# Audio manipulations with Singular Value Decomposition¶

## 1976 Film about SVD¶

Lets apply Singular Value Decomposition to an audio complex spectrogram $X \in \mathbb{C}^{n \times k}$, where rows $n$ are time bins, and columns $k$ are frequency bins.

$$ USV^* = X$$$U$,$S$ and $V$ dimensions could be arranged to be $U \in \mathbb{C}^{n \times k}$, $S \in \mathbb{C}^{k \times k}$ and $V \in \mathbb{C}^{k \times k}$.

This decomposition could be analyzed as $V$ representing a spectrum basis, $S$ as scaling values, and $U$ as the score with the weight for the linear combinations of basis vectors.

This is how the log magnitude of $V^*$ looks like for this audio