The algebraic multiplicity of an eigenvalue is the multiplicity of that scalar as a root of the characteristic polynomial of a linear operator or matrix; it counts how many times the eigenvalue appears as a factor (λ − λ0) in the characteristic polynomial.

Principle

Count eigenvalue occurrence in the characteristic polynomial: if charpoly(λ) = (λ − λ0)^m · q(λ) with q(λ0) ≠ 0, then the algebraic multiplicity of λ0 is m.

Demonstration

For A = diag(2,2,2,−1) the characteristic polynomial is (λ−2)^3(λ+1); the algebraic multiplicity of 2 is 3 because (λ−2) appears to the third power.

Misapplication

Treating algebraic multiplicity as the dimension of the eigenspace (geometric multiplicity) and concluding diagonalizability from algebraic counts alone; or assuming algebraic multiplicities are invariant under extension of the base field without checking the polynomial factorization.

Consequence

Algebraic multiplicities determine the sizes and counts of Jordan blocks in the Jordan canonical form and, combined with geometric multiplicities, decide diagonalizability and the structure of generalized eigenspaces.

Reversal

A simple eigenvalue is the inverse case: algebraic multiplicity equal to 1, implying the eigenvalue is a simple root of the characteristic polynomial.

Boundary

Defined relative to the characteristic polynomial over a chosen field; over non-algebraically closed fields an eigenvalue may not appear as a root until the polynomial is factored in an extension field. It applies to finite-dimensional endomorphisms; infinite-dimensional operators require a different spectral multiplicity theory.

Semantic Tension

Often confused with geometric multiplicity (dimension of eigenspace); algebraic multiplicity is a polynomial/root-count concept, geometric multiplicity is a linear-algebraic dimension concept — they agree only in special cases.

Synthesis

Algebraic multiplicity is the polynomial count of how many times an eigenvalue appears as a root of the characteristic polynomial; it sets algebraic constraints on Jordan block sizes and, together with geometric multiplicity, controls diagonalizability and the internal structure of an operator's spectrum.