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**:

**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.

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

,

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.

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 .

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