Linear Algebra 55 | Algebraic Multiplicity [dark version]

TL;DR

Algebraic multiplicity of an eigenvalue λ̃ is the exponent K of the factor (λ−λ̃)^K in the fully factored characteristic polynomial p_A(λ)=det(A−λI).

Briefing Cornell Notes

Briefing

Algebraic multiplicity measures how many times a given eigenvalue shows up as a repeated root of the characteristic polynomial—so it’s the “counting rule” for eigenvalues inside the polynomial factorization. Since eigenvalues are defined through the characteristic polynomial p_A(λ)=det(A−λI), the algebraic multiplicity of an eigenvalue λ̃ is exactly the exponent of the factor (λ−λ̃) in the fully factored form of p_A(λ). This matters because it quantifies repetition even when eigenvalues are complex, and it sets up the later comparison with geometric multiplicity (which counts independent eigenvectors).

The discussion starts from the characteristic polynomial’s degree. For an n×n matrix A, p_A(λ) is a polynomial of degree n, so the fundamental theorem of algebra applies: over the complex numbers, a degree-n polynomial has exactly n zeros counting multiplicity. That guarantees that every matrix has at least one eigenvalue once complex numbers are allowed, even if a real matrix’s eigenvalues are not real. A concrete example uses the 2×2 matrix [[0,−1],[1,0]], whose characteristic polynomial is λ^2+1. Its roots are ±i, so the eigenvalues are i and −i—showing why extending linear algebra to the complex realm simplifies eigenvalue theory.

Next comes the factorization viewpoint. If the characteristic polynomial factors into linear terms, repeated roots appear as powers. For instance, a diagonal matrix with entries 1,2,1,2 has characteristic polynomial (λ−1)^2(λ−2)^2, so the eigenvalue 1 has algebraic multiplicity 2 and the eigenvalue 2 has algebraic multiplicity 2. In general, if λ̃ is a root of p_A(λ) and the factor (λ−λ̃) occurs K times in the factorization, then K is the algebraic multiplicity of λ̃ (often written as α(λ̃)).

The transcript also highlights bounds and consistency checks. If λ̃ is an eigenvalue of an n×n matrix, then its algebraic multiplicity satisfies 1≤α(λ̃)≤n. Conversely, having algebraic multiplicity at least 1 means λ̃ is a zero of the characteristic polynomial, hence an eigenvalue. Summing algebraic multiplicities over all distinct eigenvalues must produce exactly n, because the characteristic polynomial has n linear factors in total when counted with multiplicity. If the sum exceeds n, something went wrong in the calculation.

Finally, the groundwork is laid for the key next step: algebraic multiplicity and geometric multiplicity need not match. The difference between them determines how many eigenvectors exist for a repeated eigenvalue, which affects diagonalizability and the structure of solutions to linear systems.

Cornell Notes

Algebraic multiplicity counts how many times an eigenvalue appears as a repeated root of the characteristic polynomial. For an n×n matrix A, the characteristic polynomial p_A(λ)=det(A−λI) has degree n, and over the complex numbers it factors into n linear terms (counting multiplicity). If p_A(λ) contains the factor (λ−λ̃)^K, then the eigenvalue λ̃ has algebraic multiplicity α(λ̃)=K. This quantity is bounded: for any eigenvalue, 1≤α(λ̃)≤n. Adding the algebraic multiplicities of all distinct eigenvalues must equal n, providing a built-in check for computations. The next topic will contrast this with geometric multiplicity, which counts independent eigenvectors and can differ from algebraic multiplicity.

Why does algebraic multiplicity come from the characteristic polynomial rather than directly from the matrix?

Eigenvalues are defined through the characteristic polynomial p_A(λ)=det(A−λI). Once p_A(λ) is factored into linear terms over the complex numbers, repeated factors show up as powers. Algebraic multiplicity is precisely the exponent of (λ−λ̃) in that factorization, so it measures repetition of the root λ̃ in p_A(λ).

How does the fundamental theorem of algebra guarantee eigenvalues exist for any matrix?

For an n×n matrix, p_A(λ) is a degree-n polynomial. The fundamental theorem of algebra says a degree-n polynomial has exactly n zeros in the complex numbers counting multiplicity. Therefore p_A(λ)=0 always has at least one complex solution, meaning every matrix has at least one eigenvalue once complex numbers are allowed (even if the matrix is real).

What does the example A=[[0,−1],[1,0]] show about algebraic multiplicity and complex eigenvalues?

That matrix has characteristic polynomial λ^2+1, whose roots are i and −i. Both are simple roots, so each appears with algebraic multiplicity 1. The example illustrates that real matrices can have non-real eigenvalues, and the algebraic multiplicity framework naturally lives over the complex numbers.

How do repeated diagonal entries translate into algebraic multiplicity?

If a diagonal matrix has repeated diagonal values, the characteristic polynomial repeats the corresponding linear factors. For example, with diagonal entries 1,2,1,2, the characteristic polynomial becomes (λ−1)^2(λ−2)^2. Hence α(1)=2 and α(2)=2, matching the number of times each value appears on the diagonal.

What consistency check should always hold for algebraic multiplicities?

Because the characteristic polynomial has degree n and factors into n linear factors counting multiplicity, the sum of algebraic multiplicities over all distinct eigenvalues must equal n. For instance, if eigenvalues are 2 and 2 (counted as repeated roots), their multiplicities add to the total degree (e.g., 2+2=4 for a 4×4 case). If the sum exceeds n, the factorization or counting is incorrect.

Review Questions

Given an n×n matrix with characteristic polynomial p_A(λ)=(λ−λ̃)^K·q(λ), what is the algebraic multiplicity of λ̃ and how do you read it off from the factorization?
Why must the sum of algebraic multiplicities of all distinct eigenvalues equal n for an n×n matrix?
How would you expect algebraic multiplicity to differ from geometric multiplicity, and what consequence might that have for the number of independent eigenvectors?

Key Points

1
Algebraic multiplicity of an eigenvalue λ̃ is the exponent K of the factor (λ−λ̃)^K in the fully factored characteristic polynomial p_A(λ)=det(A−λI).
2
For an n×n matrix, p_A(λ) has degree n, so its factorization over the complex numbers contains exactly n linear factors counting multiplicity.
3
Allowing complex numbers ensures every matrix has at least one eigenvalue, even when a real matrix’s eigenvalues are non-real (e.g., ±i).
4
Repeated eigenvalues appear as repeated roots of p_A(λ), and that repetition is quantified directly by algebraic multiplicity.
5
For any eigenvalue λ̃ of an n×n matrix, the algebraic multiplicity satisfies 1≤α(λ̃)≤n.
6
Summing algebraic multiplicities over all distinct eigenvalues must equal n; exceeding n signals a calculation error.

Highlights

Algebraic multiplicity is read straight from the characteristic polynomial factorization: it’s the power on (λ−λ̃).

A real 2×2 matrix can have purely complex eigenvalues; the example λ^2+1 yields eigenvalues i and −i.

For a diagonal matrix with diagonal entries 1,2,1,2, the characteristic polynomial becomes (λ−1)^2(λ−2)^2, so each eigenvalue’s algebraic multiplicity matches its repetition count.

The total of all algebraic multiplicities must equal the matrix size n, providing a built-in check on eigenvalue counting.

Algebraic multiplicity sets up the later comparison with geometric multiplicity, which can differ for repeated eigenvalues.