\title{Things about the Rayleigh Quotient}

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…