Matrix decompositionMatlab includes several functions for matrix decomposition or factorization.LU decomposition: the name of the built-in function is 'lu'. To get the LU factorization of a square matrix M, type the command [L,U] = lu(M). Matlab returns a lower triangular matrix L and an upper triangular matrix U such that L*U = M. Example: Type M = [2 3 4 5 3 2 6 1 2 3 1 7 9 8 0 2]; [l,u] = lu(M) l*u Matlab answer is: l = 0.2222 1.0000 0.4583 1.0000 0.3333 -0.5455 1.0000 0 0.2222 1.0000 0 0 1.0000 0 0 0 u = 9.0000 8.0000 0 2.0000 0 1.2222 1.0000 6.5556 0 0 6.5455 3.9091 0 0 0 -3.7917 ans = 2.0000 3.0000 4.0000 5.0000 3.0000 2.0000 6.0000 1.0000 2.0000 3.0000 1.0000 7.0000 9.0000 8.0000 0 2.0000 QR factorization: the name of the appropriate built-in function for this purpose is 'qr'. Typing the command [Q,R] = qr(M) returns an orthogonal matrix Q and an upper triangular matrix R such that Q*R = M. Example (assuming the same matrix M as above): Type [q,r] = qr(M) q*r and the fast Matlab answer is: q = -0.2020 0.6359 -0.4469 -0.5959 -0.3030 -0.4127 -0.8144 0.2731 -0.2020 0.6359 0.0027 0.7449 -0.9091 -0.1450 0.3702 -0.1241 r = -9.8995 -9.0914 -2.8284 -4.5457 0 1.8295 0.7028 6.9274 0 0 -6.6713 -2.2896 0 0 0 2.2595 ans = 2.0000 3.0000 4.0000 5.0000 3.0000 2.0000 6.0000 1.0000 2.0000 3.0000 1.0000 7.0000 9.0000 8.0000 -0.0000 2.0000 Cholesky decomposition: if you have a positive definite matrix A, you can factorize the matrix with the built-in function 'chol'. The command R = chol(A); produces an upper triangular matrix R such that R'*R = A for a positive definite A. This is a brief reminder of what a positive definite matrix is: An n by n matrix M is positive definite if xT Mx > 0 for all x
<> 0 and xT Mx = 0 implies x
= 0 If matrix M is positive definite, then M is diagonally dominant and has positive diagonal elements. Example: Type M = [1 1 1 1 1 2 3 4 1 3 6 10 1 4 10 20]; r = chol(M) r'*r and Matlab answer is: r = 1 1 1 1 0 1 2 3 0 0 1 3 0 0 0 1 ans = 1 1 1 1 1 2 3 4 1 3 6 10 1 4 10 20 Singular value decomposition (svd): the name of the built-in function is svd. Typing [U,S,V] = svd(M); produces a diagonal matrix S, of the same dimension as M and with nonnegative diagonal elements in decreasing order, and unitary matrices U and V so that M = U*S*V'. Example (considering the same matrix M as above): Type [u,s,v] = svd(M) u*s*v' and you get: u = -0.0602 -0.5304 0.7873 -0.3087 -0.2012 -0.6403 -0.1632 0.7231 -0.4581 -0.3918 -0.5321 -0.5946 -0.8638 0.3939 0.2654 0.1684 s = 26.3047 0 0 0 0 2.2034 0 0 0 0 0.4538 0 0 0 0 0.0380 v = -0.0602 -0.5304 0.7873 -0.3087 -0.2012 -0.6403 -0.1632 0.7231 -0.4581 -0.3918 -0.5321 -0.5946 -0.8638 0.3939 0.2654 0.1684 ans = 1.0000 1.0000 1.0000 1.0000 1.0000 2.0000 3.0000 4.0000 1.0000 3.0000 6.0000 10.0000 1.0000 4.0000 10.0000 20.0000 From 'Matrix Decomposition' to home From 'Matrix Decomposition' to 'Matlab Cookbook'
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