Matrix Diagonalization

cards/lin-alg
Diagonalizable matrix
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A square matrix $A$ is called diagonalizable if there exists a diagonal matrix $D$ and an invertible matrix $P$ such that $A=PD{P}^{-1}$.

The columns of $P$ are given by independent eigenvectors $v_1$ and $v_2$, while the diagonal values of $D$ are the corresponding eigenvalues:

$$A=\left[\begin{array}{cc}|& |\\ v_1& v_2\\ |& |\end{array}\right]\left[\begin{array}{cc}{\lambda }_1& 0\\ 0& {\lambda }_2\end{array}\right]{\left[\begin{array}{cc}|& |\\ v_1& v_2\\ |& |\end{array}\right]}^{-1}$$

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Steps to diagonalize:

1. Find eigenvalues of the matrix by solving the characteristic equation $\left|A-\lambda I\right|=0$.
2. Find the basis of ${\mathbb{R}}^n$ consisting of eigenvectors of the matrix.
3. Construct the matrices $D$ and $P$.
   • The matrix $D$ has the eigenvalues of $A$ as entries on the diagonal.
   • The columns of $P$ are the corresponding eigenvectors. Note that the columns should match.

The diagonalization of a matrix is unique up to the permutations of the order of the eigenvalues.

Powers with diagonalization

One benefit of diagonalizing a matrix is it lets us find the power of a matrix easily. For example:

$$\begin{array}{rl}{A}^2& =PD{P}^{-1}\cdot PD{P}^{-1}\\ & =PD\cancel{P^{-1}\cdot P}D{P}^{-1}\\ & =P(DD)P^{-1}\\ & =P{D}^2{P}^{-1}\end{array}$$

In general:

$$A^k=P{D}^k{P}^{-1}$$

and since $D$ is diagonal, $D^k$ is easy to obtain.