Algebraic vs. Geometric Multiplicity

SECTION 1

Historical Context & Motivation

Mathematicians have been working with systems of equations for centuries. In the 1700s and 1800s, they started studying special numbers tied to matrices — numbers we now call eigenvalues (from the German word eigen, meaning "own" or "characteristic"). These eigenvalues reveal deep truths about how a matrix stretches, shrinks, or flips space. As mathematicians explored eigenvalues further, they realized that the same eigenvalue can show up more than once, and counting how many times it appears can be done in two very different ways.

1750s

Euler and Vibrations

Leonhard Euler studied vibrating strings and rotating bodies. He noticed that certain "characteristic" values controlled how systems moved — an early encounter with eigenvalues.

1826

Cauchy's Determinant Work

Augustin-Louis Cauchy formally defined the characteristic polynomial of a matrix, giving us the tool to find eigenvalues by solving polynomial equations.

1878

Frobenius and Normal Forms

Ferdinand Georg Frobenius studied when a matrix could be simplified (diagonalized). He found that the answer depends on comparing two kinds of multiplicity for each eigenvalue.

1904

Hilbert's Spectral Theory

David Hilbert extended eigenvalue ideas to infinite-dimensional spaces, making algebraic and geometric multiplicity central concepts in modern mathematics and physics.

So here is the big question this lesson addresses: when an eigenvalue repeats in the characteristic polynomial, does it always come with enough independent eigenvectors to match? If it does, you can fully simplify the matrix. If it doesn't, things get trickier. Understanding the difference between algebraic multiplicity and geometric multiplicity is the key to answering that question.

SECTION 2

Core Principles & Definitions

Before diving into the two types of multiplicity, let's make sure the building blocks are solid. A matrix is a rectangular grid of numbers. An eigenvalue is a special number λ (the Greek letter "lambda") such that when you multiply the matrix by a particular vector, the result is just that vector scaled by λ. That special vector is called an eigenvector. In equation form: Av = λv, where A is the matrix and v is the eigenvector.

Characteristic Polynomial

Found by computing det(A − λI) = 0. This polynomial's roots are the eigenvalues. Its degree equals the size of the matrix (an n×n matrix gives a degree-n polynomial).

Algebraic Multiplicity (AM)

The number of times an eigenvalue λ appears as a root of the characteristic polynomial. If (λ − 3)² is a factor, then λ = 3 has AM = 2.

Geometric Multiplicity (GM)

The number of linearly independent eigenvectors for a given eigenvalue λ. This equals the dimension of the null space of (A − λI).

The Key Inequality

For every eigenvalue: 1 ≤ GM ≤ AM. Geometric multiplicity is always at least 1 and never exceeds algebraic multiplicity.

Diagonalizability

A matrix can be diagonalized (fully simplified) if and only if GM = AM for every eigenvalue. When they don't match, the matrix is called defective.

✦ KEY TAKEAWAY

A useful way to keep these straight: algebraic multiplicity is an algebraic property — it comes purely from factoring the characteristic polynomial. Geometric multiplicity is a geometric property — it's the actual dimension of the eigenspace {v : (A − λI)v = 0}. That eigenspace always sits inside a larger invariant subspace of dimension AM, which is why GM can never exceed AM. But nothing guarantees the eigenspace fills that entire subspace, so GM can fall short — though never below 1, since λ being a root of the characteristic polynomial guarantees at least one eigenvector exists. When the eigenspace fills its full allotted dimension for every eigenvalue, the matrix is diagonalizable.

SECTION 3

Visual Explanation

The diagram below shows two 3×3 matrices that share the same eigenvalue λ = 2 with algebraic multiplicity 2. On the left, the geometric multiplicity also equals 2 (two independent eigenvector directions), so the matrix is diagonalizable. On the right, the geometric multiplicity is only 1 (one eigenvector direction), so the matrix is defective — it cannot be diagonalized.

Left: when GM equals AM, there are enough independent eigenvector directions — the matrix is diagonalizable. Right: when GM is less than AM, there aren't enough independent eigenvectors, making the matrix defective.

Notice how on the left side, the two eigenvectors v₁ and v₂ point in genuinely different directions. They span a two-dimensional plane, which is exactly what you need when the eigenvalue has algebraic multiplicity 2. On the right side, even though the eigenvalue still repeats twice in the characteristic polynomial, the matrix only produces one independent eigenvector direction. That missing eigenvector is the root cause of the matrix being non-diagonalizable.

SECTION 4

Mathematical Framework

Let's lay out the math step by step. Suppose you have an n×n matrix A. To find its eigenvalues, you build and solve the characteristic equation. Once you have the eigenvalues, you compute each type of multiplicity.

CHARACTERISTIC EQUATION

det(A − λI) = 0

A is the matrix, λ (lambda) is the eigenvalue, and I is the identity matrix (1s on the diagonal, 0s elsewhere). The expression det(…) means the determinant.

Expanding det(A − λI) gives you a polynomial in λ. For a 2×2 matrix, it's a quadratic. For a 3×3 matrix, it's a cubic, and so on. You can then factor this polynomial to find the eigenvalues and their algebraic multiplicities.

ALGEBRAIC MULTIPLICITY

p(λ) = (λ − λ₁)^{a₁} × (λ − λ₂)^{a₂} × … × (λ − λₖ)^{aₖ}

Each exponent aᵢ is the algebraic multiplicity of eigenvalue λᵢ. The sum a₁ + a₂ + … + aₖ = n (the matrix size).

GEOMETRIC MULTIPLICITY

GM(λ) = dim(null(A − λI)) = n − rank(A − λI)

To find the geometric multiplicity, compute A − λI, then find how many free variables the system (A − λI)x = 0 has. That count is the number of independent eigenvectors — the geometric multiplicity.

FUNDAMENTAL INEQUALITY

1 ≤ GM(λ) ≤ AM(λ) for every eigenvalue λ

Every eigenvalue has at least one eigenvector (GM ≥ 1), and you can never have more independent eigenvectors than the eigenvalue's repeat count in the polynomial (GM ≤ AM).

SECTION 5

Classifying Matrices by Their Multiplicities

Matrices can be sorted into categories based on how their algebraic and geometric multiplicities compare. The diagram below shows three common scenarios for a 3×3 matrix with eigenvalues. Understanding these cases helps you quickly decide whether a matrix can be diagonalized.

Three scenarios for a 3×3 matrix. Cases 1 and 2 (green borders) are diagonalizable because GM = AM for every eigenvalue. Case 3 (red border) is defective because λ = 2 has GM = 1 but AM = 2 — one eigenvector is missing.

Case 1 is the simplest — when all eigenvalues are different, each one automatically has AM = 1 and GM = 1, so the matrix is always diagonalizable. Case 2 shows that a repeated eigenvalue isn't automatically a problem: as long as it produces enough independent eigenvectors, everything still works. Case 3 is the tricky one. The eigenvalue λ = 2 appears twice in the polynomial (AM = 2), but the matrix only gives us one eigenvector direction for it (GM = 1). That gap means the matrix is defective and needs more advanced tools (like Jordan normal form) to simplify.

SECTION 6

Worked Example

Let's walk through a complete example with a 3×3 matrix. We'll find the eigenvalues, compute their algebraic multiplicities, then determine the geometric multiplicities.

Finding AM and GM for a 3×3 Matrix

Step 1 — State the MatrixConsider the matrix A = [[2, 1, 0], [0, 2, 0], [0, 0, 5]]. This is an upper triangular matrix (all entries below the diagonal are zero). For triangular matrices, the eigenvalues are simply the entries on the main diagonal.

A is a 3×3 upper triangular matrix.

Step 2 — Find the EigenvaluesThe diagonal entries are 2, 2, and 5. So the characteristic polynomial is det(A − λI) = (2 − λ)² × (5 − λ) = 0. The eigenvalues are λ = 2 and λ = 5.

Eigenvalues: λ = 2 and λ = 5

Step 3 — Determine Algebraic MultiplicitiesThe factor (2 − λ) appears squared, so λ = 2 has AM = 2. The factor (5 − λ) appears once, so λ = 5 has AM = 1. Check: 2 + 1 = 3, which matches the 3×3 matrix size. ✓

AM(2) = 2, AM(5) = 1

Step 4 — Find GM for λ = 5Compute A − 5I = [[−3, 1, 0], [0, −3, 0], [0, 0, 0]]. Row reduce: the matrix is already in row echelon form. There are 2 pivots (in columns 1 and 2), so the rank is 2. The number of free variables is 3 − 2 = 1.

GM(5) = 1. Since AM(5) = 1, they match. ✓

Step 5 — Find GM for λ = 2Compute A − 2I = [[0, 1, 0], [0, 0, 0], [0, 0, 3]]. Row reduce: there are 2 pivots (in columns 2 and 3), so the rank is 2. The number of free variables is 3 − 2 = 1.

GM(2) = 1. But AM(2) = 2, so GM < AM!

Step 6 — ConclusionSince GM(2) = 1 < AM(2) = 2, the eigenvalue λ = 2 does not have enough independent eigenvectors. The matrix A is not diagonalizable (it is defective). The "1" in the upper-right area of the original matrix (position [1,2]) is what causes this — it prevents the second eigenvector from being independent.

Matrix A is NOT diagonalizable because GM(2) ≠ AM(2).

SECTION 7

Algebraic vs. Geometric Multiplicity — Side by Side

It's easy to confuse the two multiplicities since both relate to eigenvalues. The table below gives you a quick-reference comparison so you can keep them straight.

Side-by-side comparison of algebraic and geometric multiplicity

Feature	Algebraic Multiplicity (AM)	Geometric Multiplicity (GM)
What it counts	How many times λ appears as a root of the characteristic polynomial	How many independent eigenvectors exist for λ
How to find it	Factor det(A − λI) and read the exponent on each factor	Row reduce (A − λI) and count the free variables
Minimum value	1 (every eigenvalue appears at least once)	1 (every eigenvalue has at least one eigenvector)
Maximum value	n (the size of the matrix)	AM (never exceeds algebraic multiplicity)
What it reveals	The "weight" of the eigenvalue in the polynomial	The actual number of independent directions the matrix stretches by λ
Role in diagonalization	Sets the target: GM must reach this value	Determines success: if GM = AM for all eigenvalues, the matrix diagonalizes

✦ KEY TAKEAWAY

A helpful mental model: algebraic multiplicity tells you how many dimensions of "room" the characteristic polynomial reserves for an eigenvalue. Geometric multiplicity tells you how many of those dimensions are actually spanned by genuine eigenvectors. When every reserved dimension is filled (GM = AM for every eigenvalue), the matrix diagonalizes cleanly. When some dimensions go unfilled (GM < AM), the matrix is defective, and the missing directions must be supplied by generalized eigenvectors instead — the idea behind Jordan normal form.

SECTION 8

Connection to Advanced Theory

When a matrix is defective (GM < AM for some eigenvalue), diagonalization fails. But mathematicians didn't give up! They developed more powerful tools to handle these cases. The most important is the Jordan normal form, which replaces the diagonal matrix with one that has 1s just above the diagonal in certain spots (called Jordan blocks). This lets you still simplify and analyze any square matrix, even defective ones.

Diagonalization vs. Jordan normal form

Concept	Diagonalization	Jordan Normal Form
Requirement	GM = AM for every eigenvalue	Works for any square matrix
Resulting form	Purely diagonal (eigenvalues on diagonal, zeros elsewhere)	Block diagonal with possible 1s above diagonal
Vectors used	Only eigenvectors	Eigenvectors + generalized eigenvectors
Difficulty	Moderate — factor polynomial, row reduce	Harder — requires chains of generalized eigenvectors

The gap between AM and GM tells you exactly how many generalized eigenvectors you need to fill in. For example, if AM = 3 and GM = 1, you need 2 generalized eigenvectors to complete the picture. These ideas connect to differential equations (solving systems like dx/dt = Ax), vibration analysis in engineering, and even Google's PageRank algorithm. So understanding the AM-GM relationship isn't just abstract math — it shows up in real-world problems.

SECTION 9

Practice Problems

PROBLEM 1 — CONCEPTUAL

In your own words, explain the difference between algebraic multiplicity and geometric multiplicity. Why is it possible for GM to be less than AM, but never greater?

PROBLEM 2 — BASIC CALCULATION

Consider the matrix B = [[4, 0], [0, 4]]. Find the eigenvalues and compute both the algebraic multiplicity and the geometric multiplicity for each.

PROBLEM 3 — INTERMEDIATE

Let C = [[3, 1], [0, 3]]. Find the eigenvalues, their algebraic multiplicities, and their geometric multiplicities. Is C diagonalizable?

PROBLEM 4 — APPLIED

A 4×4 matrix D has the characteristic polynomial p(λ) = (λ − 1)³ × (λ − 7). You row reduce (D − 1·I) and find it has rank 2. Is the matrix D diagonalizable? Explain how you know, and state the AM and GM for each eigenvalue.

PROBLEM 5 — CRITICAL THINKING

True or false: if a 5×5 matrix has exactly two distinct eigenvalues, it must be defective. Justify your answer with a specific example or counterexample.

SUMMARY

Summary & Review

Every eigenvalue of a matrix has two important counts. The algebraic multiplicity (AM) tells you how many times the eigenvalue appears as a root of the characteristic polynomial det(A − λI) = 0. The geometric multiplicity (GM) tells you how many linearly independent eigenvectors exist for that eigenvalue, found by computing the null space of (A − λI).

The fundamental inequality 1 ≤ GM ≤ AM always holds. A matrix is diagonalizable if and only if GM equals AM for every eigenvalue. When GM falls short of AM, the matrix is called defective, and more advanced tools like the Jordan normal form are needed to analyze it. Understanding the AM-GM relationship is the gateway to deeper topics in linear algebra and its applications in science, engineering, and data analysis.