Let $V$ be the vector space $M_{n×n}(\mathbb{R})$, and let $B : V × V → \mathbb{R}$ be the bilinear form $B(X,Y)$ = tr$(XY^T)$. Calculate the signature of $B$.
Here, tr means the trace of the matrix.
I'm stuck here as I was only able to conclude that this is a symmetric bilinear form (correct me if I'm wrong), then how to proceed after this ?
Best Answer
By the definition of trace and product of matrices, if $x_i$ denotes the $i$th row of a matrix $X$, then $${\rm tr}(XX^T)\ =\ \sum_i x_i{x_i}^T\ =\ \sum_i\|{x_i}^T\|^2\ >\ 0$$ unless each $x_i=0$.
Here $\|v\|^2=\sum_jv_j^2$ is the square of the Euclidean norm of vector $v$.
By the way, if you calculated entrywise, you'd obtain $${\rm tr}(XY^T)\ =\ \sum_{i,j}x_{i,j}y_{i,j}$$ which is just the usual inner product on the space of flattened (vectorized) matrices (utilizing the linear isomorphism $M_{m,n}(\Bbb R)\cong\Bbb R^{m\cdot n}$).