Eigenvectors & Eigenspaces — Computing Eigenvectors and Eigenspaces - Linear Algebra

SECTION 1

Historical Context & Motivation

Imagine you push a shopping cart. Usually it rolls in the direction you push, but sometimes a wobbly wheel makes it drift sideways. In mathematics, when you apply a matrix transformation (a set of rules that moves every point in a plane), most arrows change both length and direction. However, a few special arrows only get stretched or shrunk — they keep pointing the same way. Those special arrows are called eigenvectors, and finding them turns out to be one of the most powerful ideas in all of mathematics.

The word "eigen" comes from the German word meaning "own" or "characteristic." Over several centuries, mathematicians discovered that these special directions reveal deep truths about systems — from vibrating bridges to search engines.

1743 • Euler Studies Rotation

Leonhard Euler studied rotating bodies and noticed that every rotation has an axis that stays fixed — an early hint of the eigenvector concept.

1826 • Cauchy Formalizes the Idea

Augustin-Louis Cauchy introduced the formal theory of eigenvalues for symmetric matrices while studying stress in solid objects.

1904 • Hilbert Extends to Infinite Dimensions

David Hilbert generalized eigenvalue theory, paving the way for quantum mechanics and modern physics.

1998 • Google PageRank

Larry Page and Sergey Brin used eigenvectors of a giant matrix to rank web pages, creating the Google search engine.

The central question this lesson answers is: given a square matrix, how do we actually compute its eigenvectors and organize them into eigenspaces? Let's find out step by step.

SECTION 2

Core Principles & Definitions

Before we compute anything, let's nail down the key vocabulary. Every definition below builds on the one before it, so take them in order.

Eigenvector

A nonzero vector v such that when a matrix A multiplies it, the result is just a scalar (number) times v. In symbols: Av = λv. The direction doesn't change.

Eigenvalue (λ)

The scalar λ (the Greek letter lambda) that tells you how much the eigenvector gets stretched or shrunk. If λ = 2, the vector doubles in length. If λ = −1, it flips direction.

Characteristic Equation

The equation det(A − λI) = 0, where I is the identity matrix. Solving this polynomial gives you all the eigenvalues of A.

Null Space

The set of all vectors that a matrix sends to the zero vector. To find eigenvectors for a given λ, we find the null space of (A − λI).

Eigenspace

For each eigenvalue λ, the eigenspace is the set of all eigenvectors (plus the zero vector) sharing that eigenvalue. It forms a line, plane, or higher-dimensional flat.

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KEY TAKEAWAY

Think of a matrix as a machine that pushes arrows around. Most arrows get both rotated and stretched. An eigenvector is like a train on a straight track — the machine can speed it up or slow it down, but it can't push it off the track. The eigenvalue is the speed multiplier, and the whole track (every possible length of that train) is the eigenspace.

SECTION 3

Visual Explanation

The diagram below shows what a 2×2 matrix does to several vectors. Most arrows get both rotated and stretched. The two colored arrows along the eigenvector directions only change in length — they stay on the same line through the origin.

The cyan arrow v₁ and the pink arrow v₂ are eigenvectors — after the matrix acts, they stay on the same line through the origin. The dashed arrows are ordinary vectors that get both rotated and stretched.

Notice that the eigenvectors in the "after" panel lie on exactly the same line as in the "before" panel. They may be longer, shorter, or even flipped backward, but the line through the origin doesn't change. Every scalar multiple of an eigenvector is also an eigenvector (for the same eigenvalue), and together they form the eigenspace — a whole line (or plane) of vectors that the matrix can only scale, never rotate.

SECTION 4

Mathematical Framework

Here is the step-by-step recipe for computing eigenvectors and eigenspaces of a square matrix A. We will use a 2×2 matrix, but the same process works for 3×3 and beyond.

Step 1 — Set Up the Eigenvalue Equation

EIGENVALUE EQUATION

Av = λv

A = the matrix, v = a nonzero vector, λ = a scalar (the eigenvalue). We want to find both λ and v.

Step 2 — Rearrange to (A − λI)v = 0

REARRANGED FORM

(A − λI)v = 0

I is the identity matrix (1's on the diagonal, 0's elsewhere). This form tells us that v is in the null space of the matrix (A − λI).

Step 3 — Find Eigenvalues with the Characteristic Equation

CHARACTERISTIC EQUATION

det(A − λI) = 0

"det" means determinant. For a 2×2 matrix [[a, b], [c, d]], the determinant is ad − bc. Solve this equation for λ to get the eigenvalues.

Step 4 — Find Eigenvectors by Solving (A − λI)v = 0

EIGENSPACE

E_λ = Null(A − λI) = { v ∈ Rⁿ : (A − λI)v = 0 }

For each eigenvalue λ, plug it back in, form the matrix (A − λI), and find all vectors v that satisfy the equation. This set of vectors (plus the zero vector) is the eigenspace for that eigenvalue.

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Important Note

The zero vector (0, 0) is always a solution to (A − λI)v = 0, but we do not count it as an eigenvector. Eigenvectors must be nonzero. However, the zero vector is included in the eigenspace because an eigenspace must be a subspace, which always contains the zero vector.

SECTION 5

Eigenspace Structure & Dimension

An eigenspace can be a line (1-dimensional), a plane (2-dimensional), or even higher. The dimension of the eigenspace is called the geometric multiplicity of the eigenvalue. Meanwhile, how many times the eigenvalue appears as a root of the characteristic equation is called the algebraic multiplicity. The geometric multiplicity is always less than or equal to the algebraic multiplicity.

Left: when there is one independent eigenvector, the eigenspace is a line. Right: when there are two independent eigenvectors sharing the same eigenvalue, the eigenspace is a plane. Every vector in the shaded region is an eigenvector.

Property	Algebraic Multiplicity	Geometric Multiplicity
What it counts	Times λ appears as a root of det(A − λI) = 0	Number of independent eigenvectors for λ
Also equals	Exponent of (λ − λᵢ) in the characteristic polynomial	Dimension of the eigenspace E_λ
Rule	Always ≥ 1	1 ≤ geometric ≤ algebraic

Algebraic vs. Geometric Multiplicity

SECTION 6

Worked Example

Let's walk through a complete example. We will find the eigenvalues, eigenvectors, and eigenspaces of the matrix A = [[4, 1], [2, 3]].

Find All Eigenvectors and Eigenspaces of A = [[4, 1], [2, 3]]

1
Step 1 — Form (A − λI)

Subtract λ from each diagonal entry of A. We get: A − λI = [[4 − λ, 1], [2, 3 − λ]].
2
Step 2 — Compute the Determinant

det(A − λI) = (4 − λ)(3 − λ) − (1)(2) = 12 − 4λ − 3λ + λ² − 2 = λ² − 7λ + 10.

Characteristic polynomial: λ² − 7λ + 10
3
Step 3 — Solve for Eigenvalues

Factor the polynomial: λ² − 7λ + 10 = (λ − 5)(λ − 2) = 0. So λ₁ = 5 and λ₂ = 2.

Eigenvalues: λ₁ = 5, λ₂ = 2
4
Step 4 — Find Eigenvectors for λ₁ = 5

Plug λ = 5 into (A − λI): [[4−5, 1], [2, 3−5]] = [[−1, 1], [2, −2]]. We solve [[−1, 1], [2, −2]] × [x, y]ᵀ = [0, 0]ᵀ. The first row gives −x + y = 0, so y = x. Let x = t (any number). The eigenvectors are t × [1, 1]ᵀ.

Eigenspace E₅ = span{[1, 1]ᵀ} — a line
5
Step 5 — Find Eigenvectors for λ₂ = 2

Plug λ = 2 into (A − λI): [[4−2, 1], [2, 3−2]] = [[2, 1], [2, 1]]. The first row gives 2x + y = 0, so y = −2x. Let x = t. The eigenvectors are t × [1, −2]ᵀ.

Eigenspace E₂ = span{[1, −2]ᵀ} — a line
6
Step 6 — Verify

Check: A × [1, 1]ᵀ = [4×1+1×1, 2×1+3×1]ᵀ = [5, 5]ᵀ = 5 × [1, 1]ᵀ ✓. And A × [1, −2]ᵀ = [4×1+1×(−2), 2×1+3×(−2)]ᵀ = [2, −4]ᵀ = 2 × [1, −2]ᵀ ✓.

Both eigenvalues and eigenvectors verified!

SECTION 7

Common Tips & Pitfalls

Computing eigenvectors is a straightforward process, but there are a few places where students commonly trip up. The table below highlights the most important do's and don'ts.

Common Mistake	Why It's Wrong	What to Do Instead
Calling the zero vector an eigenvector	By definition, eigenvectors must be nonzero. A × 0 = λ × 0 is always true, so it gives no useful information.	State that the zero vector is in the eigenspace but is not itself an eigenvector.
Forgetting to subtract λI (subtracting λ only from one entry)	λI subtracts λ from every diagonal entry, not just one.	Write out the identity matrix I first, multiply by λ, then subtract from A.
Stopping after finding eigenvalues	Eigenvalues alone don't tell you the direction — eigenvectors do.	For each eigenvalue, solve (A − λI)v = 0 to find the eigenvectors.
Giving only one eigenvector when the eigenspace is 2-D	If the null space has dimension 2, there are two independent eigenvectors forming a basis.	Row-reduce (A − λI) fully and identify all free variables to get a complete basis.

Mistakes to Avoid When Computing Eigenvectors

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KEY TAKEAWAY

Think of finding eigenvectors like tuning a guitar. The eigenvalue is the pitch (a number), and the eigenvector is the vibration pattern of the string (a shape/direction). You need both to fully understand the note. Skipping the eigenvector is like knowing a note is a C but not knowing which string produces it.

SECTION 8

Connection to Advanced Topics

Once you can compute eigenvectors and eigenspaces, a whole world of powerful techniques opens up. The table below shows how the ideas in this lesson connect to more advanced topics you might study later.

This Lesson	Advanced Extension
Finding eigenvectors of a 2×2 matrix	Diagonalization — rewriting A as PDP⁻¹ where D is a diagonal matrix of eigenvalues and P is a matrix of eigenvectors
Eigenspace as null space of (A − λI)	Generalized eigenspaces and Jordan normal form for matrices that can't be fully diagonalized
Eigenvalues from det(A − λI) = 0	Spectral theorem: symmetric matrices always have real eigenvalues and orthogonal eigenvectors
Two eigenspaces for a 2×2 matrix	Principal Component Analysis (PCA) in data science — finding the most important directions in high-dimensional data

From This Lesson to Advanced Linear Algebra

One especially exciting application is in physics. The Schrödinger equation in quantum mechanics is an eigenvalue problem: the eigenvalues represent energy levels of atoms, and the eigenvectors describe the shapes of electron clouds. So the technique you just learned — solving (A − λI)v = 0 — is literally the same mathematics that explains how atoms work!

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

In your own words, explain why the zero vector is not considered an eigenvector, even though A × 0 = λ × 0 is always true for any λ.

Show answer

The equation Av = λv is satisfied by v = 0 for every matrix and every value of λ. If we allowed the zero vector to be an eigenvector, then every number would be an eigenvalue of every matrix, which would make the concept meaningless. We need eigenvectors to tell us a specific direction that the matrix preserves, and the zero vector has no direction. That's why the definition requires v ≠ 0.

Question 2: BASIC CALCULATION

Find the eigenvalues of A = [[3, 0], [0, 7]]. Then find the eigenvectors and describe each eigenspace.

Show answer

Since A is a diagonal matrix, the eigenvalues are just the diagonal entries: λ₁ = 3 and λ₂ = 7. For λ₁ = 3: (A − 3I) = [[0, 0], [0, 4]], which gives 4y = 0, so y = 0 and x is free. Eigenvectors: t[1, 0]ᵀ, and E₃ = span{[1, 0]ᵀ}. For λ₂ = 7: (A − 7I) = [[−4, 0], [0, 0]], which gives −4x = 0, so x = 0 and y is free. Eigenvectors: t[0, 1]ᵀ, and E₇ = span{[0, 1]ᵀ}. Each eigenspace is a line along a coordinate axis.

Question 3: INTERMEDIATE

Find the eigenvalues and eigenspaces of A = [[1, 2], [4, 3]]. Verify one of your eigenvectors by multiplying.

Show answer

Form A − λI = [[1−λ, 2], [4, 3−λ]]. Determinant: (1−λ)(3−λ) − 8 = λ² − 4λ − 5 = (λ − 5)(λ + 1) = 0. Eigenvalues: λ₁ = 5, λ₂ = −1. For λ₁ = 5: A − 5I = [[−4, 2], [4, −2]], giving −4x + 2y = 0, so y = 2x. Eigenvectors: t[1, 2]ᵀ, E₅ = span{[1, 2]ᵀ}. For λ₂ = −1: A + I = [[2, 2], [4, 4]], giving 2x + 2y = 0, so y = −x. Eigenvectors: t[1, −1]ᵀ, E₋₁ = span{[1, −1]ᵀ}. Verification: A × [1, 2]ᵀ = [1+4, 4+6]ᵀ = [5, 10]ᵀ = 5[1, 2]ᵀ ✓.

Question 4: APPLIED

A population model uses the matrix M = [[0.5, 0.4], [0.5, 0.6]] to predict next year's population distribution between two cities. Find the eigenvalues and the eigenspace for the largest eigenvalue. What does this eigenvector tell you about the long-term population split?

Show answer

det(M − λI) = (0.5 − λ)(0.6 − λ) − 0.2 = λ² − 1.1λ + 0.1 = (λ − 1)(λ − 0.1) = 0. Eigenvalues: λ₁ = 1, λ₂ = 0.1. For λ₁ = 1: M − I = [[−0.5, 0.4], [0.5, −0.4]], giving −0.5x + 0.4y = 0, so y = (5/4)x. Eigenvectors: t[4, 5]ᵀ, E₁ = span{[4, 5]ᵀ}. The eigenvalue λ = 1 means the population distribution doesn't change once it reaches this ratio. In the long run, 4/9 ≈ 44% of the population will be in City 1 and 5/9 ≈ 56% will be in City 2. The smaller eigenvalue 0.1 shrinks toward zero, so any deviation from this ratio decays rapidly.

Question 5: CRITICAL THINKING

Consider the matrix A = [[2, 0], [0, 2]]. Find its eigenvalues and eigenspaces. What is the geometric multiplicity of the eigenvalue you find? Explain why every nonzero vector in R² is an eigenvector of this matrix.

Show answer

det(A − λI) = (2 − λ)² = 0, so λ = 2 with algebraic multiplicity 2. For λ = 2: A − 2I = [[0, 0], [0, 0]], so both x and y are free. The eigenspace E₂ = span{[1, 0]ᵀ, [0, 1]ᵀ} = R², and the geometric multiplicity is 2. Since A = 2I, multiplying any vector v by A gives Av = 2Iv = 2v. This means every nonzero vector in R² satisfies the eigenvector equation with λ = 2. Geometrically, the matrix A simply doubles every vector without rotating it, so no direction is more 'special' than any other — the entire plane is the eigenspace.

SUMMARY

Lesson Summary

Summary

An eigenvector of a matrix A is a nonzero vector v that satisfies Av = λv, where λ is the eigenvalue. To find eigenvalues, solve the characteristic equation det(A − λI) = 0. Then, for each eigenvalue, compute the null space of (A − λI) to find the eigenvectors. The collection of all eigenvectors for a given eigenvalue (along with the zero vector) forms the eigenspace, which is always a subspace of Rⁿ.

The dimension of an eigenspace is called its geometric multiplicity, and it is always between 1 and the algebraic multiplicity (how many times λ appears as a root). These ideas connect to powerful applications like diagonalization, population models, search engines, and quantum mechanics. The key steps are: (1) find eigenvalues from the characteristic equation, (2) for each eigenvalue, solve (A − λI)v = 0, and (3) describe the eigenspace using a basis of independent eigenvectors.