##  [Algebraic Multiplicity](/algebraic-multiplicity-0) 

 Definition

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

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

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

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

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

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

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

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

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.