Prove $H$ (all $m\times n$ real matrices) is a Hilbert space

Question

$H$ consists of all $m\times n$ real matrices with addition and scalar multiplication defined as the usual corresponding operations with matrices, and with the inner product of two matrices $A, B$ defined as $(A,B) = \operatorname{Trace} [A^\prime QB]$ where $A^\prime$ denotes the transpose of the matrix $A$ and $Q$ is a symmetric, positive-definite $m\times m$ matrix.

How do I prove that $H$ is a Hilbert space? I'm not even sure where to begin. I've tried to prove it from the two-part definition that $H$ must be complete and must be a linear vector space $X$ with an inner product defined on $X\times X$, but that didn't get me anywhere...

Answer 1

You want to show that

1. $H$ is a vector space. This is easy: it's the space of $m \times n$ real matrices, which is essentially just $\mathbb R^{mn}$ with a different notation.
2. $(A,B)$ is a positive definite inner product.

Since $\mathbb R^{mn}$ is finite-dimensional, it is automatically complete in any norm.

Hint for positive definiteness of the inner product: if $B$ is a real symmetric matrix, $\text{Tr}(B)$ is the sum of the eigenvalues of $B$, and if that is $0$ and $B$ is positive semidefinite then $B$ must be $0$.