eEigenValue

Example Find the eigen value of the following matrix.．

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

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 $t$

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

Then the eigen values of $A$

are $\lambda = 3, \pm i .$