Diagonalizability

To find whether $ A$ is diagonalizable, we can use the command [P,D] = eig(A) and the function det(P). If det(P) returns 0, or the magnitude of $ 10^{-8}$ (for example, 1.2608e-08), $ P$ is not invertible, and therefore, $ A$ is not diagonalizable. Otherwise, $ P$ is invertible, and $ A$ is diagonalizable.

EXAMPLE 1. Diagonalize each of the following matrices if possible.

  1. $ A = \begin{bmatrix}
1 & 3 & 3 \\
-3 &-5 &-3 \\
3 & 3 & 1
\end{bmatrix}$
  2. $ A = \begin{bmatrix}
2 & 4 & 3 \\
-4 &-6 &-3 \\
3 & 3 & 1
\end{bmatrix}$
  3. $ A = \begin{bmatrix}
5 & 0 & 0 & 0 \\
0 & 5 & 0 & 0 \\
1 & 4 &-3 & 0 \\
-1 &-2 & 0 &-3
\end{bmatrix}$

Power of diagonalizable matrices

Let $ D$ be a diagonal matrix with diagonal entities  $ \lambda_1,\ldots,\lambda_n$. Then the $ k$-th power $ D^k$ is simply given by

$\displaystyle D^k =
\begin{bmatrix}
\lambda_1^k & 0 & \cdots & 0 \\
0 & \lambda_2^k & \cdots & 0 \\
\hdotsfor{4} \\
0 & 0 & \cdots & \lambda_n^k
\end{bmatrix}$

In particular when $ A$ is diagonalizable, we can express $ A^k = P D^k P^{-1}$.



Department of Mathematics
Last modified: 2005-10-21