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Singular Value Decomposition

  1. May 7, 2007 #1
    I have a couple questions about the singular value decomposition theorem, which states that any mxn matrix A of rank r > 0 can be factored into
    [tex] A = U \Sigma V[/tex]
    into the product of an mxm matrix U with orthonormal columns, the mxn matrix ∑ with ∑ = diag([tex]\sqrt{\lambda_i}[/tex]), and the nxn matrix V with orthonormal columns.

    In case the definition doesn't provide much help, the V has the orthonormalized eigenvectors of (A^T)A, and U has the orthonormalized eigenvectors of A(A^T).

    Do the first r columns of U span A, i.e. do the first r columns of U form a basis for the range of A? Similarly, will the first r columns of V form a basis for the corng of A?

    Really what I am trying to determine is if C is a 3x3 matrix with eigenvalues of 0, 1, and 2, if the eigenvalues of C^T C can be determined with the eigenvalues of C.
  2. jcsd
  3. May 7, 2007 #2

    D H

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  4. May 7, 2007 #3
    Actually, I was really certain about the bases for rng and corng.

    Mostly what I don't see is the relation between the eigenvalues of C and the eigenvalues of C^T C, because one matrix is a symmetric positive definite matrix, while the other (C) is a general 3x3 matrix.
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