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SVDintro2011

# SVDintro2011 - MoreLinear Algebra Toward SVD A matrix A of...

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3/24/2011 1 More Linear Algebra Toward SVD Simple but important facts A matrix A of size n by n has exactly n complex eigenvalues. Assume that A is symmetric and real matrix. Then all eigenvalues of A are real numbers. Gram-Schmidt process If v 1 , v 2 ,…, v k are a set of orthogonal vectors in R n (k<n), then we can find n-k vectors v k+1 ,…, v n such that { v 1 ,…, v n } forms an orthogonal basis.

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3/24/2011 2 Real Symmetric Matrix Recall the famous result of linear algebra: Let A be any real symmetric matrix . Then there exists an orthogonal matrix U such that the following holds : (i) A = UDU T , D a diagonal matrix. (ii) The diagonal entries of D are the eigenvalues of A . (iii) The column vectors of U are the eigenvectors of the eigenvalues of A. The proof can be done using the facts mentioned in the previous slide. (Do it?) What if A is not symmetric? Singular Value Ddecomposition All eigenvalues of X T X are non-negative.
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