matrix_power_op

→TensorImplPtr

int matrix_power_op(const int & a, int n)

source edit

Python wrappers

pylucid.linalg.matrix_powerfunction

Compute $A^n$ for a square float matrix and integer exponent.

Validates that a is at least 2-D, square in the trailing two axes, and float-typed; dispatches to the backend. The output shape always matches a's shape (matrix powers preserve shape). Autograd is not wired — the result is a leaf in the gradient graph.

Math

A^n = \begin{cases} I & n = 0, \\ \underbrace{A \cdot A \cdots A}_{n\;\text{times}} & n > 0, \\ (A^{-1})^{|n|} & n < 0. \end{cases}

Repeated squaring decomposes $n = \sum_i b_i 2^i$ (its binary representation) and accumulates the result in $O(\log |n|)$ matrix multiplies.

Shape

a: (..., N, N).
Output: same shape as a.

References

Knuth, The Art of Computer Programming, Vol. 2 §4.6.3 (right-to-left binary exponentiation). Higham, Functions of Matrices (2008), §1.2.

See Also

inv_op : Used internally when n < 0. matmul_op : Single-step underlying primitive.

Parameters

aTensorImplPtr

Input matrix of shape (..., N, N).

nint

Integer exponent. Named n for parity with the reference framework's matrix_power(a, n) signature; the algorithm refers to it as

p

internally. May be zero, positive, or negative; negative values require

A

to be invertible.

Returns

TensorImplPtr

Tensor of shape (..., N, N) holding $A^n$ (and the identity matrix if n == 0).

Raises

ValueError

If a is not square, fewer than 2-D, or non-float.

LinAlgError

(At backend level, when n < 0.) If

A

is singular.

Notes

No backward is registered; matrix_power outputs are leaves.
For graph neural networks $A$ is often an adjacency or normalised Laplacian — large n is fine because cost grows only logarithmically.