Matrix $A$ is similar to diagonal matrix $D$. Determine the diagonal elements of $D$.

Question

Matrix $A =\begin{bmatrix}5/2&3/2\\-5/2&13/2\end{bmatrix}$ and $D = \begin{bmatrix}u&0\\0&v\end{bmatrix}$

I know that $D = P^{-1}AP$.

Knowing that $P^{-1}P=I$,

$D = P^{-1}AP$

$D = A$

I tried answering $u = 5/2$ and $v = 13/2$ but it was wrong.

Any help would be greatly appreciated

Answer 1

$AP = PD$

$P$ has column vectors $p_1,p_2$ and $PD = [p_1u, p_2 v]$

and $AP = [Ap_1,Ap_2]$

$Ap_1 = up_1$ and $Ap_2 = vp_1$ if this is true then

$Ap_1 - up_1 = 0\\ Ap_1 - u Ip_1 = 0\\ (A-u I)p_1 = 0$

and $(A- u I)$ is a singular matrix.

and if $(A-u I)$ is singular then $\det (A-u I) = 0$

and similarly $\det(A-v I) = 0$

Answer 2

$D = P^{-1}AP$ doesn't imply that $D=A$. The matrix multiplication isn't commutative.

As you know that $A$ is diagonalizable, you just need to find the eigenvalues of $A$ and these would be the entries $u,v$ in $D$. It's not hard to do that. In fact it boils down to solving the characteristic polynomial of $A$ and $\Delta_A = x^2 - 9x + 20$