Question 5.2.5: Let A = [ 1 2 0 3 ] and P = [ 1 1 1 2] Verify that the matri...

The characteristic equation for A is
det(A − λI) = (1 − λ)(3 − λ) = 0
so the eigenvalues of A are \lambda _{1} = 1   and  \lambda _{2} = 3.  Since

B = P^{-1} AP = \begin{bmatrix} 2 &-1 \\ -1 &1  \end{bmatrix}\begin{bmatrix} 1 &2 \\ 0 &3  \end{bmatrix}\begin{bmatrix} 1 &1 \\ 1 &2 \end{bmatrix}=\begin{bmatrix} 3 &4 \\ 0 &1  \end{bmatrix}

the characteristic equation for B is
det(B − λI) = (1 − λ)(3 − λ) = 0
and hence, the eigenvalues of B are also \lambda _{1} = 1   and   \lambda _{2} = 3.