Matrix Ops

This section contains various matrix decomposition routines.

SVD#

Description#

Singular Value Decomposition (SVD) performs a decomposition such that: $$ M = U_{n} \Sigma_{n} V^{*} $$

Returns#

• Matrix $ U_{n} $: Orthogonal matrix
• Matrix $ \Sigma_{n} $: Ranked singular values
• Matrix $ V^{*} $: Orthogonal matrix

QR#

Description#

Performs a decomposition such that: $$ M = QR $$ where $ Q $ is an orthogonal matrix and $ R $ is an upper triangular matrix.

Returns#

• Matrix Q: Orthogonal matrix
• Matrix R: Upper triangular matrix

PCA#

Description#

Suppose we have a data matrix $ X $. In Principal Component Analysis, we maximize the following relation:

$$ F(W) = \frac{1}{n} \sum_{j=0}^{q-1} W_j^{\top} X^{\top} X W_j $$

subject to the constraint $ W^{\top} W = I $ where $ W $ is a matrix of $ q $ orthonomal vectors.

Returns#

• Scores: Ranked projections
• Singular Values: Ranked eigenvalues
• W: Ranked columnwise coefficient matrix
• T2: Hotelling’s T-Squared
• % Variance: Percent variance explained by each component

NNMF#

Description#

Suppose we have a matrix $ V $ with no negative values and we want the following decomposition: $$ V = W H $$ where $ W $ and $ H $ have only non-negative elements, this is known as Non-Negative Matrix Factorization.

Returns#

• $ W_{m \mathrm{x} p} $
• $ H_{p \mathrm{x} n} $