Chapter 7 · Introduction

Chapter 7 — Eigenvalues and Eigenvectors

Most vectors are knocked off their line by a matrix — but a few special directions are only stretched. Those directions, and their stretch factors, explain long-run behaviour everywhere from web search to bridges and wildlife.

Apply a matrix to a vector and the result usually points somewhere new: the matrix rotates it, shears it and stretches it all at once. Yet almost every square matrix has a few special directions that it does not turn at all. A vector pointing along one of them comes out on the same line, simply multiplied by a number. These directions are the eigenvectors of the matrix (from the German eigen, “own”), and the multipliers are its eigenvalues. Once you know them, a complicated matrix becomes as easy to understand as a list of stretch factors.

That simple idea runs a surprising amount of the modern world. Google's original PageRank ranked web pages by finding one eigenvector of an enormous “who links to whom” matrix. Engineers designing a footbridge or a tall building compute the eigenvalues of its stiffness matrix to find the natural frequencies at which it vibrates — and keep footsteps, wind and earthquakes away from them. Ecologists predict whether a population of turtles or fish will grow or die out from the largest eigenvalue of an age-structured population matrix, and the matching eigenvector tells them the long-run mix of young and old animals. In this chapter you will learn to find eigenvalues and eigenvectors, to use them to rewrite a matrix in its simplest possible form, and to predict the long-term behaviour of systems that change step by step.

$A\mathbf{x}=\lambda\mathbf{x}$→Characteristic equation→Eigenspaces & multiplicity→Diagonalisation $A=PDP^{-1}$→Powers $A^k$→Dynamical systems & Markov chains→Power method & $\mathbf{x}'=A\mathbf{x}$
🎯 By the end of this chapter you can…
  • check whether a given vector is an eigenvector of a matrix and explain what its eigenvalue means geometrically;
  • find eigenvalues from the characteristic equation $\det(A-\lambda I)=0$ and a basis for each eigenspace $E_\lambda$;
  • compare algebraic and geometric multiplicity, and use the links between eigenvalues, the trace, the determinant, powers and inverses;
  • decide whether a matrix is diagonalisable, and when it is, write $A=PDP^{-1}$ and compute $A^k$ quickly;
  • interpret diagonalisation as a change to a basis of eigenvectors in which the matrix only stretches;
  • predict the long-run behaviour of $\mathbf{x}_{k+1}=A\mathbf{x}_k$ (population models, Markov chains) from the dominant eigenvalue, and estimate it with the power method;
  • solve systems of linear differential equations $\mathbf{x}'=A\mathbf{x}$ with eigenvectors (uses calculus).
Teaser — Which Directions Does the Matrix Only Stretch?

Twenty-four unit vectors $\mathbf{x}$ (faint) are moved smoothly to their images $A\mathbf{x}$. Most arrows swing round as they travel. The red arrows never leave their own dashed line — they are only stretched, shrunk or flipped. Those are the eigen-directions, and the stretch factor along each is an eigenvalue.

morph $t$ (0 = $\mathbf{x}$, 1 = $A\mathbf{x}$)—
eigen-directions found—
eigenvalues $\lambda$—
largest swing of any arrow—

Sections in this chapter

7.1

Eigenvalues and Eigenvectors

Directions a matrix only stretches; the characteristic equation; eigenspaces, multiplicities, and links with the trace, determinant, powers and inverses.

$A\mathbf{x}=\lambda\mathbf{x},\ \ \det(A-\lambda I)=0$
Open section →7.2

Diagonalisation

When a matrix has enough independent eigenvectors it factors as $PDP^{-1}$ — pure stretching in the eigenvector basis — and its powers become easy.

$A^k=PD^kP^{-1}$
Open section →7.3

Applications of Eigenvalues

Population and predator–prey models, Markov steady states, the power method, and systems of differential equations.

$\mathbf{x}_k=c_1\lambda_1^k\mathbf{v}_1+\dots+c_n\lambda_n^k\mathbf{v}_n$
Open section →
✅ Before you start
Eigenvalues come from determinants, so be fluent with cofactor expansion and the properties of determinants — especially “$\det M=0$ exactly when $M$ is not invertible”. Eigenvectors are found by solving homogeneous systems, so revise Gaussian elimination and null spaces. Section 7.2 uses matrix inverses, linear independence, bases, coordinates and change of basis and the matrix of a linear transformation relative to a basis; Section 7.3 builds on Markov chains. Only the last part of 7.3 uses calculus (the derivative of $e^{\lambda t}$ — see Calculus II §7.3).
← 6.4 The Geometry of Linear Maps in the Plane7.1 Eigenvalues and Eigenvectors →