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Diagonalization of a square matrix.

A diagonal matrix is a square matrix whose only non zero entries are on the diagonal, i.e. A=(aij) is diagonal if, and only if, $i \neq j \Longrightarrow a_{ij}=0$. If the diagonal entries are $\alpha_1, \alpha_2, \dots , \alpha_r$, we denote $D= \text{diag}(\alpha_1, \alpha_2, \dots , \alpha_r)$ .

For example, $\begin{pmatrix}1 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & -1 \end{pmatrix}$ is a diagonal matrix.

A square matrix A is diagonalizable if there exists an invertible matrix P such that P-1AP is a diagonal matrix.

Theorem 11.6.1   Consider the square matrix A of order n as the matrix of an endomorphism of $\bbb{R}^n$ w.r.t. the standard basis. If there exist a basis of $\bbb{R}^n$ composed only from eigenvectors of A, then A is diagonalizable.

The matrix P is the change of base matrix from the standard basis to the basis of eigenvectors.

Example 11.6.2   With the settings of  4.3, we have the eigenvectors $\overrightarrow{u_1} =\begin{pmatrix}2 \\ 1 \end{pmatrix}$ and $\overrightarrow{u_2} =\begin{pmatrix}1 \\ -2 \end{pmatrix}$. They are linearly independent; as $\dim V=2$, they form a basis of V. Define the matrix $P=P_{eu}= \begin{pmatrix}2 & 1 \\ 1 & -2 \end{pmatrix}$; then $P^{-1}= \frac 15 \begin{pmatrix}2 & 1 \\ 1 & -2 \end{pmatrix}$and $P^{-1}AP=\begin{pmatrix}5 & 0 \\ 0 & -5 \end{pmatrix}$.

Noah Dana-Picard