We know that if v is an eigenvector of is called the Rayleigh quotient and that the corresponding eigenvalue Now, suppose  is an approximation to an eigenvector. How well does the Rayleigh quotient...

We know that if v is an eigenvector of is called the Rayleigh quotient and that the

corresponding eigenvalue Now, suppose
 is an approximation to an eigenvector. How well does the Rayleigh quotient approximate
 We can answer that question when
 is symmetric.

 Without loss of generality, assume that
 is a good approximation to eigenvector

₁
of the symmetric matrix
 Argue that

<sub>1

1

2

2

n

n</sub>
 where the

_i
are an orthonormal set of eigenvectors of
 corresponding to eigenvalues

_i

 Show that

 Using the result of part
 show that

and argue that
 is a close approximation to

₁