Rayleigh quotient

For a scalar field $\mathbb{k}$ that is either $\mathbb{R}$ or $\mathbb{C}$, an inner product space $E$ over $\mathbb{k}$, and a $\mathbb{k}$-linear map $T$ from $E$ to itself, the Rayleigh quotient of $T$ at a vector $x \in E$ is defined as the real number $R_T(x) = \operatorname{Re}\langle x, Tx\rangle / \|x\|^2$, where $\langle x, Tx\rangle$ is the (possibly complex-valued) inner product of $x$ with $Tx$, $\operatorname{Re}(\cdot)$ denotes taking its real part, and $\|x\|$ is the norm of $x$. The definition places no hypothesis that $x$ be nonzero, and no hypothesis that $T$ be self-adjoint, symmetric, or satisfy any other property beyond $\mathbb{k}$-linearity. In the degenerate case $x = 0$, the denominator $\|x\|^2$ is $0$; by the ambient division convention (division by zero is defined to equal zero), $R_T(0)$ is therefore defined and equals $0$, rather than being undefined or excluded by a side condition.