Diagonalization

Diagonal Matrix

A matrix is diagonal if the only non-zero elements, if any, are on the main diagonal.

• If $A$ is diagonal, then $A^k$ is very easy to compute because you simply exponentiate every diagonal element by $k$

Diagonal matrices cannot have $\lambda =0$

Strang

Consider a matrix $A$ with some eigenvalues and eigenvectors. $A\overrightarrow{v_0}={\lambda }_0\overrightarrow{v_0}$ … $A\overrightarrow{v_n}={\lambda }_n\overrightarrow{v_n}$

$$\begin{array}{c}A[\overrightarrow{v_0},...,\overrightarrow{v_n}]=[{\lambda }_0\overrightarrow{v_0},...,{\lambda }_n\overrightarrow{v_n}]\\ A[\overrightarrow{v_0},...,\overrightarrow{v_n}]=[\overrightarrow{v_0},...,\overrightarrow{v_n}]\left[\begin{array}{ccc}{\lambda }_0& & \\ & ...& \\ & & {\lambda }_n\end{array}\right]\\ A=[\overrightarrow{v_0},...,\overrightarrow{v_n}]\left[\begin{array}{ccc}{\lambda }_0& & \\ & ...& \\ & & {\lambda }_n\end{array}\right][\overrightarrow{v_0},...,\overrightarrow{v_n}{]}^{-1}\end{array}$$

Diagonalizable

Orthogonal Diagonalization