Without using MATLAB, find the eigenvalues of Is there a basis of eigenvectors for 5 What is the spectral radius of Plot n2 for 50. Find the maximum value of n2 Build another nonsymmetric...

Without using MATLAB, find the eigenvalues of

Is there a basis of eigenvectors for

⁵

What is the spectral radius of

Plot

ⁿ

₂
for

50.

Find the maximum value of

ⁿ

₂

Build another nonsymmetric matrix with

Do parts

and

for it.

Perform the same actions with the symmetric matrix SYMMAT in the software distribution.

Attempt to explain your results. For a symmetric matrix, use the spectral theorem. We will develop Schur’s triangularization in Chapter 19. It states that very

real matrix

with real eigenvalues can be factored into

^T

where

is an orthogonal matrix and

is an upper-triangular matrix. Apply this result to the nonsymmetric case. In particular, is there a relationship between the eigenvalues of

and

that can explain what happens to

ⁿ

₂
as

increases