MATH 323 Lecture 26

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Theorem 6.3.1

If $\lambda _{1},\ldots ,\lambda _{k}$ are distinct eigenvalues of an Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \times n} matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} with corresponding eigenvectors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{x}_1, \ldots, \vec{x}_k} , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{x}_1, \ldots, \vec{x}_2} are linearly independent.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} V &= \mathrm{Span}\{ \vec{x}_1, \ldots, \vec{x}_k \} \\ \dim V &= r \end{align}}

Diagonalization

$A$ is said to be diagonalizable if there is a nonsingular matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} with

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X^{-1} A X = D = \begin{pmatrix}\lambda_1&\dots&0\\\vdots&\ddots&\vdots\\0&\dots&\lambda_n\end{pmatrix}}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D} is a diagonal matrix.

Therefore Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \sim D} by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = X D X^{-1}} .

Theorem 6.3.2

An Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \times n} matrix $A$ is diagonalizable iff Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} has Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} linearly independent eigenvectors.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} A: & \vec{x}_1, \ldots, \vec{x}_n \quad \text{linearly independent eigenvectors} \\ & \lambda_1, \ldots, \lambda_n \quad \text{eigenvalues (not necessarily distinct)} \\ X &= (\vec{x}_1, \ldots, \vec{x}_n) \\ D &= \begin{bmatrix}\lambda_1&\dots&0\\\vdots&\ddots&\vdots\\0&\dots&\lambda_n\end{bmatrix} \end{align}}

Corollary

1. If $A$ is diagonalizable and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = X D X^{-1}} , then diagonal entries of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D} are eigenvalues of $A$.
2. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X = (\vec{x}_1, \ldots, \vec{x}_n)} , but Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is not unique.
3. If eigenvalues are distinct, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is diagonalizable
4. If they are not distinct, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} may or may not be diagonalizable
5. If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = X D X^{-1}} , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^k = X D^k X^{-1} = X \begin{pmatrix}\lambda_1^k&\dots&0\\\vdots&\ddots&\vdots\\0&\dots&\lambda_n^k\end{pmatrix} X^{-1}} .

Example

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} A &= \begin{bmatrix}2 & -3 \\ 2 & -5 \end{bmatrix} \\ \lambda_1 &= 1 & \lambda_2 &= -4 \\ \vec{x}_1 &= \left\langle 3, 1 \right\rangle & \vec{x}_2 &= \left\langle 1, 2 \right\rangle \\ X &= \begin{pmatrix}3&1\\1&2\end{pmatrix} \\ X^{-1} A X = D &= \begin{pmatrix}1 & 0 \\ 0 & -4\end{pmatrix} \\ &= \frac{1}{5} \begin{bmatrix}2&-1\\-1&3\end{bmatrix} \, \begin{bmatrix}2&-3\\2&-5\end{bmatrix} \, \begin{bmatrix}3&1\\1&2\end{bmatrix} \end{align}}

Matrix Exponential

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{e}^x = \sum_{n=0}^\infty \frac{1}{n!} x^n} is a very important function.

We similarily define Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{e}^A} for a matrix $A$ to be

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{e}^A = \sum_{n=0}^\infty \frac{1}{n!} A^n}

If A is diagonalizable, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{e}^A = X \mathrm{e}^D X^{-1}} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{e}^D} is given by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{e}^D = \begin{bmatrix}\mathrm{e}^{\lambda_1} & \dots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \dots & \mathrm{e}^{\lambda_n}\end{bmatrix}}

Example

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = \begin{bmatrix}-2&-6\\1&3\end{bmatrix} = \begin{bmatrix}-2&-3\\1&1\end{bmatrix} \, \begin{bmatrix}1&0\\0&0\end{bmatrix} \, \begin{bmatrix}1&3\\-1&-2\end{bmatrix}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{e}^A = \begin{bmatrix}-2&-3\\1&1\end{bmatrix} \, \begin{bmatrix}\mathrm{e}&0\\0&1\end{bmatrix} \, \begin{bmatrix}1&3\\-1&-2\end{bmatrix} = \begin{bmatrix}3-2\mathrm{e}&6-6\mathrm{e}\\\mathrm{e}-1&3\mathrm{e}-2\end{bmatrix}}

Application: Differential Equation

Solution to differential equation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y' = ay} is $y=\mathrm {e} ^{at}\,y_{0}$.

Similarly, the system of differential equations Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{Y}' = A \vec{Y}} has the solution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{Y} = \mathrm{e}^{tA} \vec{Y}_0} .

Final Exam Review

5 problems and an 11-point bonus problem

• Linear transformations (Null Space, Range, Basis)
• Orthogonality
• Orthogonalization (Gram-Schmidt process, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle QR} factorization)
• Eigenvalues and Eigenvectors
• Least Squares Problems

Note on Factoring Cubics

For a cubic polynomial $p(\lambda )=a\lambda ^{3}+b\lambda ^{2}+c\lambda +d$, if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(m) = 0} for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \in \mathbb{Z}} , $m$ divides $d$:

${\begin{aligned}p(\lambda )&=a\lambda ^{3}+b\lambda ^{2}+c\lambda +d\\&=(\alpha \lambda ^{2}+\beta \lambda +\gamma )(\lambda -m)\end{aligned}}$