\title{Things about the Rayleigh Quotient}

The Rayleigh quotient is defined as

where is a symmetrical real matrix and is a real vector. It has the property that for any non-zero real scalar . We have

where and are the minimum and maximum eigenvalue of . The equality holds when and , where and are the corresponding eigenvectors, respectively.

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

where .

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

To be continued…

### Like this:

Like Loading...