## Singular Value Decomposition Calculator

This application calculates the singular value decomposition of an $m\times n$ real or complex matrix $A$ in the form \[ A = U\Sigma V^T\quad\mbox{or}\quad A = U\Sigma V^H \] where $U$ and $V$ are orthogonal matrices if $A$ is a real matrix or unitary matrices if $A$ is a complex matrix, $V^H$ is the conjugate transpose of $V$, with orders $m$ and $n$ respectively, and $\Sigma$ is an $m\times n$ diagonal matrix with real diagonal elements, $\sigma_i$, in the order such that \[ \sigma_1 \geqslant \sigma_2 \geqslant \ldots \geqslant \sigma_{\min(m,n)} \geqslant 0 \] The $\sigma_i$ are the singular values of $A$ and the first $\min(m,n)$ columns of $U$ and $V$ are the left and right singular vectors of A and satisfy the relations \[ Av_i = \sigma_iu_i \quad\mbox{and}\quad A^Tu_i=\sigma_iv_i \] where $u_i$ and $v_i$ are the $i^{\mbox{th}}$ columns of $U$ and $V$ respectively.

See Input Data for the description of how to enter matrix or just click Example for a simple example.