# tensorplay.linalg

Source: https://www.tensorplay.cn/docs/linalg.html

Common linear algebra operations.
See [Linear algebra (tensorplay.linalg)](/docs/notes/numerical_accuracy.html#linear-algebra-stability) for some common numerical edge-cases.

## Matrix Properties

| tensorplay.linalg.norm | |
| --- | --- |
| tensorplay.linalg.vector_norm | dim=None norms the whole tensor: the input is flattened first, and keepdim then restores the reduced axes as ones. |
| tensorplay.linalg.matrix_norm |  |
| tensorplay.linalg.diagonal | linalg_diagonal(A, *, offset=0, dim1=-2, dim2=-1) -> Tensor |
| tensorplay.linalg.det |  |
| tensorplay.linalg.slogdet |  |
| tensorplay.linalg.cond |  |
| tensorplay.linalg.matrix_rank | Computes the numerical rank of each matrix in A. |

## Decompositions

| tensorplay.linalg.cholesky | |
| --- | --- |
| tensorplay.linalg.qr |  |
| tensorplay.linalg.polar |  |
| tensorplay.linalg.lu | linalg_lu(Tensor A, *, bool pivot=True) -> (Tensor P, Tensor L, Tensor U) linalg_lu.out(Tensor A, *, bool pivot=True, Tensor(a!) P, Tensor(b!) L, Tensor(c!) U) -> (Tensor(a!) P, Tensor(b!) L, Tensor(c!) U) |
| tensorplay.linalg.lu_factor | linalg_lu_factor(Tensor A, *, bool pivot=True) -> (Tensor LU, Tensor pivots) linalg_lu_factor.out(Tensor A, *, bool pivot=True, Tensor(a!) LU, Tensor(b!) pivots) -> (Tensor(a!) LU, Tensor(b!) pivots) |
| tensorplay.linalg.eig |  |
| tensorplay.linalg.eigvals |  |
| tensorplay.linalg.eigh |  |
| tensorplay.linalg.eigvalsh |  |
| tensorplay.linalg.svd |  |
| tensorplay.linalg.svdvals |  |

## Solvers

| tensorplay.linalg.solve | |
| --- | --- |
| tensorplay.linalg.solve_triangular | linalg_solve_triangular(Tensor A, Tensor B, *, bool upper, bool left=True, bool unitriangular=False) -> Tensor linalg_solve_triangular.out(Tensor self, Tensor B, *, bool upper, bool left=True, bool unitriangular=False, Tensor(a!) out) -> Tensor(a!) |
| tensorplay.linalg.lu_solve | linalg_lu_solve(Tensor LU, Tensor pivots, Tensor B, *, bool left=True, bool adjoint=False) -> Tensor linalg_lu_solve.out(Tensor LU, Tensor pivots, Tensor B, *, bool left=True, bool adjoint=False, Tensor(a!) out) -> Tensor(a!) |
| tensorplay.linalg.lstsq |  |

## Inverses

| tensorplay.linalg.inv | |
| --- | --- |
| tensorplay.linalg.pinv | Moore-Penrose pseudo-inverse. |

## Matrix Functions

| tensorplay.linalg.matrix_exp |Square matrix exponential via the degree-13 Pade approximant with scaling and squaring: A is halved until its 1-norm falls under the approximant's accuracy threshold, then the result is squared back. |
| --- | --- |
| tensorplay.linalg.matrix_sqrth | Matrix square root via the Denman-Beavers fixed-point iteration (converges for matrices with no eigenvalues on the closed negative real axis). |
| tensorplay.linalg.matrix_power |  |

## Matrix Products

| tensorplay.linalg.cross | |
| --- | --- |
| tensorplay.linalg.matmul |  |
| tensorplay.linalg.vecdot | Dot product along dim with the first argument conjugated for complex inputs. |
| tensorplay.linalg.vdot | Conjugating dot product over 1-D operands: sum(conj(self) * other). |
| tensorplay.linalg.multi_dot | Chained matrix product evaluated in the parenthesization that minimizes the scalar multiplication count (matrix-chain dynamic program). |
| tensorplay.linalg.householder_product | linalg_householder_product(Tensor input, Tensor tau) -> Tensor linalg_householder_product.out(Tensor input, Tensor tau, *, Tensor(a!) out) -> Tensor(a!) |

## Tensor Operations

| tensorplay.linalg.tensorinv |Inverse of A seen as a square matrix over the split at ind: the product of the leading ind dimensions must equal that of the rest. |
| --- | --- |
| tensorplay.linalg.tensorsolve | Solves the tensor equation A X = B after flattening the contracted dimensions into a square matrix. |

## Misc

| tensorplay.linalg.vander | |
| --- | --- |

## Experimental Functions

| tensorplay.linalg.cholesky_ex | |
| --- | --- |
| tensorplay.linalg.inv_ex |  |
| tensorplay.linalg.solve_ex |  |
| tensorplay.linalg.lu_factor_ex | linalg_lu_factor_ex(Tensor A, *, bool pivot=True, bool check_errors=False) -> (Tensor LU, Tensor pivots, Tensor info) linalg_lu_factor_ex.out(Tensor A, *, bool pivot=True, bool check_errors=False, Tensor(a!) LU, Tensor(b!) pivots, Tensor(c!) info) -> (Tensor(a!) LU, Tensor(b!) pivots, Tensor(c!) info) |
| tensorplay.linalg.ldl_factor | linalg_ldl_factor(Tensor A, *, bool hermitian=False) -> (Tensor LD, Tensor pivots) linalg_ldl_factor.out(Tensor self, *, bool hermitian=False, Tensor(a!) LD, Tensor(b!) pivots) -> (Tensor(a!) LD, Tensor(b!) pivots) |
| tensorplay.linalg.ldl_factor_ex | linalg_ldl_factor_ex(Tensor A, *, bool hermitian=False, bool check_errors=False) -> (Tensor LD, Tensor pivots, Tensor info) linalg_ldl_factor_ex.out(Tensor self, *, bool hermitian=False, bool check_errors=False, Tensor(a!) LD, Tensor(b!) pivots, Tensor(c!) info) -> (Tensor(a!) LD, Tensor(b!) pivots, Tensor(c!) info) |
| tensorplay.linalg.ldl_solve | linalg_ldl_solve(Tensor LD, Tensor pivots, Tensor B, *, bool hermitian=False) -> Tensor linalg_ldl_solve.out(Tensor LD, Tensor pivots, Tensor B, *, bool hermitian=False, Tensor(a!) out) -> Tensor(a!) |

## Result Types

Solvers and decompositions that return several tensors package them in the
named tuples below; the _ex variants report failures through an info
field instead of raising.

| tensorplay.linalg.LinAlgError |Raised when a decomposition or solve fails on a numerically invalid input. |
| --- | --- |
| tensorplay.linalg.EigResult |  |
| tensorplay.linalg.EighResult |  |
| tensorplay.linalg.LstsqResult |  |
| tensorplay.linalg.QRResult |  |
| tensorplay.linalg.SVDResult |  |
| tensorplay.linalg.SlogdetResult |  |
| tensorplay.linalg.CholeskyExResult |  |