For A not exactly symmetric it probably takes route (1) and gives you a basis of each subspace, but not necessarily with orthogonal vectors. Matlab is probably taking route (2) (thus forcing V_a to be orthogonal) only if A is symmetric. (2) However, since every subspace has an orthonormal basis,you can find orthonormal bases for each eigenspace, so you can find an orthonormal basis of eigenvectors. Eigenvectors corresponding to the same eigenvalue need not be orthogonal to each other. This is my bet as to what's happening: as noted by Eigenvectors corresponding to distinct eigenvalues of a symmetric matrix must be orthogonal to each other. Still, it's surprising that so wildly different results are obtained for V_A when A is symmetric and when A is nearly symmetric. If you force A to be exactly symmetric you'll get an orthogonal V_A, up to numerical errrors on the order of eps: > A = (A+A.')/2 The differences are on the order of eps, as expected from numerical errors. But your input matrix A is not exactly symmetric. The eigenvectors of a real symmetric matrix are orthogonal. This seems to be a numerical precision issue. I was under the impression that if my matrix was real and symmetric then Matlab would return an orthogonal matrix of eigenvectors, so what is the issue here? V_A*V_A' is not equalling the identity matrix (taking into account rounding errors). I am finding that my matrix of eigenvectors V_A is not orthogonal i.e. % less than the given number of repeats if we ever input the same % Put one random value into another (note this sometimes will result in % Now generate a vector of eigenvalues with the given number of repeats The variable-precision counterparts are E eig (vpa (A)) and V,E eig (vpa (A)). When used, the string 'L'specifies the left eigenvector, and 'R'the right. % First generate a random matrix of eignevectors that is orthonormal Answers Trial Software Product Updates Eigenvalues The symbolic eigenvalues of a square matrix A or the symbolic eigenvalues and eigenvectors of A are computed, respectively, using the commands E eig (A) and V,E eig (A). Arguments M, Nare square matrices of equal size and contain real or complex numbers. I have a matrix symmetric 100x100 and trying to obtain eigen value and eigen vector from it. The problem is I don't match the result same as my matlab code. When I try to find the eigen-decomposition of a matrix in Matlab that has a repeated eigenvalue but is NOT defective, it is not returning an orthonormal matrix of eignevectors. Eigen asked Apr 5 '13 Alexandre Bizeau 56 1 6 updated Apr 15 '13 Hello, I need to retrieve the eigen vector from a matrix.
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