Question

let A be an m*n matrix. show that A^T *A and A*A^T are both symetric ??

Answer 1

how can m*n matrix be symmetric ?

Answer 2

A^T*A is symetric not A

Answer 3

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

Answer 4

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

Answer 5

sorry
but I think , the question changed !?

Answer 6

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

Answer 7

\[(A*A^T)^T=\\(A^T)^T*A^T=\\A*A^T\]

Answer 8

thanx

Answer 9

a useful link -
http://users.math.msu.edu/users/hhu/309/3091326.pdf

Answer 10

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

Answer 11

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.