## The Characteristic Equation of a Matrix

The characteristic equation of a matrix is a polynomial equation in the matrix equal to zero.
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{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}

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.