Homogeneous Linear SystemsExpected Educational ResultsHomogeneous Linear SystemsNormal Form of a System of Linear DEsNormal Form of a Homogenous System of Linear DEsNormal Form of a Homogenous System of Linear DEs with Constant CoefficientsSolutions to Systems of ODEsComplex EigenvaluesInvestigation 02Investigation 03CC BY-NC-SA 4.0

**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

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

where ,

, and

is the coefficient matrix.

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

otherwise, the system is nonhomogenous.

This is the above equation where all elements, , of the coefficient matrix are all constants.

If is a coefficient matrix of a system of linear DEs, then is an degree polynomial in , called the **characteristic polynomial**.

Let be the complex conjugate eigenvalues for the characteristic polynomial corresponding to the complex eigenvector . Then the solution to the system of DEs, is

where is the real part of the solution and is the imaginary part of the solution.

**NOTE**: Only one eigenvalue and its corresponding eigenvector is needed.

**Example 04**:

Solve the following system of DEs.

**Solution**:

Rewrite system as:

**Using technology**, the eigenvalues of the coefficient matrix arex1`(* Store the matrix [2,1; -1,2] *)`

2`A = {{2, 1}, {-1, 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]`

**Using technology**, the eigenvectors of the coefficient matrix are:- For ,
- For ,

Find the fundamental solution set, expand using Euler's formula (See Question 1 in CPT_11_Complex_Solutions.pdf.):

The solution to the system: which is equivalent to

For the above example, use the other eigenvalue and corresponding eigenvector and verify the equivalence of the solutions.

Use matrix operations to solve the following systems of DEs:

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