## Proof That Similar Matrices Have the Same Characteristic Equation

TheoremSimilar matrices have the same characteristic equation.

Proof

Matrices

\[A\]

and \[B\]

are similar if there is an invertible matrix \[P\]

such that \[A=P^{-1}BP\]

.The characteristic equation of the matrix

\[A\]

is the determinant of the matrix \[A- \lambda I\]

\[\begin{equation} \begin{aligned} det(A- \lambda I) &= det(P^{-1}(B- \lambda I)P) \\ &= det(P^{-1}) det (B- \lambda I) det(P) \\ &= \frac{1}{det(P)} det (B- \lambda I) det(P) \\ &= det (B- \lambda I) \end{aligned} \end{equation}\]

If matrices have the same characteristic solution it does not follow that they are similar. Matrices

\[A= \left( \begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array} \right) \: B= \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right) \]

both have the characteristic equation \[(1- \lambda )^2\]

but these matrices are not similar.