1. The Question That Starts Everything
The Central Question
Given a square matrix $A$ (viewed as a linear transformation), which nonzero vectors keep their direction — merely getting stretched, shrunk, or reversed?
When a matrix $A$ multiplies a vector $\mathbf{x}$, the result $A\mathbf{x}$ is usually a completely different vector — rotated, sheared, or otherwise redirected. But some special vectors come out parallel to themselves: the matrix only scales them. These are the eigenvectors, and the scale factor is the eigenvalue.
The Defining Equation
$$A\mathbf{x} = \lambda\mathbf{x}$$- $A$ is an $n \times n$ (square) matrix.
- $\mathbf{x}$ is a nonzero vector — the eigenvector.
- $\lambda$ is a scalar — the eigenvalue.
Why Exclude the Zero Vector?
The zero vector $\mathbf{0}$ satisfies $A\mathbf{0} = \lambda\mathbf{0}$ for every scalar $\lambda$ and every matrix $A$. It tells us nothing about the matrix — it "certifies nothing." That is why the definition insists $\mathbf{x} \neq \mathbf{0}$.
But Zero as an Eigenvalue Is Allowed!
| Concept | Status | Meaning |
|---|---|---|
| $\mathbf{x} = \mathbf{0}$ (zero vector) | Excluded | Trivially satisfies everything; gives no information. |
| $\lambda = 0$ (zero scalar) | Allowed & meaningful | Means $A\mathbf{x} = \mathbf{0}$: the matrix sends certain nonzero vectors to zero. These vectors lie in the kernel (null space) of $A$. |
If $\lambda = 0$ is an eigenvalue, then $A$ is singular (not invertible), because it collapses some nonzero vectors to zero.
2. Geometric Intuition — A Gallery of Plane Transformations
Before computing anything, let's see eigenvectors in familiar transformations. These pictures should become your permanent mental furniture.
$$A = \begin{pmatrix} 3 & 0 \\ 0 & 2 \end{pmatrix}$$
This stretches the $x$-axis by 3 and the $y$-axis by 2.
- $\mathbf{e}_1 = \begin{pmatrix}1\\0\end{pmatrix}$ is an eigenvector with $\lambda = 3$.
- $\mathbf{e}_2 = \begin{pmatrix}0\\1\end{pmatrix}$ is an eigenvector with $\lambda = 2$.
Every vector along an axis stays on that axis — only its length changes.