can you prove that for matrix A the matrix A×A^T(transpose matrix) and A^T×A are Symmetric matrixes?

Answer 1

Yes! For a matrix to be symmetric, it must equal its own transpose. Now let us consider (AA^T)^T (the transpose of your matrix). We know that in general for the product of two matrices AB, (AB)^T = B^TA^T. Furthermore, we know that the tranpose of a transpose is the original matrix. Combining these two facts, we can deduce that

(1) (AA^T)^T = (A^T)^TA^T = AA^T

Thus, we found that the transpose of AA^T equals itself; thus, it must be symmetric. We could then follow the exact same process with A^TA to get the same result. I hope that helps!