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Math Help - Finding a matrix and its transpose from a given matrix

  1. #16
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    Thanks. I will have a try and fix things up before asking for help again. Thanks again JakeD. You have been very helpful
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  2. #17
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    Quote Originally Posted by JakeD View Post
    Since if AA^T = B implies B is symmetric, there exists an orthogonal matrix P such that P^TBP = \Lambda, and \Lambda is a diagonal matrix with the eigenvalues of B along the diagonal and the columns of P are the eigenvectors of B. P^TBP = \Lambda is called a unitary transformation. Since P is orthogonal, PP^T = I and thus B = PP^T B PP^T = P\Lambda P^T.

    Further, since x^TBx = x^TAA^Tx = (A^Tx)^T A^Tx > 0 when B is nonsingular, B is positive definite and thus has positive eigenvalues. Thus \Lambda = D D^T where D has the square roots of the eigenvalues of B on the diagonal.

    Then B = P\Lambda P^T = PD D^TP^T = A A^T where A = PD. So the problem is reduced to finding the eigenvalues and eigenvectors of a symmetric matrix B. There are efficient numerical methods for this.
    Here is the issue when B = I, the identity matrix. It means that AA^T = I and thus A is orthogonal. The problem is that any orthogonal matrix can serve as the eigenvector matrix P for the identity matrix. Thus when the original A is orthogonal, there is no way to identify it by looking at B = I.

    This brings up a more general point. For any B, a matrix A such that AA^T = B is not necessarily unique. The theory of unitary transformations says nothing about uniqueness.
    Last edited by JakeD; August 10th 2007 at 01:13 AM.
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