xesn.RandomMatrix.normalize

RandomMatrix.normalize(A)

Rescale a random matrix either through simple multiplication, or by first normalizing by the matrix’s spectral radius or induced 2 norm. Given the matrix \(A\) with eigenvalues \(\{ {\lambda}_1, {\lambda}_2, ..., {\lambda}_n\}\), and singular values \(\{ {\sigma}_1, {\sigma}_2, ..., {\sigma}_n\}\), the normalization options return the following:

normaliation=”multiply”:

\[factor * A\]

normalization=”eig”:

\[\dfrac{factor}{\rho(A)} A\]

where \({ \rho(A) = \max \{ | \lambda_1 | , | \lambda_2 |, ..., | \lambda_n | \} }\) is the spectral radius of \(A\).

normalization=”svd”:

\[\dfrac{factor}{\sigma(A)} A\]

where \(\sigma(A) = \max \{\sigma_1, \sigma_2, ..., \sigma_n\}\) is the largest singular value, a.k.a. the induced 2-norm of \(A\).

Parameters:

A (array_like) – random matrix

Returns:

A (array_like) – rescaled matrix, based on class attributes