Eigenvectors and Eigenvalues

Multiplication by a square matrix $A$ moves most vectors in some new direction. For a few special nonzero vectors the direction does not change at all: $A$ only lengthens or shortens them, possibly reversing their sign. These are the eigenvectors of $A$ .

Two asymmetries in the definition matter. An eigenvector must be nonzero, since $A 0 = λ 0$ holds for every $λ$ and would make the notion vacuous. An eigenvalue, on the other hand, is allowed to be $0$ ; the number $0$ is an eigenvalue of $A$ exactly when $A$ has a nontrivial null space.

The eigenvector

v

keeps its direction under

A

(here

A v = 2 v

); the ordinary vector

u

is rotated onto a new line.

Testing a vector and testing a scalar

Deciding whether a given vector is an eigenvector, or whether a given number is an eigenvalue, needs no special theory. Both reduce to a single matrix product or a single homogeneous system.

Is $x$ an eigenvector? Compute $A x$ and check whether it is a scalar multiple of $x$ .
Is $λ$ an eigenvalue? Check whether $(A - λ I) x = 0$ has a nontrivial solution, since $A x = λ x$ is equivalent to $(A - λ I) x = 0$ . This holds precisely when $A - λ I$ is singular.

The eigenspace

The set of solutions of $(A - λ I) x = 0$ is the null space of $A - λ I$ , a subspace of $R^{n}$ . When $λ$ is an eigenvalue this subspace has more than just the zero vector in it.

For the matrix $A = [1562]$ , the eigenspace for $λ = 7$ is the line through $(1, 1)$ , and the eigenspace for $λ = - 4$ is the line through $(6, - 5)$ . On the first line $A$ acts as multiplication by $7$ ; on the second, as multiplication by $- 4$ . Each eigenspace is an invariant line: $A$ maps it into itself.

The two eigenspaces of

A

; on each line

A

acts as pure scaling, by

7

on the line through

(1, 1)

and by

- 4

on the line through

(6, - 5)

Eigenspaces can be more than one-dimensional.

Eigenvalues of a triangular matrix

For most matrices, finding eigenvalues means solving the characteristic equation, a polynomial equation in $λ$ . Triangular matrices are the one large family where the answer is immediate.

The reason is transparent from the structure of $A - λ I$ . If $A$ is upper triangular, then

A - λ I = a_{11} - λ 00 a_{12} a_{22} - λ 0 a_{13} a_{23} a_{33} - λ,

and this matrix has a free variable (so $(A - λ I) x = 0$ has a nontrivial solution) exactly when one of the diagonal entries $a_{ii} - λ$ is zero, that is, when $λ$ equals one of $a_{11}, a_{22}, a_{33}$ .

A - λ I

for a triangular matrix, a zero appears on the diagonal precisely when

λ

equals a diagonal entry, producing a free variable.

For example, the matrices

A = 300600 - 8 62, B = 4 - 2 5 013004

have eigenvalues ${3, 0, 2}$ and ${4, 1}$ respectively, read straight off their diagonals. Note that $A$ has $0$ as an eigenvalue, consistent with $A$ being singular.

Independence of eigenvectors

Eigenvectors belonging to different eigenvalues cannot combine trivially, so distinct eigenvalues supply independent directions.

Three eigenvectors for three distinct eigenvalues point along three different invariant lines; no one of them lies in the span of the others.

A useful consequence: an $n \times n$ matrix has at most $n$ distinct eigenvalues, because more than $n$ eigenvectors for distinct eigenvalues would be an independent set of more than $n$ vectors in $R^{n}$ , which is impossible.

Eigenvectors and difference equations

The reason eigenvalues govern long-term behavior appears already in a first-order difference equation

x_{k + 1} = A x_{k} (k = 0, 1, 2, \dots) .

Here ${x_{k}}$ is a sequence of vectors in $R^{n}$ produced by applying $A$ repeatedly. A solution is an explicit formula for $x_{k}$ that does not require computing the earlier terms. If $x_{0}$ happens to be an eigenvector with eigenvalue $λ$ , one solution is immediate:

x_{k} = λ^{k} x_{0},

since $A x_{k} = A (λ^{k} x_{0}) = λ^{k} (A x_{0}) = λ^{k + 1} x_{0} = x_{k + 1}$ . The dynamics along an eigenspace are pure geometric growth or decay at rate $λ$ . Linear combinations of such solutions are again solutions, so once $x_{0}$ is written in an eigenvector basis the whole trajectory decouples into independent scalar sequences. The same construction underlies difference equations and Markov chains and the classification of dynamical systems.

Summary

Question	Reduces to	Tool
Is $x$ an eigenvector?	Is $A x$ a multiple of $x$ ?	one matrix product
Is $λ$ an eigenvalue?	Is $A - λ I$ singular?	test $Nul (A - λ I) \neq = {0}$
Find eigenvectors for known $λ$	Solve $(A - λ I) x = 0$	row reduction
Eigenvalues of a triangular matrix	Read the diagonal	none
Eigenvalue $0$ present?	Is $A$ singular?	invertibility test

Finding the eigenvalues of a general (non-triangular) matrix requires turning $det (A - λ I) = 0$ into a polynomial equation, the characteristic equation.¹

Lay, Linear Algebra and Its Applications, 5th ed., §5.1 — Eigenvectors and Eigenvalues: the definitions, the eigenspace as the null space of $A - λ I$ , Theorem 1 on triangular matrices, and Theorem 2 on independence of eigenvectors for distinct eigenvalues. ↩