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"Let $A$ be a $5\times 5$ random matrix and let $B = A^TA$ (note that the entries of the matrix $B$ are symmetric with respect to the diagonal. Such a matrix is called a symmetric matrix). Find a basis of eigenvectors for the matrix $B$, and check that this basis is orthogonal."

Does anyone know how to do this, especially in Matlab? I am only really familiar with how to find eigenvalues.

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1 Answer 1

Given some A, there are two ways to find the eigenvectors of B=A'*A:

  1. [V, D] = eig(A'*A) and V is the required matrix of eigenvectors.

  2. (better!) [U, S, V] = svd(A) and U is the required matrix of eigenvectors.

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So the basis is just the evects themselves? How do I check they are orthonormal in Matlab? –  Randy Dec 3 '11 at 3:39
    
You know the usual $\mathbf Q^\top\mathbf Q=\mathbf I$ condition, no? –  J. M. Dec 3 '11 at 3:44
    
Can Q be formed from the eigenvectors in any convenient manner? Is Q just each column divided by its length? –  Randy Dec 3 '11 at 3:49
    
Orthogonal* sorry –  Randy Dec 3 '11 at 3:55
    
V in the first, and U in the second, are what you perform the Q'*Q operation on... –  J. M. Dec 3 '11 at 3:56

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