Linear Algebra: Eigenvalues

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Eigenvector Definition

When a linear transformation reshapes space, most vectors twist and turn, changing both their magnitude and direction. Yet, a select few vectors possess a remarkable property: they maintain their original direction, only stretching or shrinking. These special vectors are called eigenvectors, and their behavior reveals fundamental characteristics of the transformation itself.

The Core Idea: Direction Unchanged

A linear transformation (represented by a matrix $A$ ) maps vectors from one space to another. For a generic vector $v$ , the transformed vector $Av$ will typically point in a different direction than $v$ . An eigenvector, however, is a non-zero vector $v$ such that when $A$ acts upon it, the resulting vector $Av$ is simply a scalar multiple of $v$ . This means $Av$ points in the exact same direction as $v$ , or in the exact opposite direction if the scalar is negative.

Vectors Under Linear Transformation

This plot shows how a matrix

A = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}

transforms the standard basis vectors

e_1, e_2

and a specific eigenvector

v_1

. Notice that

e_1

and

e_2

change direction, but

v_1

only scales.

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Key Insight: Eigenvectors like

v_1

are special because their direction remains unchanged after transformation, only their magnitude is affected.

The Eigenvalue Equation: $Av = \lambda v$

The defining characteristic of an eigenvector is captured by a simple yet powerful equation. For a square matrix $A$ , a non-zero vector $v$ is an eigenvector if there exists a scalar $\lambda$ (lambda) such that the product of $A$ and $v$ is equal to the product of $\lambda$ and $v$ . This scalar $\lambda$ is known as the eigenvalue corresponding to the eigenvector $v$ .

📐 The Eigenvalue Equation

The fundamental relationship between a matrix $A$ , its eigenvector $v$ , and its eigenvalue $\lambda$ is:

Av = \lambda v

pythonVerifying an Eigenvector and Eigenvalue

Check Your Understanding

What does a negative eigenvalue imply about the eigenvector's direction after transformation?

The Role of the Eigenvalue ( $\lambda$ )

The eigenvalue $\lambda$ quantifies the scaling effect of the transformation on its corresponding eigenvector. If $\lambda > 1$ , the eigenvector is stretched; if $0 < \lambda < 1$ , it's shrunk. A negative $\lambda$ means the eigenvector's direction is reversed (scaled in the opposite direction). If $\lambda = 0$ , the eigenvector is mapped to the zero vector, meaning it lies in the null space (kernel) of the transformation.

pythonEigenvector with a Negative Eigenvalue

The Null Space Connection: $(A - \lambda I)v = 0$

To find eigenvectors and eigenvalues, we rearrange the eigenvalue equation $Av = \lambda v$ . By subtracting $\lambda v$ from both sides, we get $Av - \lambda v = 0$ . Since $\lambda v$ can be written as $\lambda I v$ (where $I$ is the identity matrix), we can factor out $v$ to obtain $(A - \lambda I)v = 0$ . This form is critical because it tells us that $v$ must be a non-zero vector in the null space (or kernel) of the matrix $(A - \lambda I)$ .

💡 Condition for Non-Trivial Solutions

For a non-zero vector $v$ to exist in the null space of $(A - \lambda I)$ , the matrix $(A - \lambda I)$ must be singular (non-invertible). This implies its determinant must be zero:

\det(A - \lambda I) = 0

Properties of Eigenvectors

Eigenvectors have a few important properties. First, by definition, an eigenvector must be a non-zero vector. The zero vector $0$ would always satisfy $A0 = \lambda 0$ , but it provides no information about direction. Second, if $v$ is an eigenvector, then any non-zero scalar multiple of $v$ (e.g., $2v$ , $-5v$ ) is also an eigenvector corresponding to the same eigenvalue $\lambda$ . This means eigenvectors are not unique in their exact values, but rather represent a direction or a 'line' in space.

pythonScalar Multiples are also Eigenvectors

Check Your Understanding

Why is the zero vector explicitly excluded from being an eigenvector?

Conceptual Flow: Identifying an Eigenvector

Process for Identifying an Eigenvector

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This diagram illustrates the conceptual steps to determine if a given vector is an eigenvector for a specific linear transformation.

Key Takeaways

An eigenvector is a non-zero vector whose direction remains unchanged after a linear transformation, only its magnitude is scaled.
The eigenvalue equation is $Av = \lambda v$ , where $A$ is the matrix, $v$ is the eigenvector, and $\lambda$ is the eigenvalue.
The eigenvalue $\lambda$ is the scalar factor by which the eigenvector is scaled; a negative $\lambda$ indicates direction reversal.
Eigenvectors are found by solving $(A - \lambda I)v = 0$ , meaning $v$ lies in the null space of $(A - \lambda I)$ .
For non-trivial eigenvectors to exist, the determinant $\det(A - \lambda I)$ must be zero.
Any non-zero scalar multiple of an eigenvector is also an eigenvector for the same eigenvalue, representing the same invariant direction.

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Eigenvector Definition

The Core Idea: Direction Unchanged

The Eigenvalue Equation: Av=λvAv = \lambda vAv=λv

The Role of the Eigenvalue (λ\lambdaλ)

The Null Space Connection: (A−λI)v=0(A - \lambda I)v = 0(A−λI)v=0

Properties of Eigenvectors

Conceptual Flow: Identifying an Eigenvector

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The Eigenvalue Equation: $Av = \lambda v$

The Role of the Eigenvalue ( $\lambda$ )

The Null Space Connection: $(A - \lambda I)v = 0$