symmetric transpose matrix

• Nov 19th 2009, 06:07 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, 06:13 PM
lvleph
Rember that if $\displaystyle A_{ij}$ is the $\displaystyle ij$ entry of A then $\displaystyle A^T$ has an $\displaystyle ij$ entry $\displaystyle (A^T)_{ij} = A_{ji}$. Now write the multiplication $\displaystyle A*A^T = \sum_{k=1}^n A_{ik}*A^T_{kj}$ What can you say about these entries above and below the diagonal?
• Nov 19th 2009, 06:18 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, 06:27 PM
lvleph
So the other thing you must remember is that if a matrix is symmetrix $\displaystyle B = B^T$. Therefore, if $\displaystyle B = A*A^T$ then $\displaystyle B^T = A^T*A = A*A^T = B$

EDIT: One last hint: $\displaystyle (\sum_{k=1}^n A_{ik} A^T_{kj})^T = \sum_{k=1}^n (A_{ik} A^T_{kj})^T \dots$
• Nov 19th 2009, 06:59 PM
ecc5
I figured it out... I was definitely over-analyzing it. Thanks!