Homogeneous Linear Systems

Homogeneous Linear SystemsExpected Educational ResultsHomogeneous Linear SystemsSystem of Linear DEsSystem of Two Linear DEsSolve for eigenvalues, :Solve for eigenvectors:CC BY-NC-SA 4.0

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

Expected Educational Results

Homogeneous Linear Systems

System of Linear DEs

Given the following homogeneous system of linear DEs:

where is the coefficient matrix of [constant] coefficients of .

Let’s try the functions , where is a vector of arbitrary constants and is a number (it is called the eigenvalue of the coefficient matrix), as solutions to the homogeneous system of linear DEs (see CPT_10c_Characteristic_Equation.html). So,

This simplifies to:

We need to solve for , i.e., root of the characteristic equation. So,

, where

System of Two Linear DEs

Solve for eigenvalues, :

,

The above has a solution only if

The -degree polynomial in below is called the characteristic polynomial.

Solve for and using any valid algebraic method.

Solve for eigenvectors:

Use to solve for eigenvector :

,

Using matrix multiplication [there are other procedures to solve for the components ]:

Solve the above system of two equations and two unknowns for and .

So, the eigenvector that corresponds to the eigenvalue, is any vector for which

or any multiple of this vector.

Similarly, use to solve for eigenvector :

,

Verify and by showing the equivalence:

Similarly, verify and .

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