Definition of Eigenvalues and Eigenvectors

When matrices represent linear transformations, they change vectors by rotating, scaling, or shearing them. A fascinating aspect of these transformations involves certain vectors that, when transformed by a specific matrix, do not change their direction. These vectors might get longer, shorter, or even reverse direction, but they consistently remain aligned along their original line. Such distinct vectors and their associated scaling factors constitute primary properties of the matrix itself.

Let's formalize this. For a given square matrix $A$ of size $n \times n$, we are looking for non-zero vectors $x$ and scalars $\lambda$ that satisfy the following equation:

$Ax = \lambda x$

This equation is the heart of the matter. Let's break down its components:

• $A$: An $n \times n$ square matrix. It represents the linear transformation we're analyzing. Eigenvalues and eigenvectors are defined for square matrices because the transformation maps vectors from $\mathbb{R}^n$ back to $\mathbb{R}^n$, allowing the output $Ax$ to be potentially parallel to the input $x$.
• $x$: A non-zero vector in $\mathbb{R}^n$. This is called an eigenvector of the matrix $A$. It's the vector whose direction is preserved (up to a sign change) by the transformation $A$. We require $x$ to be non-zero because if $x$ were the zero vector, the equation $A0 = \lambda 0$ would hold true ($0=0$) for any value of $\lambda$. This trivial case doesn't tell us anything specific about the matrix $A$ or its characteristic directions.
• $\lambda$: A scalar (a real or complex number). This is called the eigenvalue associated with the eigenvector $x$. It represents the factor by which the eigenvector $x$ is scaled when transformed by $A$.
  • If $\lambda > 1$, the eigenvector is stretched.
  • If $0 < \lambda < 1$, the eigenvector is shrunk.
  • If $\lambda < 0$, the eigenvector's direction is reversed, and its magnitude is scaled by $|\lambda|$.
  • If $\lambda = 1$, the eigenvector remains unchanged by the transformation.
  • If $\lambda = 0$, the eigenvector is mapped to the zero vector (meaning it lies in the null space of $A$).

In simple terms, applying the matrix $A$ to its eigenvector $x$ has the same effect as just scaling $x$ by the eigenvalue $\lambda$. The matrix multiplication $Ax$ results in a vector that is parallel to the original vector $x$.

The word "eigen" comes from German and means "own" or "characteristic". Eigenvalues and eigenvectors are indeed characteristic properties intrinsic to the matrix $A$, revealing fundamental aspects of the transformation it represents.

An eigenvalue $\lambda$ and its corresponding eigenvector $x$ form an eigenpair $(\lambda, x)$. A matrix can have multiple distinct eigenvalues, and each eigenvalue can be associated with one or more linearly independent eigenvectors. However, a single non-zero eigenvector $x$ corresponds to exactly one eigenvalue.

Consider a simple example. Let $A$ be the identity matrix $I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$. Applying $A$ to any vector $x = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}$ gives:

$Ax = Ix = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = 1x$

Here, $Ax = 1x$. This fits the definition $Ax = \lambda x$ with $\lambda = 1$. This means that for the identity matrix, every non-zero vector is an eigenvector with an eigenvalue of 1. This makes sense geometrically: the identity transformation doesn't change any vector.

Now consider a scaling matrix $B = \begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix}$. Applying $B$ to any vector $x$:

$Bx = \begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \begin{pmatrix} 3x_1 \\ 3x_2 \end{pmatrix} = 3 \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = 3x$

In this case, $Bx = 3x$. Again, every non-zero vector is an eigenvector, but this time the eigenvalue is $\lambda = 3$. The transformation uniformly scales every vector by a factor of 3, preserving all directions.

These examples are straightforward. For most matrices, only specific directions (the eigenvectors) are preserved, and finding them requires more systematic methods. The next sections will explore the geometric meaning of eigenvectors more deeply and introduce the standard techniques for calculating eigenvalues and eigenvectors, starting with the characteristic equation.