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

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

Linear Algebra
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At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

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how can m*n matrix be symmetric ?
A^T*A is symetric not A
\[A_{m*n} \rightarrow (A_{m*n} )^T=A'_{n*m} \] transpose of m*n matrix is a n*m matrix ,and when \[m \neq n\] \[A^T \neq A\]

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A^T isnt symmetric but A*A^T is symmetric read it again !
sorry but I think , the question changed !?
it suffice to apply transpose , to check symmetry \[(A^T*A)^T=\\(A^T)*(A^T)^T=\\A^T*A\]
\[(A*A^T)^T=\\(A^T)^T*A^T=\\A*A^T\]
thanx
a useful link - http://users.math.msu.edu/users/hhu/309/3091326.pdf
\[(AA^T)_{m*m}=(A_{m*n} \times A^T_{n*m})^T=\\(A^T_{n*m})^T \times (A_{m*n})^T=\\(A_{m*n})\times A^T_{n*m}=\\(AA^T)_{m*m}\]
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.

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