Engineering Math Series 13: Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors may feel abstract at first, but they are actually the concept that most concisely tells you how a matrix transforms space. Some vectors do not change direction after the transformation -- only their magnitude changes -- and that direction is precisely the eigenvector.

Definition

For a matrix $A$ and a nonzero vector $\mathbf{v}$, if

$$A\mathbf{v} = \lambda \mathbf{v}$$

then $\lambda$ is called an eigenvalue and $\mathbf{v}$ is the corresponding eigenvector.

This means that for certain special directions, the matrix does not rotate but only stretches or shrinks.

Why Are They Important

Eigenvalues reveal the core modes of a system.

- Growth and decay in dynamical systems

- Natural modes in vibration systems

- Principal directions of data

All of these are connected to eigenvalue problems.

How to Find Them

From the defining equation, rearranging

$$A\mathbf{v} = \lambda \mathbf{v}$$

gives

$$ (A - \lambda I)\mathbf{v} = 0 $$

For a nontrivial solution to exist,

$$\det(A - \lambda I) = 0$$

This equation is called the **characteristic equation**.

Worked Example

Consider the matrix

$$

A =

\begin{pmatrix}

2 & 1 \\

0 & 3

\end{pmatrix}

$$

The characteristic equation is

$$

\det

\begin{pmatrix}

2-\lambda & 1 \\

0 & 3-\lambda

\end{pmatrix}

= (2-\lambda)(3-\lambda)=0

$$

so the eigenvalues are

$$\lambda_1 = 2, \quad \lambda_2 = 3$$

For $\lambda = 2$

$$

\begin{pmatrix}

0 & 1 \\

0 & 1

\end{pmatrix}

\begin{pmatrix}

x \\

y

\end{pmatrix}

=

\begin{pmatrix}

0 \\

0

\end{pmatrix}

$$

so $y=0$, and the eigenvector is a multiple of

$$

\mathbf{v}_1 =

\begin{pmatrix}

1 \\

0

\end{pmatrix}

$$

For $\lambda = 3$

$$

\begin{pmatrix}

-1 & 1 \\

0 & 0

\end{pmatrix}

\begin{pmatrix}

x \\

y

\end{pmatrix}

=

\begin{pmatrix}

0 \\

0

\end{pmatrix}

$$

so $y=x$, and the eigenvector is a multiple of

$$

\mathbf{v}_2 =

\begin{pmatrix}

1 \\

1

\end{pmatrix}

$$

Interpretation

This matrix scales certain vectors by 2 and others by 3 while keeping their direction unchanged. A general vector can be viewed as a combination of eigendirections, so the matrix action can be understood much more structurally.

Engineering Applications

Vibration Analysis

The natural frequencies and mode shapes of a structure appear as eigenvalue problems.

Control Systems

The eigenvalues of a linear system matrix determine stability and response speed.

Machine Learning

PCA uses the eigenvalues and eigenvectors of the covariance matrix to find the principal directions of data.

Common Mistakes

Only computing eigenvalues and stopping

You need to compute eigenvectors too in order to understand the actual directional structure of the system.

Assuming every matrix decomposes neatly

Some matrices do not have enough eigenvectors for diagonalization. We will continue this in the next post.

Separating computation from interpretation

Whether eigenvalues are positive or negative, large or small, is directly connected to the meaning of the system.

One-Line Summary

Eigenvalues and eigenvectors are the core concept showing how a matrix stretches and shrinks specific directions.

Next Post Preview

In the next post, we will push the eigenvalue idea one step further and connect it to **diagonalization and dynamic system analysis**.

References

- Erwin Kreyszig, _Advanced Engineering Mathematics_, 10th Edition

- Gilbert Strang, _Linear Algebra for Everyone_

- Carl D. Meyer, _Matrix Analysis and Applied Linear Algebra_