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Math Help - Help with Positive Definite Proof

  1. #1
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    Help with Positive Definite Proof

    Problem: Let A and B be symmetric nxn matrices whose eigenvalues are all positive. Show that the eigenvalues of A + B are all positive.

    I really don't know where to begin, a hint would be greatly appreciated.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by crymorenoobs View Post
    Problem: Let A and B be symmetric nxn matrices whose eigenvalues are all positive. Show that the eigenvalues of A + B are all positive.

    I really don't know where to begin, a hint would be greatly appreciated.
    What machinery do you have to work with?
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  3. #3
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    Well, the rest of the questions were about Quadratic forms and their corresponding symmetric matrices, but I don't see that helping me on this proof.

    I had to prove that Q is positive definite if Det(A) > 0 and A11 > 0 for the corresponding 2x2 matrix in the previous question.

    Should I be checking if det(A+B) > 0 and A11 + B11 > 0 in this question?
    If so, how can I go about checking the determinant of an arbitrary sized square matrix without lots of excessive writing.
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