Is tr(A A^T) Equal to tr(A^T A)?

[ha9981]

to begin I am wondering if its even true that they are equal. As i lost the sheet with that on it. If that is not true it could have been tr(A B^T) = tr(B A^T) but it doubt it.

I have tried proving it in so many ways, but I am stuck.

 

[lstellyl]

well so the proof is easy if you know that the trace is cyclic (then it is just one line actually...)

assuming that you are not allowed to use this property, or at least must prove it first...

to prove that it is cyclical, notice that the trace of a matrix A is the sum a_{i,i}

try writing out the formula for matrix multiplication of two arbitrary matrices A and B (ie, what is the i,jth element of the product AB?) and then think about the case i=j

if tr(AB) = tr(BA), then the trace is cyclic... and then tr(AA^T) = tr(A^TA)