# symmetric transpose matrix

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• Nov 19th 2009, 07:07 PM
ecc5
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!
• Nov 19th 2009, 07:13 PM
lvleph
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?
• Nov 19th 2009, 07:18 PM
ecc5
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.
• Nov 19th 2009, 07:27 PM
lvleph
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: $(\sum_{k=1}^n A_{ik} A^T_{kj})^T = \sum_{k=1}^n (A_{ik} A^T_{kj})^T \dots$
• Nov 19th 2009, 07:59 PM
ecc5
I figured it out... I was definitely over-analyzing it. Thanks!