freealg.trace

freealg.trace(A, N=None, p=1.0)

Estimate the trace of a power of a Hermitian matrix.

This function estimates the trace of the matrix power \(\mathbf{A}^p\) or that of a larger matrix containing \(\mathbf{A}\).

Parameters:

Anumpy.ndarray

The symmetric real-valued matrix \(\mathbf{A}\) whose trace of a power (or that of a matrix containing \(\mathbf{A}\)) is to be computed.

Nint, default=None

The size of the matrix containing \(\mathbf{A}\) to estimate eigenvalues of. If None, returns estimates of the eigenvalues of \(\mathbf{A}\) itself.

pfloat, default=1.0

The exponent \(p\) in \(\mathbf{A}^p\).

Returns:

tracefloat

matrix trace

See also

eigh

cond

slogdet

norm

Notes

The trace is highly amenable to subsampling: under free decompression the average eigenvalue is assumed constant, so the trace increases linearly. Traces of powers fall back to freealg.eigh().

Examples

>>> from freealg import norm
>>> from freealg.distributions import MarchenkoPastur

>>> mp = MarchenkoPastur(1/50)
>>> A = mp.matrix(3000)
>>> trace(A, 100_000)