# torch.linalg.lu_solve¶

torch.linalg.lu_solve(LU, pivots, B, *, left=True, adjoint=False, out=None)

Computes the solution of a square system of linear equations with a unique solution given an LU decomposition.

Letting $\mathbb{K}$ be $\mathbb{R}$ or $\mathbb{C}$, this function computes the solution $X \in \mathbb{K}^{n \times k}$ of the linear system associated to $A \in \mathbb{K}^{n \times n}, B \in \mathbb{K}^{n \times k}$, which is defined as

$AX = B$

where $A$ is given factorized as returned by lu_factor().

If left= False, this function returns the matrix $X \in \mathbb{K}^{n \times k}$ that solves the system

$XA = B\mathrlap{\qquad A \in \mathbb{K}^{k \times k}, B \in \mathbb{K}^{n \times k}.}$

If adjoint= True (and left= True), given an LU factorization of :math:A this function function returns the $X \in \mathbb{K}^{n \times k}$ that solves the system

$A^{\text{H}}X = B\mathrlap{\qquad A \in \mathbb{K}^{k \times k}, B \in \mathbb{K}^{n \times k}.}$

where $A^{\text{H}}$ is the conjugate transpose when $A$ is complex, and the transpose when $A$ is real-valued. The left= False case is analogous.

Supports inputs of float, double, cfloat and cdouble dtypes. Also supports batches of matrices, and if the inputs are batches of matrices then the output has the same batch dimensions.

Parameters:
• LU (Tensor) – tensor of shape (*, n, n) (or (*, k, k) if left= True) where * is zero or more batch dimensions as returned by lu_factor().

• pivots (Tensor) – tensor of shape (*, n) (or (*, k) if left= True) where * is zero or more batch dimensions as returned by lu_factor().

• B (Tensor) – right-hand side tensor of shape (*, n, k).

Keyword Arguments:
• left (bool, optional) – whether to solve the system $AX=B$ or $XA = B$. Default: True.

• adjoint (bool, optional) – whether to solve the system $AX=B$ or $A^{\text{H}}X = B$. Default: False.

• out (Tensor, optional) – output tensor. Ignored if None. Default: None.

Examples:

>>> A = torch.randn(3, 3)
>>> LU, pivots = torch.linalg.lu_factor(A)
>>> B = torch.randn(3, 2)
>>> X = torch.linalg.lu_solve(LU, pivots, B)
>>> torch.allclose(A @ X, B)
True

>>> B = torch.randn(3, 3, 2)   # Broadcasting rules apply: A is broadcasted
>>> X = torch.linalg.lu_solve(LU, pivots, B)
>>> torch.allclose(A @ X, B)
True

>>> B = torch.randn(3, 5, 3)
>>> X = torch.linalg.lu_solve(LU, pivots, B, left=False)
>>> torch.allclose(X @ A, B)
True

>>> B = torch.randn(3, 3, 4)   # Now solve for A^T
>>> X = torch.linalg.lu_solve(LU, pivots, B, adjoint=True)
>>> torch.allclose(A.mT @ X, B)
True
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