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Math Help - Matrix problem, help please

  1. #1
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    Matrix problem, help please

    Let A be a non-square matrix, say m*n for m not equal to n, then C=A*A^t and D =A^t*A are both square matrics, but of different sizes. Show that C and D both have the same non-zero eignvalues.
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  2. #2
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    Quote Originally Posted by pdnhan View Post

    Let A be a non-square matrix, say m*n for m not equal to n, then C=A*A^t and D =A^t*A are both square matrics, but of different sizes. Show that C and D both have the same non-zero eignvalues.
    let \lambda \neq 0 be an eigenvalue of C. then AA^Tv=Cv=\lambda v, for some non-zero vector v. note that w=A^Tv \neq 0, because otherwise we'd have 0=Aw=\lambda v, which is impossible.

    now Dw=A^TAw=\lambda A^Tv=\lambda w, which means that \lambda is an eigenvalue of D. so every non-zero eigenvalue of C is an eigenvalue of D.

    proving that every non-zero eigenvalue of D is an eigenvalue of C is identical and is left for you.
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