I've come across the following notation in a paper:
Let $L : \mathbb{R}^n \to \mathbb{R}^n$ be a self-adjoint operator. Suppose $X$ is a random vector in $\mathbb{R}^n$ such that $\langle LX, X \rangle$ is a random variable.
How exactly is $\langle LX, X \rangle$ a random variable defined on an inner product space here? We can represent $L$ as a $n\times n$ matrix. Is it then correct to assume that
$$\langle LX, X \rangle = \langle X, LX \rangle =(LX)^TX,$$
which would give a random variable in this sense.