 19, 20],
...                        [21, 22, 23],
...                        [24, 25, 26]]])
>>> d = ivy.diagonal(x, offset=1, axis1=1, axis2=-1)
>>> print(d)
ivy.array([[ 1,  5],
[10, 14],
[19, 23]])
```
```>>> x = ivy.native_array([[0, 1, 2],
...                       [3, 4, 5],
...                       [6, 7, 8]])
>>> d = ivy.diagonal(x)
>>> print(d)
ivy.array([0, 4, 8])
```

With `ivy.Container` inputs:

```>>> x = ivy.Container(
...        a = ivy.array([[7, 1, 2],
...                       [1, 3, 5],
...                       [0, 7, 4]]),
...        b = ivy.array([[4, 3, 2],
...                       [1, 9, 5],
...                       [7, 0, 6]])
...    )
>>> d = ivy.diagonal(x)
>>> print(d)
{
a: ivy.array([7, 3, 4]),
b: ivy.array([4, 9, 6])
}
```
ivy.eigh(x, /, *, UPLO='L', out=None)[source]

Returns an eigendecomposition x = QLQᵀ of a symmetric matrix (or a stack of symmetric matrices) `x`, where `Q` is an orthogonal matrix (or a stack of matrices) and `L` is a vector (or a stack of vectors).

Note

The function `eig` will be added in a future version of the specification, as it requires complex number support, once complex numbers are supported, each square matrix must be Hermitian.

Note

Whether an array library explicitly checks whether an input array is a symmetric matrix (or a stack of symmetric matrices) is implementation-defined.

Parameters

x (`Union`[`Array`, `NativeArray`]) – input array having shape `(..., M, M)` and whose innermost two dimensions form square matrices. Must have a floating-point data type.

Return type

`Tuple`[`Union`[`Array`, `NativeArray`]]

Returns

• ret – a namedtuple (`eigenvalues`, `eigenvectors`) whose

• first element must have the field name `eigenvalues` (corresponding to `L` above) and must be an array consisting of computed eigenvalues. The array containing the eigenvalues must have shape `(..., M)`.

• second element have have the field name `eigenvectors` (corresponding to `Q` above) and must be an array where the columns of the inner most matrices contain the computed eigenvectors. These matrices must be orthogonal. The array containing the eigenvectors must have shape `(..., M, M)`.

• Each returned array must have the same floating-point data type as `x`.

• .. note:: – Eigenvalue sort order is left unspecified and is thus implementation-dependent.

This function conforms to the Array API Standard. This docstring is an extension of the docstring in the standard.

Both the description and the type hints above assumes an array input for simplicity, but this function is nestable, and therefore also accepts `ivy.Container` instances in place of any of the arguments.

ivy.eigvalsh(x, /, *, UPLO='L', out=None)[source]

Return the eigenvalues of a symmetric matrix (or a stack of symmetric matrices) x.

Note

The function `eig` will be added in a future version of the specification, as it requires complex number support, once complex numbers are supported, each square matrix must be Hermitian.

Note

Whether an array library explicitly checks whether an input array is a symmetric matrix (or a stack of symmetric matrices) is implementation-defined.

Parameters
• x (`Union`[`Array`, `NativeArray`]) – input array having shape (…, M, M) and whose innermost two dimensions form square matrices. Must have floating-point data type.

• UPLO (`Optional`[`str`]) – optional string being ‘L’ or ‘U’, specifying whether the calculation is done with (default: `'L'`) the lower triangular part of x (‘L’, default) or the upper triangular part (‘U’).

• out (`Optional`[`Array`]) – optional output array, for writing the result to. It must have a shape that the (default: `None`) inputs broadcast to.

Return type

`Array`

Returns

ret – an array containing the computed eigenvalues. The returned array must have shape (…, M) and have the same data type as x.

This function conforms to the Array API Standard. This docstring is an extension of the docstring in the standard.

Both the description and the type hints above assumes an array input for simplicity, but this function is nestable, and therefore also accepts `ivy.Container` instances in place of any of the arguments.

Examples

With `ivy.Array` inputs:

```>>> x = ivy.array([[[1.0,2.0],[2.0,1.0]]])
>>> y = ivy.eigvalsh(x)
>>> print(y)
ivy.array([[-1.,  3.]])
```

Using out

```>>> x = ivy.array([[[3.0,2.0],[2.0,3.0]]])
>>> y = ivy.zeros([1,2])
>>> ivy.eigvalsh(x, out=y)
>>> print(y)
ivy.array([[1., 5.]])
```
```>>> x = ivy.array([[[3.0,2.0],[2.0,3.0]]])
>>> ivy.eigvalsh(x, out=x)
>>> print(x)
ivy.array([[1., 5.]])
```

Using UPLO

```>>> x = ivy.array([[[2.0,3.0,6.0],[3.0,4.0,5.0],[6.0,5.0,9.0]],
... [[1.0,1.0,1.0],[1.0,2.0,2.0],[1.0,2.0,2.0]]])
>>> y = ivy.eigvalsh(x, UPLO="U")
>>> print(y)
ivy.array([[-1.45033181e+00,  1.02829754e+00,  1.54220343e+01],
[-1.12647155e-15,  4.38447177e-01,  4.56155300e+00]])
```

With `ivy.NativeArray` inputs:

```>>> x = ivy.native_array([[[1., 1., 2.], [1., 2., 1.], [1., 1., 2]]])
>>> y = ivy.eigvalsh(x)
>>> print(y)
ivy.array([[0.26794919, 1.        , 3.7320509 ]])
```

With `ivy.Container` inputs:

```>>> x = ivy.Container(a=ivy.array([[[1.,2.,3.],[2.,4.,5.],[3.,5.,6.]]]),
...                      b=ivy.array([[[1.,1.,2.],[1.,2.,1.],[2.,1.,1.]]]),
...                      c=ivy.array([[[2.,2.,2.],[2.,3.,3.],[2.,3.,3.]]]))
>>> y = ivy.eigvalsh(x)
>>> print(y)
{
a: ivy.array([[-0.51572949, 0.17091519, 11.3448143]]),
b: ivy.array([[-1., 1., 4.]]),
c: ivy.array([[-8.88178420e-16, 5.35898387e-01, 7.46410179e+00]])
}
```
ivy.inner(x1, x2, /, *, out=None)[source]

Returns the inner product of two vectors `x1` and `x2`.

Parameters
• x1 (`Union`[`Array`, `NativeArray`]) – first one-dimensional input array of size N. Should have a numeric data type. a(N,) array_like First input vector. Input is flattened if not already 1-dimensional.

• x2 (`Union`[`Array`, `NativeArray`]) – second one-dimensional input array of size M. Should have a numeric data type. b(M,) array_like Second input vector. Input is flattened if not already 1-dimensional.

• out (`Optional`[`Array`]) – optional output array, for writing the result to. It must have a shape that the (default: `None`) inputs broadcast to.

Return type

`Array`

Returns

ret – a two-dimensional array containing the inner product and whose shape is (N, M). The returned array must have a data type determined by Type Promotion Rules.

Returns the multiplicative inverse of a square matrix (or a stack of square matrices) `x`.

Parameters
• x (`Union`[`Array`, `NativeArray`]) – input array having shape `(..., M, M)` and whose innermost two dimensions form square matrices. Should have a floating-point data type.

• out (`Optional`[`Array`]) – optional output array, for writing the result to. It must have a shape that the (default: `None`) inputs broadcast to.

Return type

`Array`

Returns

ret – an array containing the multiplicative inverses. The returned array must have a floating-point data type determined by type-promotion and must have the same shape as `x`.

This function conforms to the Array API Standard. This docstring is an extension of the docstring in the standard.

Both the description and the type hints above assumes an array input for simplicity, but this function is nestable, and therefore also accepts `ivy.Container` instances in place of any of the arguments.

Examples

With `ivy.Array` inputs:

```>>> x = ivy.array([[1.0, 2.0], [3.0, 4.0]])
>>> y = ivy.zeros(3)
>>> ivy.inv(x, out=y)
>>> print(y)
ivy.array([[-2., 1.],[1.5, -0.5]])
```

Using inplace

```>>> x = ivy.array([[1.0, 2.0], [5.0, 5.0]])
>>> ivy.inv(x, out=x)
>>> print(x)
ivy.array([[-1., 0.4],[1., -0.2]])
```

Inverses of several matrices can be computed at once:

```>>> x = ivy.array([[[1.0, 2.0],[3.0, 4.0]],
...                [[1.0, 3.0], [3.0, 5.0]]])
>>> y = ivy.inv(x)
>>> print(y)
ivy.array([[[-2., 1.],[1.5, -0.5]],
[[-1.25, 0.75],[0.75, -0.25]]])
```

With `ivy.Container` inputs

```>>> x = ivy.Container(a=ivy.array([[11., 100., 10.],
...                                [300., 40., 20.], [25., 30, 100.]]),
...                   b=ivy.array([[4., 400., 50.], [10., 10., 15.],
...                               [50., 5000., 40.]]), c=ivy.array([[25., 22., 100.],
...                               [55, 20., 20.], [55., 50., 100.]]))
>>> y = ivy.Container.static_inv(x)
>>> print(y)
{
a: ivy.array([[-0.0012, 0.00342, -0.000565],
[0.0104, -0.0003, -0.000981],
[-0.00282, -0.000766, 0.0104]]),
b: ivy.array([[-0.0322, 0.101, 0.00237],
[0.000151, -0.00101, 0.00019],
[0.0214, 0., -0.00171]]),
c: ivy.array([[0.0107, 0.03, -0.0167],
[-0.0472, -0.0322, 0.0536],
[0.0177, -0.000429, -0.00762]])
```

}

ivy.matmul(x1, x2, /, *, transpose_a=False, transpose_b=False, out=None)[source]

Computes the matrix product.

Parameters
• x1 (`Union`[`Array`, `NativeArray`]) – first input array. Should have a numeric data type. Must have at least one dimension.

• x2 (`Union`[`Array`, `NativeArray`]) – second input array. Should have a numeric data type. Must have at least one dimension.

• out (`Optional`[`Array`]) – optional output array, for writing the result to. It must have a shape that the (default: `None`) inputs broadcast to.

Return type

`Array`

Returns

ret

• if both x1 and x2 are one-dimensional arrays having shape (N,), a zero-dimensional array containing the inner product as its only element.

• if x1 is a two-dimensional array having shape (M, K) and x2 is a two-dimensional array having shape (K, N), a two-dimensional array

containing the conventional matrix product and having shape (M, N).

• if x1 is a one-dimensional array having shape (K,) and x2 is an array having shape (…, K, N), an array having shape (…, N) (i.e., prepended dimensions during vector-to-matrix promotion must be removed) and containing the conventional matrix product.

• if x1 is an array having shape (…, M, K) and x2 is a one-dimensional array having shape (K,), an array having shape (…, M) (i.e., appended dimensions during vector-to-matrix promotion must be removed) and containing the conventional matrix product.

• if x1 is a two-dimensional array having shape (M, K) and x2 is an array having shape (…, K, N), an array having shape (…, M, N) and containing the conventional matrix product for each stacked matrix.

• if x1 is an array having shape (…, M, K) and x2 is a two-dimensional array having shape (K, N), an array having shape (…, M, N) and containing the conventional matrix product for each stacked matrix.

• if either x1 or x2 has more than two dimensions, an array having a shape determined by Broadcasting shape(x1)[:-2] against shape(x2)[:-2] and containing the conventional matrix product for each stacked matrix.

Raises

• if either x1 or x2 is a zero-dimensional array.

• if x1 is a one-dimensional array having shape (K,), x2 is a one-dimensional

array having shape (L,), and K != L.

• if x1 is a one-dimensional array having shape (K,), x2 is an array having shape (…, L, N), and K != L.

• if x1 is an array having shape (…, M, K), x2 is a one-dimensional array having shape (L,), and K != L.

• if x1 is an array having shape (…, M, K), x2 is an array having shape (…, L, N), and K != L.

This function conforms to the Array API Standard. This docstring is an extension of the docstring in the standard.

Both the description and the type hints above assumes an array input for simplicity, but this function is nestable, and therefore also accepts `ivy.Container` instances in place of any of the arguments.

Examples

With `ivy.Array` inputs:

```>>> x = ivy.array([2., 0., 3.])
>>> y = ivy.array([4., 1., 8.])
>>> z = ivy.matmul(x, y)
>>> print(z)
ivy.array(32.)
```

With `ivy.NativeArray` inputs:

```>>> x = ivy.native_array([[1., 2.], [0., 1.]])
>>> y = ivy.native_array([[2., 0.], [0., 3.]])
>>> z = ivy.matmul(x, y)
>>> print(z)
ivy.array([[2., 6.],[0., 3.]])
```

With `ivy.Container` inputs:

```>>> x = ivy.Container(a=ivy.array([5., 1.]), b=ivy.array([1., 0.]))
>>> y = ivy.Container(a=ivy.array([4., 7.]), b=ivy.array([3., 0.]))
>>> z = ivy.matmul(x,y)
>>> print(z)
{
a: ivy.array(27.),
b: ivy.array(3.)
}
```

With a combination of `ivy.Array` and `ivy.Container` inputs:

```>>> x = ivy.array([9., 0.])
>>> y = ivy.Container(a=ivy.array([2., 1.]), b=ivy.array([1., 0.]))
>>> z = ivy.matmul(x, y)
>>> print(z)
{
a: ivy.array(18.),
b: ivy.array(9.)
}
```

With a combination of `ivy.NativeArray` and `ivy.Array` inputs:

```>>> x = ivy.native_array([[1., 2.], [0., 3.]])
>>> y = ivy.array([[1.], [3.]])
>>> z = ivy.matmul(x, y)
>>> print(z)
ivy.array([[7.],[9.]])
```
ivy.matrix_norm(x, /, *, ord='fro', axis=(-2, -1), keepdims=False, out=None)[source]

Compute the matrix p-norm.

Parameters
• x (`Union`[`Array`, `NativeArray`]) – Input array having shape (…, M, N) and whose innermost two dimensions form MxN matrices. Should have a floating-point data type.

• ord (`Optional`[`Union`[`int`, `float`, `Literal`[inf, -inf, ‘fro’, ‘nuc’]]]) –

order of the norm. The following mathematical norms must be supported: (default: `'fro'`)

ord

description

’fro’

Frobenius norm

’nuc’

nuclear norm

1

max(sum(abs(x), axis=0))

2

largest singular value

inf

max(sum(abs(x), axis=1))

The following non-mathematical “norms” must be supported:

ord

description

-1

min(sum(abs(x), axis=0))

-2

smallest singular value

-inf

min(sum(abs(x), axis=1))

If `ord=1`, the norm corresponds to the induced matrix norm where `p=1` (i.e., the maximum absolute value column sum).

If `ord=2`, the norm corresponds to the induced matrix norm where `p=inf` (i.e., the maximum absolute value row sum).

If `ord=inf`, the norm corresponds to the induced matrix norm where `p=2` (i.e., the largest singular value).

Default: “fro”.

• axis (`Optional`[`Tuple`[`int`, `int`]]) – specifies the axes that hold 2-D matrices. Default: (-2, -1). (default: `(-2, -1)`)

• keepdims (`bool`) – If this is set to True, the axes which are normed over are left in the result as (default: `False`) dimensions with size one. With this option the result will broadcast correctly against the original x. Default is `False`.

• out (`Optional`[`Array`]) – optional output array, for writing the result to. It must have a shape that the (default: `None`) inputs broadcast to.

Return type

`Array`

Returns

ret – Matrix norm of the array at specified axes.

This function conforms to the Array API Standard. This docstring is an extension of the docstring in the standard.

Both the description and the type hints above assumes an array input for simplicity, but this function is nestable, and therefore also accepts `ivy.Container` instances in place of any of the arguments.

Examples

With :class:’ivy.Array’ inputs: >>> x = ivy.array([[1., 2.], [3., 4.]]) >>> y = ivy.matrix_norm(x) >>> print(y) ivy.array(5.47722558)

```>>> x = ivy.arange(8, dtype=float).reshape((2, 2, 2))
>>> y = ivy.zeros(2)
>>> ivy.matrix_norm(x, ord=1, out=y)
>>> print(y)
ivy.array([ 4., 12.])
```
```>>> x = ivy.arange(12, dtype=float).reshape((3, 2, 2))
>>> ivy.matrix_norm(x, ord=ivy.inf, axis=(2, 1), out=x)
>>> print(x)
ivy.array([ 4., 12., 20.])
```
```>>> x = ivy.array([[1.1, 2.2], [3.3, 4.4], [5.5, 6.6]])
>>> y = ivy.matrix_norm(x, ord='nuc', keepdims=True)
>>> print(y)
ivy.array([[11.]])
```
```>>> x = ivy.array([[[1.1, 2.2, 3.3], [4.4, 5.5, 6.6]],                         [[1., 0., 1.1], [1., 1., 0.]]])
>>> ivy.matrix_norm(x, ord='fro', out=x)
>>> print(x)
ivy.array([10.5 ,  2.05])
```

With :class:’ivy.Container’ input: >>> x = ivy.Container(a=ivy.array([[0.666, 9.11], [42.69, 9.23]]), b=ivy.array([[1.1, 2.2, 3.3], [4.4, 5.5, 6.6]])) >>> y = ivy.matrix_norm(x, ord=-ivy.inf) >>> print(y) {

a: ivy.array(9.776), b: ivy.array(6.6000004)

}

With multiple :class:’ivy:Container’ inputs: >>> x = ivy.Container(a=ivy.arange(12, dtype=float).reshape((3, 2, 2)), b=ivy.arange(8, dtype=float).reshape((2, 2, 2))) >>> ord = ivy.Container(a=1, b=float(‘inf’)) >>> axis = ivy.Container(a=(1, 2), b=(2, 1)) >>> k = ivy.Container(a=False, b=True) >>> y = ivy.matrix_norm(x, ord=ord, axis=axis, keepdims=k) >>> print(y) {

a: ivy.array([4., 12., 20.]), b: ivy.array([[[4.]],

[[12.]]])

}

ivy.matrix_power(x, n, /, *, out=None)[source]

Raises a square matrix (or a stack of square matrices) x to an integer power n.

Parameters
• x (`Union`[`Array`, `NativeArray`]) – input array having shape (…, M, M) and whose innermost two dimensions form square matrices. Should have a floating-point data type.

• n (`int`) – integer exponent.

Return type

`Array`

Returns

ret – if n is equal to zero, an array containing the identity matrix for each square matrix. If n is less than zero, an array containing the inverse of each square matrix raised to the absolute value of n, provided that each square matrix is invertible. If n is greater than zero, an array containing the result of raising each square matrix to the power n. The returned array must have the same shape as x and a floating-point data type determined by Type Promotion Rules.

This function conforms to the Array API Standard. This docstring is an extension of the docstring in the standard.

Both the description and the type hints above assumes an array input for simplicity, but this function is nestable, and therefore also accepts `ivy.Container` instances in place of any of the arguments.

ivy.matrix_rank(x, /, *, atol=None, rtol=None, out=None)[source]

Returns the rank (i.e., number of non-zero singular values) of a matrix (or a stack of matrices).

Parameters
• x (`Union`[`Array`, `NativeArray`]) – input array having shape `(..., M, N)` and whose innermost two dimensions form `MxN` matrices. Should have a floating-point data type.

• atol (`Optional`[`Union`[`float`, `Tuple`[`float`]]]) – absolute tolerance. When None it’s considered to be zero. (default: `None`)

• rtol (`Optional`[`Union`[`float`, `Tuple`[`float`]]]) – relative tolerance for small singular values. Singular values approximately less (default: `None`) than or equal to `rtol * largest_singular_value` are set to zero. If a `float`, the value is equivalent to a zero-dimensional array having a floating-point data type determined by type-promotion (as applied to `x`) and must be broadcast against each matrix. If an `array`, must have a floating-point data type and must be compatible with `shape(x)[:-2]` (see broadcasting). If `None`, the default value is `max(M, N) * eps`, where `eps` must be the machine epsilon associated with the floating-point data type determined by type-promotion (as applied to `x`). Default: `None`.

• out (`Optional`[`Array`]) – optional output array, for writing the result to. It must have a shape that the (default: `None`) inputs broadcast to.

Return type

`Array`

Returns

ret – an array containing the ranks. The returned array must have a floating-point data type determined by type-promotion and must have shape `(...)` (i.e., must have a shape equal to `shape(x)[:-2]`).

This function conforms to the Array API Standard. This docstring is an extension of the docstring in the standard.

Both the description and the type hints above assumes an array input for simplicity, but this function is nestable, and therefore also accepts `ivy.Container` instances in place of any of the arguments.

Examples

With :code: ‘ivy.Array’ inputs:

1. Full Matrix

```>>> x = ivy.array([[1., 2.], [3., 4.]])
>>> ivy.matrix_rank(x)
ivy.array(2.)
```
1. Rank Deficient Matrix

```>>> x = ivy.array([[1., 0.], [0., 0.]])
>>> ivy.matrix_rank(x)
ivy.array(1.)
```
1. 1 Dimension - rank 1 unless all 0

```>>> x = ivy.array([[1., 1.])
>>> ivy.matrix_rank(x)
ivy.array(1.)
```
```>>> x = ivy.array([[0., 0.])
>>> ivy.matrix_rank(x)
ivy.array(0)
```

With :code: ‘ivy.NativeArray’ inputs:

```>>> x = ivy.native_array([[1., 2.], [3., 4.]], [[1., 0.], [0., 0.]])
>>> ivy.matrix_rank(x)
ivy.array([2., 1.])
```

With :code: ‘ivy.Container’ inputs: >>> x = ivy.Container(a = ivy.array([[1., 2.], [3., 4.]]), b = ivy.array([[1., 0.], [0., 0.]])) >>> ivy.matrix_rank(x) {

a:ivy.array(2.), b:ivy.array(1.)

}

ivy.matrix_transpose(x, /, *, out=None)[source]

Transposes a matrix (or a stack of matrices) `x`.

Parameters
• x (`Union`[`Array`, `NativeArray`]) – input array having shape `(..., M, N)` and whose innermost two dimensions form `MxN` matrices.

• out (`Optional`[`Array`]) – optional output array, for writing the result to. It must have a (default: `None`) shape that the inputs broadcast to.

Return type

`Array`

Returns

ret – an array containing the transpose for each matrix and having shape `(..., N, M)`. The returned array must have the same data type as `x`.

This function conforms to the Array API Standard. This docstring is an extension of the docstring in the standard.

Both the description and the type hints above assumes an array input for simplicity, but this function is nestable, and therefore also accepts `ivy.Container` instances in place of any of the arguments.

Examples

With :code: ‘ivy.Array’ inputs:

```>>> x = ivy.array([[0., 2.], [1., 3.]])
>>> y = ivy.matrix_transpose(x)
>>> print(y)
ivy.array([[0., 1.],
[2., 3.]])
```
```>>> x = ivy.array([[1., 4.], [2., 5.], [3., 1.]])
>>> y = ivy.zeros((3, 2))
>>> ivy.matrix_transpose(x, out=y)
ivy.array([[1., 2., 3.],
[4., 5., 1.]])
```
```>>> x = ivy.array([[2., 3.], [1., 2.]])
>>> ivy.matrix_transpose(x, out=x)
ivy.array([[2., 1.],
[3., 2.]])
```

With `ivy.NativeArray` input:

```>>> x = ivy.native_array([[0., 1., 2.], [1., 2., 3.]])
>>> y = ivy.matrix_transpose(x)
>>> print(y)
ivy.array([[0., 1.],
[1., 2.],
[2., 3.]])
```

With :code: ‘ivy.Container’ inputs:

```>>> x = ivy.Container(a=ivy.array([[0., 1.], [0., 2.]]),                           b=ivy.array([[3., 4.], [3., 5.]]))
>>> y = ivy.matrix_transpose(x)
>>> print(y)
{
a: ivy.array([[0., 0.],
[1., 2.]]),
b: ivy.array([[3., 3.],
[4., 5.]])
}
```
ivy.outer(x1, x2, /, *, out=None)[source]

Returns the outer product of two vectors `x1` and `x2`.

Parameters
• x1 (`Union`[`Array`, `NativeArray`]) – first one-dimensional input array of size N. Should have a numeric data type. a(N,) array_like First input vector. Input is flattened if not already 1-dimensional.

• x2 (`Union`[`Array`, `NativeArray`]) – second one-dimensional input array of size M. Should have a numeric data type. b(M,) array_like Second input vector. Input is flattened if not already 1-dimensional.

• out (`Optional`[`Array`]) – optional output array, for writing the result to. It must have a shape that the (default: `None`) inputs broadcast to.

Return type

`Array`

Returns

ret – a two-dimensional array containing the outer product and whose shape is (N, M). The returned array must have a data type determined by Type Promotion Rules.

This function conforms to the Array API Standard. This docstring is an extension of the docstring in the standard.

Both the description and the type hints above assumes an array input for simplicity, but this function is nestable, and therefore also accepts `ivy.Container` instances in place of any of the arguments.

Examples

```>>> x = ivy.array([[1., 2.],[3., 4.]])
>>> y = ivy.array([[5., 6.],[7., 8.]])
>>> d = ivy.outer(x,y)
>>> print(d)
ivy.array([[ 5.,  6.,  7.,  8.],
[10., 12., 14., 16.],
[15., 18., 21., 24.],
[20., 24., 28., 32.]])
```
```>>> d = ivy.outer(x, 1)
>>> print(d)
ivy.array([[1.],
[2.],
[3.],
[4.]])
```

A 3-D Example

```>>> x = ivy.array([[[1., 2.],
[3., 4.]],
[[5., 6.],
[7., 8.]]])
>>> y = ivy.array([[[9., 10.],
[11., 12.]],
[[13., 14.],
[15., 16.]]])
>>> d = ivy.outer(x, y)
>>> print(d)
ivy.array([[  9.,  10.,  11.,  12.,  13.,  14.,  15.,  16.],
[ 18.,  20.,  22.,  24.,  26.,  28.,  30.,  32.],
[ 27.,  30.,  33.,  36.,  39.,  42.,  45.,  48.],
[ 36.,  40.,  44.,  48.,  52.,  56.,  60.,  64.],
[ 45.,  50.,  55.,  60.,  65.,  70.,  75.,  80.],
[ 54.,  60.,  66.,  72.,  78.,  84.,  90.,  96.],
[ 63.,  70.,  77.,  84.,  91.,  98., 105., 112.],
[ 72.,  80.,  88.,  96., 104., 112., 120., 128.]])
```
ivy.pinv(x, /, *, rtol=None, out=None)[source]

Returns the (Moore-Penrose) pseudo-inverse of a matrix (or a stack of matrices) `x`.

Parameters
• x (`Union`[`Array`, `NativeArray`]) – input array having shape `(..., M, N)` and whose innermost two dimensions form `MxN` matrices. Should have a floating-point data type.

• rtol (`Optional`[`Union`[`float`, `Tuple`[`float`]]]) – relative tolerance for small singular values. Singular values approximately less (default: `None`) than or equal to `rtol * largest_singular_value` are set to zero. If a `float`, the value is equivalent to a zero-dimensional array having a floating-point data type determined by type-promotion (as applied to `x`) and must be broadcast against each matrix. If an `array`, must have a floating-point data type and must be compatible with `shape(x)[:-2]` (see broadcasting). If `None`, the default value is `max(M, N) * eps`, where `eps` must be the machine epsilon associated with the floating-point data type determined by type-promotion (as applied to `x`). Default: `None`.

• out (`Optional`[`Array`]) – optional output array, for writing the result to. It must have a shape that the (default: `None`) inputs broadcast to.

Return type

`Array`

Returns

ret – an array containing the pseudo-inverses. The returned array must have a floating-point data type determined by type-promotion and must have shape `(..., N, M)` (i.e., must have the same shape as `x`, except the innermost two dimensions must be transposed).

This function conforms to the Array API Standard. This docstring is an extension of the docstring in the standard.

Both the description and the type hints above assumes an array input for simplicity, but this function is nestable, and therefore also accepts `ivy.Container` instances in place of any of the arguments.

Examples

```>>> x = ivy.array([[1., 2.],[3., 4.]])
>>> y = ivy.pinv(x)
>>> print(y)
ivy.array([[-2., 1.],[1.5, -0.5]])
```
```>>> x = ivy.array([[1., 2.],[3., 4.]])
>>> out = ivy.zeros(x.shape)
>>> ivy.pinv(x, out=out)
>>> print(out)
ivy.array([[-2., 1.],[1.5, -0.5]])
```
ivy.qr(x, /, *, mode='reduced')[source]

Returns the qr decomposition x = QR of a full column rank matrix (or a stack of matrices), where Q is an orthonormal matrix (or a stack of matrices) and R is an upper-triangular matrix (or a stack of matrices).

Parameters
• x (`Union`[`Array`, `NativeArray`]) – input array having shape (…, M, N) and whose innermost two dimensions form MxN matrices of rank N. Should have a floating-point data type.

• mode (`str`) –

decomposition mode. Should be one of the following modes: (default: `'reduced'`) - ‘reduced’: compute only the leading K columns of q, such that q and r have

dimensions (…, M, K) and (…, K, N), respectively, and where K = min(M, N).

• ’complete’: compute q and r with dimensions (…, M, M) and (…, M, N), respectively.

Default: ‘reduced’.

Return type

`Tuple`[`Union`[`Array`, `NativeArray`], `Union`[`Array`, `NativeArray`]]

Returns

ret – a namedtuple (Q, R) whose - first element must have the field name Q and must be an array whose shape

depends on the value of mode and contain matrices with orthonormal columns. If mode is ‘complete’, the array must have shape (…, M, M). If mode is ‘reduced’, the array must have shape (…, M, K), where K = min(M, N). The first x.ndim-2 dimensions must have the same size as those of the input array x.

• second element must have the field name R and must be an array whose shape depends on the value of mode and contain upper-triangular matrices. If mode is ‘complete’, the array must have shape (…, M, N). If mode is ‘reduced’, the array must have shape (…, K, N), where K = min(M, N). The first x.ndim-2 dimensions must have the same size as those of the input x.

This function conforms to the Array API Standard. This docstring is an extension of the docstring in the standard.

Both the description and the type hints above assumes an array input for simplicity, but this function is nestable, and therefore also accepts `ivy.Container` instances in place of any of the arguments.

ivy.slogdet(x, /)[source]

Computes the sign and natural logarithm of the determinant of an array.

Parameters

x (`Union`[`Array`, `NativeArray`]) – input array having shape (…, M, M) and whose innermost two dimensions form square matrices. Should have a floating-point data type.

Return type

`Tuple`[`Union`[`Array`, `NativeArray`], `Union`[`Array`, `NativeArray`]]

Returns

ret

This function returns NamedTuple with two values -

sign: An array containing a number representing the sign of the determinant for each square matrix.

logabsdet: An array containing natural log of the absolute determinant of each square matrix.

Both the description and the type hints above assumes an array input for simplicity, but this function is nestable, and therefore also accepts `ivy.Container` instances in place of any of the arguments.

Examples

With `ivy.Array` input:

```>>> x = ivy.array([[1.0, 2.0],
...                [3.0, 4.0]])
>>> y = ivy.slogdet(x)
>>> print(y)
slogdet(sign=ivy.array(-1.), logabsdet=ivy.array(0.6931472))
```

With `ivy.Container` input:

```>>> x = ivy.Container(a=ivy.array([[1.0, 2.0],
...                                [3.0, 4.0]]),
...                   b=ivy.array([[1.0, 2.0],
...                                [2.0, 1.0]]))
>>> y = ivy.slogdet(x)
>>> print(y)
{
a: (list[2], <class ivy.array.array.Array> shape=[]),
b: (list[2], <class ivy.array.array.Array> shape=[])
}
```

This function conforms to the Array API Standard. This docstring is an extension of the docstring in the standard.

Both the description and the type hints above assumes an array input for simplicity, but this function is nestable, and therefore also accepts `ivy.Container` instances in place of any of the arguments.

ivy.solve(x1, x2, /, *, out=None)[source]

Returns the solution to the system of linear equations represented by the well- determined (i.e., full rank) linear matrix equation AX = B.

Parameters
• x1 (`Union`[`Array`, `NativeArray`]) – coefficient array A having shape (…, M, M) and whose innermost two dimensions form square matrices. Must be of full rank (i.e., all rows or, equivalently, columns must be linearly independent). Should have a floating-point data type.

• x2 (`Union`[`Array`, `NativeArray`]) – ordinate (or “dependent variable”) array B. If x2 has shape (M,), x2 is equivalent to an array having shape (…, M, 1). If x2 has shape (…, M, K), each column k defines a set of ordinate values for which to compute a solution, and shape(x2)[:-1] must be compatible with shape(x1)[:-1] (see Broadcasting). Should have a floating-point data type.

• out (`Optional`[`Array`]) – optional output array, for writing the result to. It must have a shape that the (default: `None`) inputs broadcast to.

Return type

`Array`

Returns

ret – an array containing the solution to the system AX = B for each square matrix. The returned array must have the same shape as x2 (i.e., the array corresponding to B) and must have a floating-point data type determined by Type Promotion Rules.

This function conforms to the Array API Standard. This docstring is an extension of the docstring in the standard.

Both the description and the type hints above assumes an array input for simplicity, but this function is nestable, and therefore also accepts `ivy.Container` instances in place of any of the arguments.

ivy.svd(x, /, *, compute_uv=True, full_matrices=True)[source]

Returns a singular value decomposition A = USVh of a matrix (or a stack of matrices) `x`, where `U` is a matrix (or a stack of matrices) with orthonormal columns, `S` is a vector of non-negative numbers (or stack of vectors), and `Vh` is a matrix (or a stack of matrices) with orthonormal rows.

Parameters
• x (`Union`[`Array`, `NativeArray`]) – input array having shape `(..., M, N)` and whose innermost two dimensions form matrices on which to perform singular value decomposition. Should have a floating-point data type.

• full_matrices (`bool`) – If `True`, compute full-sized `U` and `Vh`, such that `U` has shape (default: `True`) `(..., M, M)` and `Vh` has shape `(..., N, N)`. If `False`, compute on the leading `K` singular vectors, such that `U` has shape `(..., M, K)` and `Vh` has shape `(..., K, N)` and where `K = min(M, N)`. Default: `True`.

• compute_uv (`bool`) – If `True` then left and right singular vectors will be computed and returned (default: `True`) in `U` and `Vh`, respectively. Otherwise, only the singular values will be computed, which can be significantly faster.

• note:: (..) – with backend set as torch, svd with still compute left and right singular vectors irrespective of the value of compute_uv, however Ivy will still only return the singular values.

Return type

`Union`[`Array`, `Tuple`[`Array`, `...`]]

Returns

• .. note:: – once complex numbers are supported, each square matrix must be Hermitian.

• ret – a namedtuple `(U, S, Vh)` whose

• first element must have the field name `U` and must be an array whose shape depends on the value of `full_matrices` and contain matrices with orthonormal columns (i.e., the columns are left singular vectors). If `full_matrices` is `True`, the array must have shape `(..., M, M)`. If `full_matrices` is `False`, the array must have shape `(..., M, K)`, where `K = min(M, N)`. The first `x.ndim-2` dimensions must have the same shape as those of the input `x`.

• second element must have the field name `S` and must be an array with shape `(..., K)` that contains the vector(s) of singular values of length `K`, where `K = min(M, N)`. For each vector, the singular values must be sorted in descending order by magnitude, such that `s[..., 0]` is the largest value, `s[..., 1]` is the second largest value, et cetera. The first `x.ndim-2` dimensions must have the same shape as those of the input `x`.

• third element must have the field name `Vh` and must be an array whose shape depends on the value of `full_matrices` and contain orthonormal rows (i.e., the rows are the right singular vectors and the array is the adjoint). If `full_matrices` is `True`, the array must have shape `(..., N, N)`. If `full_matrices` is `False`, the array must have shape `(..., K, N)` where `K = min(M, N)`. The first `x.ndim-2` dimensions must have the same shape as those of the input `x`.

Each returned array must have the same floating-point data type as `x`.

This function conforms to the Array API Standard. This docstring is an extension of the docstring in the standard.

Both the description and the type hints above assumes an array input for simplicity, but this function is nestable, and therefore also accepts `ivy.Container` instances in place of any of the arguments.

Examples

With `ivy.Array` input:

```>>> x = ivy.random_normal(shape = (9, 6))
>>> U, S, Vh = ivy.svd(x)
>>> print(U.shape, S.shape, Vh.shape)
(9, 9) (6,) (6, 6)
```

Reconstruction from SVD, result is numerically close to x

```>>> reconstructed_x = ivy.matmul(U[:,:6] * S, Vh)
>>> print((reconstructed_x - x > 1e-3).sum())
ivy.array(0)
```
```>>> U, S, Vh = ivy.svd(x, full_matrices = False)
>>> print(U.shape, S.shape, Vh.shape)
(9, 6) (6,) (6, 6)
```

With `ivy.NativeArray` input:

```>>> x = ivy.native_array([[3.4,-0.7,0.9],[6.,-7.4,0.],[-8.5,92,7.]])
>>> U, S, Vh = ivy.svd(x)
>>> print(U.shape, S.shape, Vh.shape)
(3, 3) (3,) (3, 3)
```

With `ivy.Container` input:

```>>> x = ivy.Container(a=ivy.array([[2.0, 3.0, 6.0], [5.0, 3.0, 4.0],
...                                [1.0, 7.0, 3.0], [3.0, 2.0, 5.0]]),
...                   b=ivy.array([[7.0, 1.0, 2.0, 3.0, 9.0],
...                                [2.0, 5.0, 3.0, 4.0, 10.0],
...                                [2.0, 11.0, 6.0, 1.0, 3.0],
...                                [8.0, 3.0, 4.0, 5.0, 9.0]]))
>>> U, S, Vh = ivy.svd(x)
>>> print(U.shape)
{
a: [
4,
4
],
b: [
4,
4
]
}
```
ivy.svdvals(x, /, *, out=None)[source]

Returns the singular values of a matrix (or a stack of matrices) `x`.

Parameters
• x (`Union`[`Array`, `NativeArray`]) – input array having shape `(..., M, N)` and whose innermost two dimensions form `MxN` matrices.

• out (`Optional`[`Array`]) – optional output array, for writing the result to. It must have a shape that the (default: `None`) inputs broadcast to.

Return type

`Array`

Returns

ret – array with shape `(..., K)` that contains the vector(s) of singular values of length `K`, where K = min(M, N). The values are sorted in descending order by magnitude.

This function conforms to the Array API Standard. This docstring is an extension of the docstring in the standard.

Both the description and the type hints above assumes an array input for simplicity, but this function is nestable, and therefore also accepts `ivy.Container` instances in place of any of the arguments.

Functional Examples

With `ivy.Array` input:

```>>> x = ivy.array([[5.0, 7.0], [4.0, 3.0]])
>>> S = ivy.svdvals(x)
>>> print(S.shape)
(2,)
```

Compare the singular value S by ivy.svdvals() with the result by ivy.svd().

```>>> _, SS, _ = ivy.svd(x)
>>> print(SS.shape)
(2,)
```
```>>> error = (SS - S).abs()
>>> print(error)
ivy.array([0.,0.])
```

With `ivy.NativeArray` input:

```>>> x = ivy.native_array([[1.0, 2.0, 3.0], [2.0, 3.0, 4.0],
...                       [2.0, 1.0, 3.0], [3.0, 4.0, 5.0]])
>>> print(x.shape)
(4, 3)
```
```>>> S = ivy.svdvals(x)
>>> print(S)
ivy.array([10.3, 1.16, 0.615])
```
```>>> _, SS, _ = ivy.svd(x)
>>> print(SS)
ivy.array([10.3, 1.16, 0.615])
```

Compare the singular value S by ivy.svdvals() with the result by ivy.svd().

```>>> error = (SS - S).abs()
>>> print(error)
ivy.array([0., 0., 0.])
```

With `ivy.Container` input:

```>>> x = ivy.Container(a=ivy.array([[2.0, 3.0], [3.0, 4.0],
...                                [1.0, 3.0], [3.0, 5.0]]),
...                   b=ivy.array([[7.0, 1.0, 2.0, 3.0],
...                                [2.0, 5.0, 3.0, 4.0],
...                                [2.0, 6.0, 1.0, 3.0],
...                                [3.0, 4.0, 5.0, 9.0]]))
>>> y = ivy.svdvals(x)
>>> print(y)
{
a: ivy.array([9.01, 0.866]),
b: ivy.array([15.8, 5.56, 4.17, 0.864])
}
```

Instance Method Examples

Using `ivy.Array` instance method:

```>>> x = ivy.array([[8.0, 3.0], [2.0, 3.0],
...                [2.0, 1.0], [3.0, 4.0],
...                [4.0, 1.0], [5.0, 6.0]])
>>> y = x.svdvals()
>>> print(y)
ivy.array([13.4, 3.88])
```

Using `ivy.Container` instance method:

```>>> x = ivy.Container(a=ivy.array([[2.0, 3.0, 6.0], [5.0, 3.0, 4.0],
...                                [1.0, 7.0, 3.0], [3.0, 2.0, 5.0]]),
...                   b=ivy.array([[7.0, 1.0, 2.0, 3.0, 9.0],
...                                [2.0, 5.0, 3.0, 4.0, 10.0],
...                                [2.0, 11.0, 6.0, 1.0, 3.0],
...                                [8.0, 3.0, 4.0, 5.0, 9.0]]))
>>> y = x.svdvals()
>>> print(y)
{
a: ivy.array([13., 4.64, 2.55]),
b: ivy.array([23.2, 10.4, 4.31, 1.36])
}
```
ivy.tensordot(x1, x2, /, *, axes=2, out=None)[source]

Returns a tensor contraction of x1 and x2 over specific axes.

Parameters
• x1 (`Union`[`Array`, `NativeArray`]) – First input array. Should have a numeric data type.

• x2 (`Union`[`Array`, `NativeArray`]) – second input array. Must be compatible with x1 for all non-contracted axes. Should have a numeric data type.

• axes (`Union`[`int`, `Tuple`[`List`[`int`], `List`[`int`]]]) – The axes to contract over. (default: `2`) Default is 2.

• out (`Optional`[`Array`]) – optional output array, for writing the result to. It must have a shape that the (default: `None`) inputs broadcast to.

Return type

`Array`

Returns

ret – The tensor contraction of x1 and x2 over the specified axes.

This function conforms to the Array API Standard. This docstring is an extension of the docstring in the standard.

Both the description and the type hints above assumes an array input for simplicity, but this function is nestable, and therefore also accepts `ivy.Container` instances in place of any of the arguments.

Functional Examples

With `ivy.Array` input:

```>>> x = ivy.array([[1., 2.], [2., 3.]])
>>> y = ivy.array([[3., 4.], [4., 5.]])
>>> res = ivy.tensordot(x, y, axes =0)
>>> print(res)
ivy.array([[[[3.,4.],[4.,5.]],[[6.,8.],[8.,10.]]],[[[6.,8.],[8.,10.]],[[9.,12.],[12.,15.]]]])
```

With a mix of `ivy.Array` and `ivy.NativeArray` inputs:

```>>> x = ivy.array([[1., 0., 1.], [2., 3., 6.], [0., 7., 2.]])
>>> y = ivy.native_array([[1.], [2.], [3.]])
>>> res = ivy.tensordot(x, y, axes = 1)
>>> print(res)
ivy.array([[ 4.],
[26.],
[20.]])
```

With `ivy.Container` input:

```>>> x = ivy.Container(a=ivy.array([[1., 0., 3.], [2., 3., 4.]]),
...                   b=ivy.array([[5., 6., 7.], [3., 4., 8.]]))
>>> y = ivy.Container(a=ivy.array([[2., 4., 5.], [9., 10., 6.]]),
...                   b=ivy.array([[1., 0., 3.], [2., 3., 4.]]))
>>> res = ivy.tensordot(x, y)
>>> print(res)
{
a: ivy.array(89.),
b: ivy.array(76.)
}
```
ivy.trace(x, /, *, offset=0, axis1=0, axis2=1, out=None)[source]

Returns the sum along the specified diagonals of a matrix (or a stack of matrices) `x`.

Parameters
• x (`Union`[`Array`, `NativeArray`]) – input array having shape `(..., M, N)` and whose innermost two dimensions form `MxN` matrices. Should have a numeric data type.

• offset (`int`) –

offset specifying the off-diagonal relative to the main diagonal. (default: `0`) - `offset = 0`: the main diagonal. - `offset > 0`: off-diagonal above the main diagonal. - `offset < 0`: off-diagonal below the main diagonal.

Default: `0`.

• out (`Optional`[`Array`]) – optional output array, for writing the result to. It must have a shape that the (default: `None`) inputs broadcast to.

Return type

`Array`

Returns

ret – an array containing the traces and whose shape is determined by removing the last two dimensions and storing the traces in the last array dimension. For example, if `x` has rank `k` and shape `(I, J, K, ..., L, M, N)`, then an output array has rank `k-2` and shape `(I, J, K, ..., L)` where

```out[i, j, k, ..., l] = trace(a[i, j, k, ..., l, :, :])
```

The returned array must have the same data type as `x`.

Examples

With `ivy.Array` inputs:

```>>> x = ivy.array([[2., 0., 3.],
...                [3., 5., 6.]])
>>> y = ivy.trace(x, offset=0)
>>> print(y)
ivy.array(7.)
```
```>>> x = ivy.array([[[1., 2.],
...                 [3., 4.]],
...                [[5., 6.],
...                 [7., 8.]]])
>>> y = ivy.trace(x, offset=1)
>>> print(y)
ivy.array([3., 4.])
```

With `ivy.NativeArray` inputs:

```>>> x = ivy.native_array([[2., 0., 3.],[3., 5., 6.]])
>>> y = ivy.trace(x, offset=0)
>>> print(y)
ivy.array(7.)
```
```>>> x = ivy.native_array([[0, 1, 2],
...                       [3, 4, 5],
...                       [6, 7, 8]])
>>> y = ivy.trace(x, offset=0)
>>> print(y)
ivy.array(12)
```

With `ivy.Container` inputs:

```>>> x = ivy.Container(
...        a = ivy.array([[7, 1, 2],
...                       [1, 3, 5],
...                       [0, 7, 4]]),
...        b = ivy.array([[4, 3, 2],
...                       [1, 9, 5],
...                       [7, 0, 6]])
...    )
>>> y = ivy.trace(x, offset=0)
>>> print(y)
{
a: ivy.array(14),
b: ivy.array(19)
}
```
```>>> x = ivy.Container(
...        a = ivy.array([[7, 1, 2],
...                       [1, 3, 5],
...                       [0, 7, 4]]),
...        b = ivy.array([[4, 3, 2],
...                       [1, 9, 5],
...                       [7, 0, 6]])
...    )
>>> y = ivy.trace(x, offset=1)
>>> print(y)
{
a: ivy.array(6),
b: ivy.array(8)
}
```
ivy.vander(x, /, *, N=None, increasing=False, out=None)[source]

Generates a Vandermonde matrix. The columns of the output matrix are elementwise powers of the input vector x^{(N-1)}, x^{(N-2)}, …, x^0x. If increasing is True, the order of the columns is reversed x^0, x^1, …, x^{(N-1)}. Such a matrix with a geometric progression in each row is named for Alexandre-Theophile Vandermonde.

Parameters
• x (`Union`[`Array`, `NativeArray`]) – 1-D input array.

• N (`Optional`[`int`]) – Number of columns in the output. If N is not specified, (default: `None`) a square array is returned (N = len(x))

• increasing (`bool`) – Order of the powers of the columns. If True, the powers increase (default: `False`) from left to right, if False (the default) they are reversed.

• out (`Optional`[`Array`]) – optional output array, for writing the result to. (default: `None`)

Return type

`Array`

Returns

ret – Vandermonde matrix.

Examples

With `ivy.Array` inputs:

```>>> x = ivy.array([1, 2, 3, 5])
>>> ivy.vander(x)
ivy.array(
[[  1,   1,   1,   1],
[  8,   4,   2,   1],
[ 27,   9,   3,   1],
[125,  25,   5,   1]]
)
```
```>>> x = ivy.array([1, 2, 3, 5])
>>> ivy.vander(x, N=3)
ivy.array(
[[ 1,  1,  1],
[ 4,  2,  1],
[ 9,  3,  1],
[25,  5,  1]]
)
```
```>>> x = ivy.array([1, 2, 3, 5])
>>> ivy.vander(x, N=3, increasing=True)
ivy.array(
[[ 1,  1,  1],
[ 1,  2,  4],
[ 1,  3,  9],
[ 1,  5, 25]]
)
```
ivy.vecdot(x1, x2, /, *, axis=-1, out=None)[source]

Computes the (vector) dot product of two arrays.

Parameters
• x1 (`Union`[`Array`, `NativeArray`]) – first input array. Should have a numeric data type.

• x2 (`Union`[`Array`, `NativeArray`]) – second input array. Must be compatible with `x1` (see broadcasting). Should have a numeric data type.

• axis (`int`) – axis over which to compute the dot product. Must be an integer on the interval (default: `-1`) `[-N, N)`, where `N` is the rank (number of dimensions) of the shape determined according to broadcasting. If specified as a negative integer, the function must determine the axis along which to compute the dot product by counting backward from the last dimension (where `-1` refers to the last dimension). By default, the function must compute the dot product over the last axis. Default: `-1`.

• out (`Optional`[`Array`]) – optional output array, for writing the result to. It must have a shape that the (default: `None`) inputs broadcast to.

Return type

`Array`

Returns

ret – if `x1` and `x2` are both one-dimensional arrays, a zero-dimensional containing the dot product; otherwise, a non-zero-dimensional array containing the dot products and having rank `N-1`, where `N` is the rank (number of dimensions) of the shape determined according to broadcasting. The returned array must have a data type determined by type-promotion.

Raises

• if provided an invalid `axis`.

• if the size of the axis over which to compute the dot product is not the same for both `x1` and `x2`.

This function conforms to the Array API Standard. This docstring is an extension of the docstring in the standard.

Both the description and the type hints above assumes an array input for simplicity, but this function is nestable, and therefore also accepts `ivy.Container` instances in place of any of the arguments.

ivy.vector_norm(x, /, *, axis=None, keepdims=False, ord=2, out=None)[source]

Computes the vector norm of a vector (or batch of vectors) `x`.

Parameters
• x (`Union`[`Array`, `NativeArray`]) – input array. Should have a floating-point data type.

• axis (`Optional`[`Union`[`int`, `Sequence`[`int`]]]) – If an integer, `axis` specifies the axis (dimension) along which to compute (default: `None`) vector norms. If an n-tuple, `axis` specifies the axes (dimensions) along which to compute batched vector norms. If `None`, the vector norm must be computed over all array values (i.e., equivalent to computing the vector norm of a flattened array). Negative indices must be supported. Default: `None`.

• keepdims (`bool`) – If `True`, the axes (dimensions) specified by `axis` must be included in the (default: `False`) result as singleton dimensions, and, accordingly, the result must be compatible with the input array (see broadcasting). Otherwise, if `False`, the axes (dimensions) specified by `axis` must not be included in the result. Default: `False`.

• ord (`Union`[`int`, `float`, `Literal`[inf, -inf]]) –

order of the norm. The following mathematical norms must be supported: (default: `2`)

ord

description

1

L1-norm (Manhattan)

2

L2-norm (Euclidean)

inf

infinity norm

(int,float >= 1)

p-norm

The following non-mathematical “norms” must be supported:

ord

description

0

sum(a != 0)

-1

1./sum(1./abs(a))

-2

1./sqrt(sum(1./abs(a)**2))

-inf

min(abs(a))

(int,float < 1)

sum(abs(a)**ord)**(1./ord)

Default: `2`.

• out (`Optional`[`Array`]) – optional output array, for writing the result to. It must have a shape that the (default: `None`) inputs broadcast to.

Return type

`Array`

Returns

ret – an array containing the vector norms. If `axis` is `None`, the returned array must be a zero-dimensional array containing a vector norm. If `axis` is a scalar value (`int` or `float`), the returned array must have a rank which is one less than the rank of `x`. If `axis` is a `n`-tuple, the returned array must have a rank which is `n` less than the rank of `x`. The returned array must have a floating-point data type determined by type-promotion.

This function conforms to the Array API Standard. This docstring is an extension of the docstring in the standard.

Both the description and the type hints above assumes an array input for simplicity, but this function is nestable, and therefore also accepts `ivy.Container` instances in place of any of the arguments.

ivy.vector_to_skew_symmetric_matrix(vector, /, *, out=None)[source]
Given vector, return the associated `Skew-symmetric matrix
Parameters
• vector (`Union`[`Array`, `NativeArray`]) – Vector to convert (batch_shape,3).

• out (`Optional`[`Array`]) – optional output array, for writing the result to. It must have a shape that the (default: `None`) inputs broadcast to.

Return type

`Array`

Returns

ret – Skew-symmetric matrix (batch_shape,3,3).

Both the description and the type hints above assumes an array input for simplicity, but this function is nestable, and therefore also accepts `ivy.Container` instances in place of any of the arguments.

This should have hopefully given you an overview of the linear algebra submodule,If you have any questions, please feel free to reach out on our discord in the linear algebra channel or in the linear algebra forum!