## anonymous one year ago let A be an m*n matrix. show that A^T *A and A*A^T are both symetric ??

1. amoodarya

how can m*n matrix be symmetric ?

2. anonymous

A^T*A is symetric not A

3. amoodarya

$A_{m*n} \rightarrow (A_{m*n} )^T=A'_{n*m}$ transpose of m*n matrix is a n*m matrix ,and when $m \neq n$ $A^T \neq A$

4. anonymous

A^T isnt symmetric but A*A^T is symmetric read it again !

5. amoodarya

sorry but I think , the question changed !?

6. amoodarya

it suffice to apply transpose , to check symmetry $(A^T*A)^T=\\(A^T)*(A^T)^T=\\A^T*A$

7. amoodarya

$(A*A^T)^T=\\(A^T)^T*A^T=\\A*A^T$

8. anonymous

thanx

9. imqwerty

10. amoodarya

$(AA^T)_{m*m}=(A_{m*n} \times A^T_{n*m})^T=\\(A^T_{n*m})^T \times (A_{m*n})^T=\\(A_{m*n})\times A^T_{n*m}=\\(AA^T)_{m*m}$

11. Empty

I don't understand these subscripts being introduced, it seems sorta extra information, we already know that A is an mxn matrix so it doesn't really tell us anything we wouldn't already know. Really this whole thing just hinges on the fact of how the transpose has this property: $(AB)^T = B^TA^T$ So if we want to prove something is symmetric we just have to show: $M=M^T$ so we take the matrix: $(AA^T)^T=A^{TT}A^T=AA^T$ And we're done once we know transposing twice is the same as not having transposed at all.