Homogeneous Linear Systems

Author: John J Weber III, PhD Corresponding Textbook Sections:

• Section 5.4 – Differential Operators and the Elimination Method for Systems
• Section 9.4 -- Linear Systems in Nominal Form
• Section 9.5 -- Homogeneous Linear Systems with Constant Coefficients
• Section 9.6 -- Complex Eigenvalues

Expected Educational Results

• Objective 16–1: I can homogeneous linear systems using eigenvalues and eigenvectors.

Homogeneous Linear Systems

Normal Form of a System of $n$ Linear DEs

$\pmb{x}^{\,\prime}(t)=\pmb{A}(t)\pmb{x}(t)+\pmb{f}(t)$

where $\displaystyle \pmb{x}(t)=\left[ \begin{array}{c} x_1(t) \\ x_2(t) \\ \vdots \\ x_n(t) \end{array} \right]$,

$\displaystyle \pmb{f}(t)=\left[ \begin{array}{c} f_1(t) \\ f_2(t) \\ \vdots \\ f_n(t) \end{array} \right]$, and

$\displaystyle \pmb{A}(t)=\left[a_{ij}\right]$ is the $n\times n$ coefficient matrix.

NOTE: Lowercase variables in boldface are vectors; uppercase variables in boldface are matrices.

Normal Form of a Homogenous System of $n$ Linear DEs

$\pmb{x}^{\,\prime}(t)=\pmb{A}(t)\pmb{x}(t)+\pmb{0}$

otherwise, the system is nonhomogenous.

Normal Form of a Homogenous System of $n$ Linear DEs with Constant Coefficients

This is the above equation where all elements, $a_{ij}$, of the coefficient matrix $\pmb{A}$ are all constants.

Solutions to Systems of ODEs

If $\pmb{A}$ is a $n \times n$ coefficient matrix of a system of $n$ linear DEs, then $\displaystyle \text{det}\left(\pmb{A}\pmb{x}-\lambda \pmb{I}\right)=\pmb{0}$ is an $\displaystyle n^{\text{th}}$ degree polynomial in $\lambda$, called the characteristic polynomial.

Repeated Real Eigenvalues

Case 1: Repeated Real Eigenvalues with Linearly Independent Eigenvectors

Example 02:

Solve the following system of DEs.

$\begin{array}{ll} x^{\,\prime}_1 &= 3x_1-x_2+2x_3 \\ x^{\,\prime}_2 &= 3x_1-x_2+6x_3 \\ x^{\,\prime}_3 &= -2x_1+2x_2-2x_3 \end{array}$

Solution:

1. Rewrite system as: $\pmb{x}^{\,\prime}=\left[\begin{array}{ccc} 3 & -1 & 2 \\ 3 & -1 & 6 \\ -2 & 2 & -2 \end{array}\right]\pmb{x}$

2. Using technology, find the eigenvalues and corresponding eigenvectors of the coefficient matrix:

   ​x
   1
   (* Store the matrix [3, -1, 3; 3, -1, 6; -2, 2, -2] *)
   2
   A = {{3, -1, 3}, {3, -1, 6}, {-2, 2, -2}};
   3
   ​
   4
   (* Display matrix *)
   5
   (* To check that we entered the matrix correctly *)
   6
   MatrixForm[A]
   7
   ​
   8
   (* Method 1 *)
   9
   (* Compute eigenvalues of matrix A *)
   10
   Eigenvalues[A]
   11
   ​
   12
   (* Compute eigenvectors of matrix A *)
   13
   Eigenvectors[A]
   14
   ​
   15
   (* Method 2 *)
   16
   (* Compute eigenvalues and eigenvectors *)
   17
   Eigensystem[A]
   

   Warnings:

   1. Be very careful with the syntax. Syntax is the set of rules on how to write computer code. Every software program has its own unique syntax. Some basic Mathematica syntax is located at: http://www.jjw3.com/TECH_Common_Functions.pdf.
   2. To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.
   3. The eigenvector and corresponding eigenvalues will be returned in order.

3. Using technology, the eigenvectors of the coefficient matrix are:

   • For $\displaystyle \lambda_1=-4$, $\displaystyle \pmb{v}_1=\left[\begin{array}{c} -1 \\ -3 \\ 2 \end{array}\right]$
   • For $\displaystyle \lambda_2=2$, $\displaystyle \pmb{v}_2=\left[\begin{array}{c} -2 \\ 0 \\ 1 \end{array}\right]$
   • For $\displaystyle \lambda_2=2$, $\displaystyle \pmb{v}_3=\left[\begin{array}{c} 1 \\ 1 \\ 0 \end{array}\right]$

4. The solution to the systems of DEs is $\displaystyle \pmb{x}(t)=c_1e^{-4t}\left[\begin{array}{c} -1 \\ -3 \\ 2 \end{array}\right]+e^{2t}\left(c_2\left[\begin{array}{c} -2 \\ 0 \\ 1 \end{array}\right]+c_3\left[\begin{array}{c} 1 \\ 1 \\ 0 \end{array}\right]\right)$ which is equivalent to

   $\displaystyle \begin{array}{l} x_1(t)=-c_1e^{-4t}+\left(-2c_2+c_3\right)e^{2t} \\ x_2(t)=-3c_1e^{-4t}+c_3e^{2t} \\ x_3(t)=2c_1e^{-4t}+c_2e^{2t}\end{array}$

Case 2: Repeated Real Eigenvalues with Linearly Dependent Eigenvectors

Example 03:

Solve the following system of DEs:

$\begin{array}{ll} x^{\,\prime}_1 &= 10x_1-2x_2 \\ x^{\,\prime}_2 &= 18x_1-2x_2 \end{array}$

Solution

1. Rewrite system as: $\pmb{x}^{\,\prime}=\left[\begin{array}{ccc} 10 & -2 \\ 18 & -2 \end{array}\right]\pmb{x}$

2. Using technology, the eigenvalues of the coefficient matrix are $\displaystyle \lambda=4,4$

3. Using technology, the [linearly dependent] eigenvectors of the coefficient matrix are:

   • For $\displaystyle \lambda_1=4$, $\displaystyle \pmb{v}_1=\left[\begin{array}{c} 1 \\ 3 \end{array}\right]$
   • For $\displaystyle \lambda_2=4$, $\displaystyle \pmb{v}_2=\left[\begin{array}{c} 0 \\ 0 \end{array}\right]$

4. The solution to the systems of DEs is which is equivalent to

   $\displaystyle \begin{array}{l} x_1(t)=c_1e^{4t}+c_2te^{4t}+Ac_2e^{4t} \\ x_2(t)=3c_1e^{4t}+3c_2te^{4t}+Bc_2e^{4t} \end{array}$

   Substitute the above solution into the system of DEs to solve for $A$ and $B$.

Investigation 04

Use matrix operations to solve the following systems of DEs:

1. $\displaystyle \begin{array}{ll} x_1^{\,\prime} &= -2x_1+x_2 \\ x_2^{\,\prime} &= -x_1 \end{array}$

2. ---

   $\displaystyle \pmb{x}^{\,\prime}=\left[ \begin{array}{cc} 2 & 1 \\ 2 & 0 \end{array} \right]\pmb{x}$

CC BY-NC-SA 4.0

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License [http://creativecommons.org/licenses/by-nc-sa/4.0/].

Last Modified: Monday, 19 October 2020 8:23 EDT