is a matrix $A*A^{t}$ or $A^{t}* A $ Symmetric?
if $\mathbb{K}$ is a field and $A\in M_{m\times n}(K)$ proof or give a counterexample that $A\cdot A^t $ and $A^t\cdot A $ are Symmetric matrix
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$\begingroup$Yeah they are symmetric
If A is an m × n matrix and A^t is its transpose, then the result of matrix multiplication with these two matrices gives two square matrices: AA^t is m × m and A^tA is n × n. Furthermore, these products are symmetric matrices. Indeed, the matrix product A.A^t has entries that are the inner product of a row of A with a column of A^t. But the columns of A^t are the rows of A, so the entry corresponds to the inner product of two rows of A. If pi j is the entry of the product, it is obtained from rows i and j in A. The entry P(ij) is also obtained from these rows, thus p(ij)= p(ji), and the product matrix p(i,j) is symmetric. Similarly, the product A^t*A is a symmetric matrix.
A quick proof of the symmetry of A AT results from the fact that it is its own transpose:
(A * A^t)^t = (A^t)^t * A^t = A * A^t
refer this for more info
$\endgroup$ $\begingroup$By the property $(AB)^t = B^tA^t$ you have $(A^tA)^t = A^t (A^t)^t = A^tA$, so $A^tA$ is symmetric. The same works for the second case.
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