Can someone help me understand eigenvalues?

Question

This question has two parts.

1. What does an eigenvalue tell you about a matrix from a pure math standpoint?
2. Now, if the matrix represented some set of data, lets say pressure data, what do the eigenvalues tell us, and can you relate the pure mathematical definition of eigenvalues back to this application-based eigenvalue?

Sorry if this is a loaded question, I just constantly see eigenvalues and eigenvectors being used in my field of study and I feel like I have very little understanding of what this mathematical tool represents.

Answer 1

I can only try to answer 1.

"From a pure math standpoint", matrix calculus is only one computational aspect of the study of linear maps; before returning to the matrices, I therefore propose to consider an example of a simple linear map : $s:\Bbb R^2 \to \Bbb R^2, (x,y) \mapsto (2y,2x)$. To understand what $s$ is, especially do not hesitate to place points $(x,y)$ in an orthonormal reference frame for example and their images. $s$ should then appear to you as the symmetry with respect to the line $d$ of equation $y=x$ composed with the homothety of center $0$ and ratio $2$. On $d$, vectors are just multiplied by $2$ and on the perpendicular to $d$ passing through $0$, through $-2$ : here are the eigenvalues.

The matrix of $s$ in the canonical basis is $\begin{bmatrix}0 & 2 \\2 & 0\end{bmatrix}$. $p(\lambda)=\begin{vmatrix}-\lambda & 2\\2 & -\lambda\end{vmatrix}=\lambda^2-4=(\lambda-2)(\lambda+2)=0 \iff \lambda \in \{2,-2\}$.

Here are the eigenvalues.

I hope this will help.

Answer 2

Let's start with eigenvectors.

For a linear transformation, there will be vectors whose direction will not be changed by that transformation. Those are eigenvectors.

While the direction of eigenvectors might not change, the magnitude does. The eigenvalue is the magnitude of that change.

What is the relevance of eigenvalues from a pure math point of view? For one, all matrices with the same eigenvalues are similar. Finding the eigenvalues of a matrix allows us to treat our matrix as something similar to a diagonal matrix. And diagonal matrices are really easy to work with.

What are some of the applications?

Quadratic forms -- A quadratic form takes on a few basic shapes, (parabola, hyperbola, eclipse) with higher-dimensional analogs. The eigenvalues tell you what basic shape you have. differential equations.

System of differential equations -- the solution to the differential equation will often be some combination of trigonometric or exponential functions. The eigenvalues determine those functions.

Principal component analysis -- The eigenvalues show the size/sensitivity of these components.

In a Markov process, a transformation is executed repeatedly. Eigenvalues with absolute values less than one disappear to insignificance. The largest eigenvalue will come to dominate the process.

This is really just scratching the surface, but I hope give you some idea where this line of study is going.