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Math Help - symmetric transpose matrix

  1. #1
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    symmetric transpose matrix

    I am stuck on this problem. Let A be an arbitrary mxn matrix. Show that A transpose times A is symmetric. My teacher has been doing proofs using i and j to denote entries of the matrix, so if you could use that notation when explaining it that would be great. Thanks!
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  2. #2
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    Rember that if A_{ij} is the ij entry of A then A^T has an ij entry (A^T)_{ij} = A_{ji}. Now write the multiplication A*A^T = \sum_{k=1}^n A_{ik}*A^T_{kj} What can you say about these entries above and below the diagonal?
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  3. #3
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    Well, the entries above and the below the diagonal must be the same, because in order for a matrix to be symmetric, A transpose must equal A. I understand the concept, but I just don't know how to write the proof.
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  4. #4
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    So the other thing you must remember is that if a matrix is symmetrix B = B^T. Therefore, if B = A*A^T then  B^T = A^T*A = A*A^T = B

    EDIT: One last hint: (\sum_{k=1}^n A_{ik} A^T_{kj})^T = \sum_{k=1}^n (A_{ik} A^T_{kj})^T \dots
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  5. #5
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    I figured it out... I was definitely over-analyzing it. Thanks!
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