Orthogonal Diagonalization

Matrix diagonalization

An $n\times n$ matrix $A$ is diagonalizable if there exists an invertible matrix, $P$, such that $D=P^{-1}AP$ is a diagonal matrix. The matrix, $P$, is said to diagonalize $A$.

Example

Consider $A=\begin{bmatrix}3&2\\3&4\end{bmatrix}$, $P=\begin{bmatrix}-1&2\\1&3\end{bmatrix}$ , and $D=\begin{bmatrix}1&0\\0&6\end{bmatrix}$. We can verify that $P^{-1}AP=D$.