If A is any real $m \times n$ matrix, then $A^TA$ is called the Gram matrix of $A$. Show that the Gram matrix of $A$ is always symmetric with nonnegative diagonal elements.
I have tried several matrices to prove that this is true but im not sure how I go about showing for a general matrix that this always holds true. Please help.
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$\begingroup$Hints:
Prove that the Gram matrix is always symmetric: What does symmetry have to do with transposes? What happens when you transpose $A^TA$?
Prove that the diagonal elements are nonnegative: Let $a_{ij}$ be the $(i,j)$ element of A. Can you find an expression for the $k$-th diagonal element of $A^TA$ in terms of the $a_{ij}$?
Some hints: for first part, use $(AB)^T=B^TA^T$
for second part, use $a_{ii}=e_i^TA^TAe_i=(Ae_i)^T(Ae_i)=||Ae_i||^2\geq0$. ($e_i$ is $i^{th}$ column of identity matrix)
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