Example Find the eigen value of the following matrix.．

$$A = \left(\begin{array}{rrr} 3&0&0\\ 0&2&-5\\ 0&1&-2 \end{array}\right)$$

Answer If $A{\bf C} = t{\bf C}$, then the value $t$ is called the eigen value of the matrix $A$ and the nonzero vector $C$ is called the eigen vector of $A$. We note that $A{\bf C} = t{\bf C}$ has the nonzero vector $C$ if and only if $A -tI\vert = 0$. Thus, we solve $\vert A - tI\vert = 0$ for $t$.

$$\Phi_{A}(t) = \left \vert\begin{array}{rrr} 3-t & 0 & 0\\ 0 & 2-t & -5\\ 0 & 1 & -2-t \end{array}\right \vert = (3-t)(t^{2} + 1) = 0．$$

Then the eigen values of $A$ are $\lambda = 3, \pm i .$