The Large Six Matrix Factorizations – Nick Higham

September 14, 2022

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Any matrix $Ainmathbb{C}^{mtimes n}$ has a singular worth decomposition (SVD)

$notag A = USigma V^*, quad Sigma = mathrm{diag}(sigma_1,sigma_2,dots,sigma_p) in mathbb{R}^{mtimes n}, quad p = min(m,n),$

the place $Uinmathbb{C}^{mtimes m}$ and $Vinmathbb{C}^{ntimes n}$ are unitary and $sigma_1gesigma_2gecdotsgesigma_pge0$ . The $sigma_i$ are the singular values of $A$ , and they’re the nonnegative sq. roots of the $p$ largest eigenvalues of $A^*A$ . The columns of $U$ and $V$ are the left and proper singular vectors of $A$ , respectively. The rank of $A$ is the same as the variety of nonzero singular values. If $A$ is actual, $U$ and $V$ may be taken to be actual. The important SVD data is contained within the compact or financial system dimension SVD $A = USigma V^*$ , the place $Uinmathbb{C}^{mtimes r}$ , $Sigma = mathrm{diag}(sigma_1,dots,sigma_r)$ , $Vinmathbb{C}^{ntimes r}$ , and $r = mathrm{rank}(A)$ .