Linear Algebra and Its Applications · Problem 6.4.8

Computations with Matrices — Gilbert Strang

Mathematics · rayleigh-quotient-eigenvalues

Problem

\lambda_1

and

\mu_1

are the smallest eigenvalues of

A

and

B

, show that the smallest eigenvalue

\theta_1

A + B

is at least as large as

\lambda_1 + \mu_1

. (Try the corresponding eigenvector

x

in the Rayleigh quotients.) Note. Problems 7 and 8 are perhaps the most typical and most important results that come easily from Rayleigh's principle, but only with great difficulty from the eigenvalue equations themselves.

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