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Thread: 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 $\displaystyle AA^T = B$ implies $\displaystyle B$ is symmetric, there exists an orthogonal matrix $\displaystyle P$ such that $\displaystyle P^TBP = \Lambda,$ and $\displaystyle \Lambda$ is a diagonal matrix with the eigenvalues of $\displaystyle B$ along the diagonal and the columns of $\displaystyle P$ are the eigenvectors of $\displaystyle B$. $\displaystyle P^TBP = \Lambda$ is called a unitary transformation. Since $\displaystyle P$ is orthogonal, $\displaystyle PP^T = I$ and thus $\displaystyle B = PP^T B PP^T = P\Lambda P^T.$

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

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

    This brings up a more general point. For any $\displaystyle B,$ a matrix $\displaystyle A$ such that $\displaystyle AA^T = B$ is not necessarily unique. The theory of unitary transformations says nothing about uniqueness.
    Last edited by JakeD; Aug 10th 2007 at 12:13 AM.
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