- #1

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## Homework Statement

Part (a): Find the eigenvalues and eigenvectors of matrix A:

[tex]

\left(

\begin{array}{cc}

2 & 0 & -1\\

0 & 2 & -1\\

-1 & -1 & 3 \\

\end{array}

\right)

[/tex]

Part(b): Find the eigenvalues and eigenvectors of matrix ##B = e^{3A} + 5I##.

## Homework Equations

## The Attempt at a Solution

**Part (a)**[tex]\lambda = 1, 2, 4[/tex]

[tex] u_1 = \frac{1}{\sqrt3}(1,1,1)[/tex]

[tex]u_2 = \frac{1}{\sqrt 2}(1,-1,0)[/tex]

[tex]u_3 = \frac{1}{\sqrt 5}(1,1,-2)][/tex]

**Part(b)**Realize A is a hermitian matrix.

Diagonalize A:

[tex] A'=

\left(

\begin{array}{cc}

1 & 0 & 0\\

0 & 2 & 0\\

0 & 0 & 4 \\

\end{array}

\right)

[/tex]

[tex] B' = exp(3A') + 5I[/tex]

Therefore, eigenvalues of ##B'= e+5, e^2 + 5, e^4+5##. Also, eigenvalues of B = B'.

How do I find the eigenvectors of B? Do I need to undiagonalize B' using the transformation matrix made up of eigenvectors of A?