For example, the characteristic equation of
\[\mathbf{M} = \left( \begin{array}{cc} 2 & 1 \\ 1 & 2 \\ \end{array} \right)\]
has the characteristic matrix \[\mathbf{M}^2 - 4 \mathbf{M} +3=0\]
.The characteristic equation of a matrix is related to it eigenvalues. In fact the equation for the matrix and the eigenvalues is the same.
The eigenvalues of
\[\mathbf{M}\]
obey \[ \mathbf{M} \mathbf{v} = \lambda \mathbf{v} \rightarrow (\mathbf{M} - \lambda \mathbf{I}) \mathbf{v} = \mathbf{0} \rightarrow det(\mathbf{M} - \lambda \mathbf{I})=0\]
With
\[M\]
as above we have\[\begin{equation} \begin{aligned} det(\left( \begin{array}{cc} 2 & 1 \\ 1 & 2 \end{array} \right) - \lambda \mathbf{I}) &= det \left( \begin{array}{cc} 2- \lambda & 1 \\ 1 & 2- \lambda \\ \end{array} \right) \\ =& (2- \lambda)^2 - 1 \times 1 \\ &= \lambda^2 - 4 \lambda +3 \\ &= (\lambda -3)(\lambda -1)=0 \end{aligned} \end{equation}\]
This is the same equation as for
\[M\]
above.In fact
\[det(\mathbf{M}^2 - 4 \mathbf{M} +3 \mathbf{I})=det((\mathbf{M} - \mathbf{I})(\mathbf{M}- \mathbf{I}))=det(\mathbf{M} - \mathbf{I}) det(\mathbf{M}- \mathbf{I})=0\]
so that
\[det(\mathbf{M} - \mathbf{I})=0\]
or \[det(\mathbf{M} -3\mathbf{I})\]
.In general to find the characteric equation for the matrix, we find the equation for the eigenavlues.