Linear Algebra 55 | Algebraic Multiplicity

Algebraic multiplicity measures how many times a particular eigenvalue shows up as a repeated root of the characteristic polynomial, and that repetition count is exactly what determines the power of the corresponding linear factor in the factorization over the complex numbers. In practice, once the characteristic polynomial is written in factored form, the exponent on (Λ − λ̃) tells how often the eigenvalue λ̃ occurs—so multiplicity is not an abstract idea, it’s the literal exponent in the polynomial’s linear-factor decomposition.

The discussion starts from the characteristic polynomial p_A(Λ) = det(A − ΛI). Because p_A is a polynomial of degree n for an n×n matrix, the fundamental theorem of algebra guarantees that p_A(Λ)=0 has exactly n complex roots counted with multiplicity. Those roots can be labeled Λ1, Λ2, …, Λn, and the polynomial can be factorized into linear terms over the complex numbers. If some roots coincide, the factorization reflects that coincidence by repeating the same linear factor, which is where multiplicity enters.

A key consequence is that every square matrix has at least one eigenvalue once complex numbers are allowed, even if the matrix entries are real. The example A = [[0, −1],[1, 0]] has characteristic polynomial Λ^2 + 1, whose roots are +i and −i—no real eigenvalues, but two complex eigenvalues. The point is that extending to the complex field makes eigenvalue theory cleaner because characteristic polynomials always split into linear factors there.

To define algebraic multiplicity, the transcript uses a diagonal example with diagonal entries 1 and 2: diag(1,2,1,2). Its characteristic polynomial factors as (Λ − 1)^2(Λ − 2)^2, so the eigenvalue 1 has algebraic multiplicity 2 and the eigenvalue 2 also has algebraic multiplicity 2. More generally, if λ̃ is an eigenvalue of A and (Λ − λ̃) appears K times in the factorization, then λ̃ has algebraic multiplicity K. The notation α(λ̃) is introduced for this exponent.

The algebraic multiplicity of any eigenvalue must lie between 1 and n, since the characteristic polynomial has degree n. Summing algebraic multiplicities over all distinct eigenvalues gives exactly n, because the characteristic polynomial has exactly n linear factors when counted with repetition. The transcript also flags a practical check: if the total exceeds n, something went wrong in the calculation.

Finally, the material sets up a contrast: algebraic multiplicity will later be compared with geometric multiplicity, which can differ. That difference is important because it reveals how eigenvalues’ algebraic repetition relates to the structure of eigenspaces.