\[ \begin{aligned} D &= \begin{bmatrix} a_1 & \\ & \ddots &\\ & & a_n \end{bmatrix} \end{aligned} \]

\[ \begin{aligned} D &= \begin{bmatrix} a_1 & \\ & \ddots &\\ & & a_n \end{bmatrix}\\\\ &= \text{diag}(a_1, \cdots, a_n) \end{aligned} \]

\[ \begin{aligned} De_1 &= \begin{bmatrix} a_1 & \\ & a_2 &\\ & & a_3 \end{bmatrix} \begin{bmatrix} 1\\ 0\\ 0 \end{bmatrix} \end{aligned} \]

\[ \begin{aligned} De_2 &= \begin{bmatrix} a_1 & \\ & a_2 &\\ & & a_3 \end{bmatrix} \begin{bmatrix} 0\\ 1\\ 0 \end{bmatrix} \end{aligned} \]

\[ \begin{aligned} De_3 &= \begin{bmatrix} a_1 & \\ & a_2 &\\ & & a_3 \end{bmatrix} \begin{bmatrix} 0\\ 0\\ 1 \end{bmatrix} \end{aligned} \]

\[ \begin{aligned} De_1 &= \begin{bmatrix} a_1 & \\ & a_2 &\\ & & a_3 \end{bmatrix} \begin{bmatrix} 1\\ 0\\ 0 \end{bmatrix}\\\\ &= \begin{bmatrix} a_1\\ 0\\ 0 \end{bmatrix} \end{aligned} \]

\[ \begin{aligned} De_2 &= \begin{bmatrix} a_1 & \\ & a_2 &\\ & & a_3 \end{bmatrix} \begin{bmatrix} 0\\ 1\\ 0 \end{bmatrix}\\\\ &= \begin{bmatrix} 0\\ a_2\\ 0 \end{bmatrix} \end{aligned} \]

\[ \begin{aligned} De_3 &= \begin{bmatrix} a_1 & \\ & a_2 &\\ & & a_3 \end{bmatrix} \begin{bmatrix} 0\\ 0\\ 1 \end{bmatrix}\\\\ &= \begin{bmatrix} 0\\ 0\\ a_3 \end{bmatrix} \end{aligned} \]

\[ \begin{aligned} De_1 &= \begin{bmatrix} a_1 & \\ & a_2 &\\ & & a_3 \end{bmatrix} \begin{bmatrix} 1\\ 0\\ 0 \end{bmatrix}\\\\ &= \begin{bmatrix} a_1\\ 0\\ 0 \end{bmatrix}\\\\ &= a_1 \cdot e_1 \end{aligned} \]

\[ \begin{aligned} De_2 &= \begin{bmatrix} a_1 & \\ & a_2 &\\ & & a_3 \end{bmatrix} \begin{bmatrix} 0\\ 1\\ 0 \end{bmatrix}\\\\ &= \begin{bmatrix} 0\\ a_2\\ 0 \end{bmatrix}\\\\ &= a_2 \cdot e_2 \end{aligned} \]

\[ \begin{aligned} De_3 &= \begin{bmatrix} a_1 & \\ & a_2 &\\ & & a_3 \end{bmatrix} \begin{bmatrix} 0\\ 0\\ 1 \end{bmatrix}\\\\ &= \begin{bmatrix} 0\\ 0\\ a_3 \end{bmatrix}\\\\ &= a_3 \cdot e_3 \end{aligned} \]

\[ \begin{aligned} D^2 &= \begin{bmatrix} a_1 & \\ & a_2 &\\ & & a_3 \end{bmatrix} \begin{bmatrix} a_1 & \\ & a_2 &\\ & & a_3 \end{bmatrix} \end{aligned} \]

\[ \begin{aligned} D^2 &= \begin{bmatrix} a_1 & \\ & a_2 &\\ & & a_3 \end{bmatrix} \begin{bmatrix} a_1 & \\ & a_2 &\\ & & a_3 \end{bmatrix}\\\\ &= \begin{bmatrix} a_1^2 & \\ & a_2^2 &\\ & & a_3^2 \end{bmatrix} \end{aligned} \]

\[ \begin{aligned} D^k &= \begin{bmatrix} a_1 & \\ & a_2 &\\ & & a_3 \end{bmatrix} \cdots \begin{bmatrix} a_1 & \\ & a_2 &\\ & & a_3 \end{bmatrix} \end{aligned} \]

\[ \begin{aligned} D^k &= \begin{bmatrix} a_1 & \\ & a_2 &\\ & & a_3 \end{bmatrix} \cdots \begin{bmatrix} a_1 & \\ & a_2 &\\ & & a_3 \end{bmatrix}\\\\ &= \begin{bmatrix} a_1^k & \\ & a_2^k &\\ & & a_3^k \end{bmatrix} \end{aligned} \]

\[ D = \begin{bmatrix} a_1 & \\ & a_2 &\\ & & a_3 \end{bmatrix} \]

\[ |D| = \]

\[ D = \begin{bmatrix} a_1 & \\ & a_2 &\\ & & a_3 \end{bmatrix} \]

\[ |D| = a_1 a_2a_3 \]

\[ D = \begin{bmatrix} a_1 & \\ & a_2 &\\ & & a_3 \end{bmatrix} \]

\[ D^{-1} = \]

\[ D = \begin{bmatrix} a_1 & \\ & a_2 &\\ & & a_3 \end{bmatrix} \]

\[ \begin{aligned} D^{-1} &= \begin{bmatrix} \frac{1}{a_1} & \\ & \frac{1}{a_2} &\\ & & \frac{1}{a_3} \end{bmatrix} \end{aligned} \]

The vectors in the standard basis are ————— of a diagonal matrix.

The vectors in the standard basis are eigenvectors of a diagonal matrix.

Definition-1: An \(n \times n\) matrix \(A\) is diagonalizable if there is a basis of eigenvectors for \(\mathbb{R}^n\)

Definition-1: An \(n \times n\) matrix \(A\) is diagonalizable if there is a basis of eigenvectors for \(\mathbb{R}^n\)

Ok, so what?

\[ Av_1 = \lambda_1v_1 \]

\[ Av_2 = \lambda_2v_2 \]

\[ Av_3 = \lambda_3v_3 \]

\[ A\begin{bmatrix} \vert\\ v_1\\ \vert \end{bmatrix} = \begin{bmatrix} \vert\\ \lambda_1 v_1\\ \vert \end{bmatrix} \]

\[ A\begin{bmatrix} \vert\\ v_2\\ \vert \end{bmatrix} = \begin{bmatrix} \vert\\ \lambda_2 v_2\\ \vert \end{bmatrix} \]

\[ A\begin{bmatrix} \vert\\ v_3\\ \vert \end{bmatrix} = \begin{bmatrix} \vert\\ \lambda_3 v_3\\ \vert \end{bmatrix} \]

\[ A\begin{bmatrix} \vert & \vert & \vert\\ v_1 & v_2 & v_3\\ \vert & \vert & \vert\\ \end{bmatrix} = \begin{bmatrix} \vert & \vert & \vert\\ \lambda_1 v_1 & \lambda_2 v_2 & \lambda_3 v_3\\ \vert & \vert & \vert \end{bmatrix} \]

\[ A\begin{bmatrix} \vert & \vert & \vert\\ v_1 & v_2 & v_3\\ \vert & \vert & \vert\\ \end{bmatrix} = \begin{bmatrix} \vert & \vert & \vert\\ v_1 & v_2 & v_3\\ \vert & \vert & \vert \end{bmatrix} \begin{bmatrix} \lambda_1 & &\\ & \lambda_2 & \\ & & \lambda_3 \end{bmatrix} \]

\[ A\begin{bmatrix} \vert & \vert & \vert\\ v_1 & v_2 & v_3\\ \vert & \vert & \vert\\ \end{bmatrix} = \begin{bmatrix} \vert & \vert & \vert\\ v_1 & v_2 & v_3\\ \vert & \vert & \vert \end{bmatrix} \begin{bmatrix} \lambda_1 & &\\ & \lambda_2 & \\ & & \lambda_3 \end{bmatrix} \]

\[ Q = \begin{bmatrix} \vert & \vert & \vert\\ v_1 & v_2 & v_3\\ \vert & \vert & \vert\\ \end{bmatrix}, D = \begin{bmatrix} \lambda_1 & &\\ & \lambda_2 & \\ & & \lambda_3 \end{bmatrix} \]

\[ A\begin{bmatrix} \vert & \vert & \vert\\ v_1 & v_2 & v_3\\ \vert & \vert & \vert\\ \end{bmatrix} = \begin{bmatrix} \vert & \vert & \vert\\ v_1 & v_2 & v_3\\ \vert & \vert & \vert \end{bmatrix} \begin{bmatrix} \lambda_1 & &\\ & \lambda_2 & \\ & & \lambda_3 \end{bmatrix} \]

\[ Q = \begin{bmatrix} \vert & \vert & \vert\\ v_1 & v_2 & v_3\\ \vert & \vert & \vert\\ \end{bmatrix}, D = \begin{bmatrix} \lambda_1 & &\\ & \lambda_2 & \\ & & \lambda_3 \end{bmatrix} \]

\[ A Q = QD \]

\[ A\begin{bmatrix} \vert & \vert & \vert\\ v_1 & v_2 & v_3\\ \vert & \vert & \vert\\ \end{bmatrix} = \begin{bmatrix} \vert & \vert & \vert\\ v_1 & v_2 & v_3\\ \vert & \vert & \vert \end{bmatrix} \begin{bmatrix} \lambda_1 & &\\ & \lambda_2 & \\ & & \lambda_3 \end{bmatrix} \]

\[ Q = \begin{bmatrix} \vert & \vert & \vert\\ v_1 & v_2 & v_3\\ \vert & \vert & \vert\\ \end{bmatrix}, D = \begin{bmatrix} \lambda_1 & &\\ & \lambda_2 & \\ & & \lambda_3 \end{bmatrix} \]

\[ A Q = QD \]
\[ \beta = \{v_1, \cdots, v_n\} \]

\[ A\begin{bmatrix} \vert & \vert & \vert\\ v_1 & v_2 & v_3\\ \vert & \vert & \vert\\ \end{bmatrix} = \begin{bmatrix} \vert & \vert & \vert\\ v_1 & v_2 & v_3\\ \vert & \vert & \vert \end{bmatrix} \begin{bmatrix} \lambda_1 & &\\ & \lambda_2 & \\ & & \lambda_3 \end{bmatrix} \]

\[ Q = \begin{bmatrix} \vert & \vert & \vert\\ v_1 & v_2 & v_3\\ \vert & \vert & \vert\\ \end{bmatrix}, D = \begin{bmatrix} \lambda_1 & &\\ & \lambda_2 & \\ & & \lambda_3 \end{bmatrix} \]

\[ A Q = QD \]
\[ \beta = \{v_1, \cdots, v_n\} \]
\[ A = QDQ^{-1} \]

\[ A\begin{bmatrix} \vert & \vert & \vert\\ v_1 & v_2 & v_3\\ \vert & \vert & \vert\\ \end{bmatrix} = \begin{bmatrix} \vert & \vert & \vert\\ v_1 & v_2 & v_3\\ \vert & \vert & \vert \end{bmatrix} \begin{bmatrix} \lambda_1 & &\\ & \lambda_2 & \\ & & \lambda_3 \end{bmatrix} \]

\[ Q = \begin{bmatrix} \vert & \vert & \vert\\ v_1 & v_2 & v_3\\ \vert & \vert & \vert\\ \end{bmatrix}, D = \begin{bmatrix} \lambda_1 & &\\ & \lambda_2 & \\ & & \lambda_3 \end{bmatrix} \]

\[ A Q = QD \]
\[ \beta = \{v_1, \cdots, v_n\} \]
\[ A = QDQ^{-1} \]

\(A\) and \(D\) are similar matrices

Definition-2: An \(n \times n\) matrix \(A\) is diagonalizable if it is similar to a diagonal matrix.

\[ A = Q D Q^{-1} \]

\[ A = Q D Q^{-1} \]

\[ \begin{aligned} |A - \lambda I| &= \end{aligned} \]

\[ A = Q D Q^{-1} \]

\[ \begin{aligned} |A - \lambda I| &= |QDQ^{-1} - \lambda I|\\\\ \end{aligned} \]

\[ A = Q D Q^{-1} \]

\[ \begin{aligned} |A - \lambda I| &= |QDQ^{-1} - \lambda I|\\\\ &= |QDQ^{-1} - \lambda Q Q^{-1}| \end{aligned} \]

\[ A = Q D Q^{-1} \]

\[ \begin{aligned} |A - \lambda I| &= |QDQ^{-1} - \lambda I|\\\\ &= |QDQ^{-1} - \lambda Q Q^{-1}|\\\\ &= |Q(D - \lambda I)Q^{-1}| \end{aligned} \]

\[ A = Q D Q^{-1} \]

\[ \begin{aligned} |A - \lambda I| &= |QDQ^{-1} - \lambda I|\\\\ &= |QDQ^{-1} - \lambda Q Q^{-1}|\\\\ &= |Q(D - \lambda I)Q^{-1}|\\\\ &= |Q| \cdot |D - \lambda I| \cdot |Q^{-1}| \end{aligned} \]

\[ A = Q D Q^{-1} \]

\[ \begin{aligned} |A - \lambda I| &= |QDQ^{-1} - \lambda I|\\\\ &= |QDQ^{-1} - \lambda Q Q^{-1}|\\\\ &= |Q(D - \lambda I)Q^{-1}|\\\\ &= |Q| \cdot |D - \lambda I| \cdot |Q^{-1}|\\\\ &= |D - \lambda I| \end{aligned} \]

\[ A = Q D Q^{-1} \]

\[ \begin{aligned} A^k &= \end{aligned} \]

\[ A = Q D Q^{-1} \]

\[ \begin{aligned} A^k &= (QDQ^{-1})^k \end{aligned} \]

\[ A = Q D Q^{-1} \]

\[ \begin{aligned} A^k &= (QDQ^{-1})^k\\\\ &= \underbrace{(QDQ^{-1}) \cdots (QDQ^{-1})}_{k \text{ blocks}} \end{aligned} \]

\[ A = Q D Q^{-1} \]

\[ \begin{aligned} A^k &= (QDQ^{-1})^k\\\\ &= \underbrace{(QDQ^{-1}) \cdots (QDQ^{-1})}_{k \text{ blocks}}\\\\ &= Q D^k Q^{-1} \end{aligned} \]