Matrix Diagonalization

The diagonalization of $A$ is the diagonal matrix $D$ where each non-zero entry (the diagonal entry) is an eigenvalue of $A$.

$$D=\left[\begin{array}{cccc}{\lambda }_1& 0& ...& 0\\ 0& {\lambda }_2& ...& 0\\ .& .& ...& .\\ 0& 0& ...& {\lambda }_n\end{array}\right]$$

A matrix, $A$, is diagonalizable if it can be transformed into a diagonal matrix through a Similarity Transformation.

$$\begin{array}{r}{P}^{-1}AP=D\\ \\ A=PD{P}^{-1}\end{array}$$

• Where $P$ is invertible and $D$ is diagonal.
• We say $P$ diagonalizes $A$ to $D$.
• $P$ and $P^{-1}$ do not cancel out cuz Matrix Product is not commutative.

Here, $A$ and $D$ are Similar Matrices, giving us useful properties. Combining similar matrices and diagonal matrices, we can do lots of stuff more efficiently.