MATH 308 Lecture 29

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Eigenvalues and Eigenvectors

(See MATH 323 Lecture 24#Eigenvalues and Eigenvectors→)

Just to mention it here,

Eigenvalues $\lambda$ are solutions to $\left|A-\lambda \,I\right|=0$
Eigenvectors are nonzero solutions ${\vec {x}}$ to $\left(A-\lambda \,I\right)\,{\vec {x}}=0\rightarrow A\,X=\lambda \,X$

Exercise 5

Find eigenvalues and eigenvectors of $A={\begin{bmatrix}5&-1\\3&1\end{bmatrix}}$

${\begin{aligned}{\begin{vmatrix}5-\lambda &-1\\3&1-\lambda \end{vmatrix}}=5-6\lambda +\lambda ^{2}+3&=0\\\lambda &=2,4\\{\begin{bmatrix}3&-1\\3&-1\end{bmatrix}}\,{\vec {x}}_{1}&={\vec {0}}\\{\begin{bmatrix}-3&1\\0&0\end{bmatrix}}\,{\vec {x}}_{1}&={\vec {0}}\\{\vec {x}}_{1}&=\alpha \left\langle 1,3\right\rangle \\{\begin{bmatrix}1&-1\\3&-3\end{bmatrix}}\,{\vec {x}}_{2}&={\vec {0}}\\{\begin{bmatrix}1&-1\\0&0\end{bmatrix}}\,{\vec {x}}_{2}&={\vec {0}}\\{\vec {x}}_{2}&=\alpha \end{aligned}}$

Find eigenvalues and eigenvectors of $B={\begin{bmatrix}1&0&0\\2&1&2\\3&2&1\end{bmatrix}}$

${\begin{aligned}{\begin{vmatrix}1-\lambda &0&0\\2&1-\lambda &2\\3&2&1-\lambda \end{vmatrix}}\to \lambda &=-1,1,3\\{\vec {x}}_{2}&=\left\langle 0,1,-1\right\rangle \\{\vec {x}}_{1}&=\left\langle 2,-3,-2\right\rangle \\{\vec {x}}_{3}&=\left\langle 0,1,1\right\rangle \\\end{aligned}}$

Solving Systems of Equations

Theorem 7.4.1

Let $X_{1},\ldots ,X_{p}$ be $p$ solutions to the homogeneous system

$X'(t)=A(t)\,X(t)$

on the interval $I$, where $A(t)$ is a continuous $n\times n$ matrix function on $I$,

then for any real numbers $c_{1},\ldots ,cp$, the vector

${\vec {x}}(t)=\sum _{i=1}^{p}c_{i}\,X_{i}$

is a solution to the homogeneous system.

Theorem 7.4.2

Let $X_{1},\ldots ,X_{n}$ be $n$ linearly independent solutions to the homogeneous system

$X'(t)=A(t)\,X(t)$

on the interval $I$, where $A(t)$ is a continuous $n\times n$ matrix function on 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 I} ,

Any 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{x}(t)} can be expressed in the form

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}(t) = \sum_{i=1}^p c_i \, X_i}

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 c_1, \ldots, c_n} are constants.

Theorem 7.4.3

The Wronskian $W(X_{1},\ldots ,X_{n})$ is either 0 on 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 I} or never vanishes on 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 I} .

In other words, 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 W(X_1, \ldots, X_n) = 0} , then the solutions are linearly dependent and do not describe all possible solutions.

Otherwise, we have found all solutions