Matrix proofs

[Guest]

Help!

I'm having trouble with these matrix proofs/explanations.

Let A be an m x n matrix.

a) Explain why the matrix multiplications A^T * A and A * A^T are possible.
b) Show that A^T * A and A * A^T are both symmetric.

T = transpose

 

[daon]

a) What is the condition that must be satisfied in order to be able to multiply two matricies? Something about certain numbers matching? Hmm...
b) Take an arbitraty matrix \(\displaystyle \L [a_{ij}]_{i,j=1}^n\), transpose it, multiply it by the original, and see if it meets the criteria for symmetry (that is, the resulting matrix is equal to its transpose).

 

[Unco]

b) Recall that \(\displaystyle \mbox{(AB)^T = B^TA^T}\) and \(\displaystyle \mbox{(A^T)^T = A}\). A matrix \(\displaystyle \mbox{A}\) is symmetric if \(\displaystyle \mbox{A^T = A}\).