The Rayleigh quotient ${R(M,x)}$ is defined as

$\displaystyle R(M,x) = \dfrac{x^TMx}{x^Tx},$

where ${M}$ is a symmetrical real matrix and ${x}$ is a real vector. It has the property that ${R(M,cx) = R(M,x)}$ for any non-zero real scalar ${c}$. We have

$\displaystyle \lambda_{min} \leq R(M,x) \leq \lambda_{max},$

where ${\lambda_{min}}$ and ${\lambda_{max}}$ are the minimum and maximum eigenvalue of ${M}$. The equality holds when ${x = v_{min}}$ and ${x = v_{max}}$, where ${v_{min}}$ and ${v_{max}}$ are the corresponding eigenvectors, respectively.

The range of the Rayleigh quotient is called the spectrum (in functional analysis), and ${\lambda_{max}}$ is known as the spectral radius. The Rayleigh quotient is used to obtain an eigenvalue approximation from an eigenvector approximation, since it gives

$\displaystyle \|Ax\| \leq \lambda_{max} \|x\|,$

where ${M = A^TA}$.

I came across this concept in the Dinur’s proof of the Probabilistically Checkable Proof (PCP) theorem.

To be continued…