matrix_exp

→Tensor

matrix_exp(A: Tensor)

source

Matrix exponential $e^A$ .

Returns the matrix exponential

e^A \,=\, \sum_{k=0}^{\infty} \frac{A^k}{k!},

which solves the matrix ODE $\dot Y = A Y$ with initial condition $Y(0) = I$ . Computed by the scaling-and-squaring algorithm of Higham (2005) with a Padé [6/6] rational approximant:

Scale $A' = A / 2^s$ so that $\|A'\| \le \theta_6$ .
Approximate $R \approx e^{A'}$ via Padé [6/6] using the even/odd polynomial splitting $R = D^{-1} N$ .
Square $R$ a total of $s$ times to recover $e^A = R^{2^s}$ .

Parameters

ATensor

Square matrix of shape (*, n, n).

Returns

Tensor

$e^A$ , shape (*, n, n).

Notes

Computational cost is $O((s + 6)\, n^3)$ where the scaling parameter $s = \lceil \log_2 (\|A\| / \theta_6) \rceil$ . The Frobenius norm is used here as a (conservative) proxy for the 1-norm — this may add one extra squaring step but keeps the result correct.

Differentiable: built entirely on matmul and inv, so autograd flows naturally. Batched inputs (ndim > 2) are handled element-wise.

Examples

>>> import lucid
>>> from lucid.linalg import matrix_exp
>>> A = lucid.tensor([[0.0, 1.0], [-1.0, 0.0]])  # 90-deg rotation generator
>>> matrix_exp(A)
Tensor([[ 0.5403,  0.8415],
        [-0.8415,  0.5403]])