Originally Posted by

**bigwave** $\textsf{ Diagonalize the matrix:}$

$$A=\left[\begin{array}{rrrr}

2& 2& -1\\1& 3& -1\\-1& -2& 2

\end{array}\right]$$

$\textit{ a. Charateristic equation:}$

$$A-\lambda I=

\left[\begin{array}{rrrr}

2& 2& -1\\1& 3& -1\\-1& -2& 2

\end{array}\right]-\left[\begin{array}{rrrr}

\lambda& 0& 0\\0& \lambda& 0\\0& 0& \lambda

\end{array}\right]$$

$$-\lambda^3+7\lambda^2-\lambda+5$$

$\textit{ b. $D=$,$P=$} $

$$v_1=\begin{bmatrix}-1\\-1\\1\end{bmatrix}

v_2=\begin{bmatrix}1\\0\\1\end{bmatrix}

v_3=\begin{bmatrix}-2\\1\\0\end{bmatrix}$$

$$\therefore P=(v_1,v_2,v_3)\begin{bmatrix}

\,\, 1&1&0\\-1&0&1\\ \,\,1&1&0

\end{bmatrix}$$

$\textit{ c. Verify}\\$

ok I got the Characteristic equation and $v_1,v_2,v_3$ from W|A

but didn't understand the example?

And didn't know how to verify it