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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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