Let $A = \begin{bmatrix} 2 & 2 \\ 1 & 3 \end{bmatrix}$ . Find a non-singular matrix $P$ such that $P^{-1}AP$ is a diagonal matrix.

Step 1: Calculate Eigenvalues

$|A-\lambda I| = 0 \implies (2-\lambda)(3-\lambda) - 2 = 0 \implies \lambda = 1, 4$

Step 2: Calculate Eigenvectors corresponding to the eigenvalues

For $\lambda = 1, (A-I)X = 0 \implies \begin{bmatrix} 1 & 2 \\ 1 & 2 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \\ \implies x_1 + 2x_2 = 0. \\ For x_2 = 1, x_1 = -2 \implies \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} -2 \\ 1 \end{bmatrix}$

For $\lambda = 4, (A-I)X = 0 \implies \begin{bmatrix} -2 & 2 \\ 1 & -1 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \\ \implies x_1 - x_2 = 0. \\ For x_2 = 1, x_1 = 1 \implies \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$

Step 3: Form the matrix

The required non-singular matrix $P$ such that $P^{-1}AP$ is a diagonal matrix is the matrix with the eigenvectors as its columns. Hence, $P = \begin{bmatrix} -2 & 1 \\ 1 & 1 \end{bmatrix}$.

answered Nov 11, 2017 by (1,920 points)