Eigenvalues and Eigenvectors — Mathematics Knowledge System

Overview

An eigenvector of a linear transformation is a nonzero vector that doesn't change direction when the transformation is applied — it only gets scaled. The scaling factor is the eigenvalue. Eigenvectors reveal the 'natural axes' of a transformation: directions along which its behavior is as simple as multiplication by a single number.

Intuition

Most vectors get pushed AND rotated by a linear transformation. But for any given transformation, there are usually a few special directions where a vector only gets stretched or squished — never rotated. Those special directions are the eigenvectors, and how much they stretch is the eigenvalue. Think of stretching a sheet of rubber: most points move in complicated diagonal paths, but along the axis of stretching, points move in a perfectly straight line, just farther from or closer to the origin.

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Formal Definition

Definition

For a linear transformation represented by matrix A, a nonzero vector v is an eigenvector with eigenvalue λ if:

Av = \lambda v

Applying A to v is the same as scaling v by the scalar λ

Eigenvalue equation

\det(A - \lambda I) = 0

The equation whose roots are the eigenvalues of A

Characteristic equation

Notation

Notation	Meaning
$\lambda$	Eigenvalue (Greek lambda), the scaling factor
$v$	Eigenvector, a nonzero vector satisfying Av = λv
$I$	The identity matrix

Derivation

Deriving the characteristic equation from the eigenvalue equation:

Av = \lambda v \iff Av - \lambda v = 0

Move everything to one side

\iff (A - \lambda I)v = 0

Factor out v, inserting the identity matrix so the subtraction is well-defined

\text{For a nonzero } v \text{ to exist, } (A-\lambda I) \text{ must be singular} \iff \det(A-\lambda I)=0

A matrix has a nontrivial null space exactly when its determinant is zero

Properties

Number of eigenvalues

An n \times n matrix has at most n eigenvalues (counted with multiplicity).

Trace equals sum of eigenvalues

\operatorname{tr}(A) = \sum_i \lambda_i

Determinant equals product of eigenvalues

\det(A) = \prod_i \lambda_i

Symmetric matrices

A real symmetric matrix always has real eigenvalues and orthogonal eigenvectors.

Applications

Principal Component Analysis (PCA) finds the eigenvectors of a data covariance matrix — the directions of maximum variance in the data.

Worked Examples

Set up the characteristic equation det(A - λI) = 0.
$\det\begin{pmatrix}2-\lambda & 1\\ 1 & 2-\lambda\end{pmatrix} = (2-\lambda)^2 - 1 = 0$
Expand and solve the quadratic.
$\lambda^2 - 4\lambda + 3 = 0 \Rightarrow (\lambda-1)(\lambda-3)=0$

Answer: λ = 1 and λ = 3

Practice Problems

Difficulty 6/10

Find the eigenvector of A = [[2,1],[1,2]] corresponding to λ = 3.

Difficulty 7/10

A building's free vibration is modelled by K·φ = ω²·M·φ. For a single-storey frame the effective stiffness is k = 8×10⁶ N/m and the mass is m = 20000 kg. Find the natural frequency ω and the period T (this is the eigenvalue of the 1×1 system).

Difficulty 7/10

In structural stability analysis, critical buckling loads are the eigenvalues of the equation K·φ = λ·K_G·φ (geometric stiffness). Conceptually, what does the SMALLEST eigenvalue represent, and why does it matter to a civil engineer?

Difficulty 5/10

A 2×2 matrix has trace 7 and determinant 12. Find its eigenvalues without forming the characteristic matrix explicitly.

Common Mistakes

Common Mistake

Believing every matrix has a full set of real eigenvalues.

A real matrix can have complex eigenvalues (e.g. rotation matrices), and some matrices are 'defective' — they lack enough independent eigenvectors to diagonalize.

Quiz

Flashcards

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Historical Background

The word 'eigen' is German for 'own' or 'characteristic', introduced by David Hilbert in 1904 in the context of integral equations. The underlying ideas trace back further: Euler studied the principal axes of rotating rigid bodies in the 18th century (a physical eigenvector problem), and Cauchy studied related questions for quadratic forms in the 1820s. The modern linear-algebraic framing solidified in the early 20th century.

18th century
Euler studies principal axes of rotation for rigid bodies
Leonhard Euler
1820s
Cauchy studies diagonalization of quadratic forms
Augustin-Louis Cauchy
1904
Hilbert coins the term 'eigenvalue' (Eigenwert)
David Hilbert

Summary

An eigenvector of A is a nonzero vector v where Av = λv — the direction doesn't change, only the length scales by λ.
Eigenvalues are found by solving the characteristic equation det(A - λI) = 0.
trace(A) = sum of eigenvalues; det(A) = product of eigenvalues.
PCA, quantum mechanics measurement, and vibration analysis are all direct applications.
Symmetric matrices always have real eigenvalues and orthogonal eigenvectors.

References

WebsiteWikipedia — Eigenvalues and eigenvectors