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HomeMatlabThe Large Six Matrix Factorizations – Nick Higham

# The Large Six Matrix Factorizations – Nick Higham

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 $r = mathrm{rank}(A)$.

Value: $14mn^2+8n^3$ for $P(:,1colon n)$, $Sigma$, and $Q$ by the Golub–Reinsch algorithm, or $6mn^2+20n^3$ with a preliminary QR factorization.

Use: figuring out matrix rank, fixing rank-deficient least squares issues, computing every kind of subspace data.

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