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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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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?2011-12-03
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    You know the usual $\mathbf Q^\top\mathbf Q=\mathbf I$ condition, no?2011-12-03
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    Can Q be formed from the eigenvectors in any convenient manner? Is Q just each column divided by its length?2011-12-03
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    Orthogonal* sorry2011-12-03
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    `V` in the first, and `U` in the second, are what you perform the `Q'*Q` operation on...2011-12-03
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    The just proved the dot products between some entries were all zero. Thanks!2011-12-03
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    Well, you did get something resembling the result of `eye()`, no?2011-12-03