Algebraic multiplicity - Definition and Overview

In linear algebra, a scalar λ is called an eigenvalue (in some older texts, a characteristic value) of a linear mapping A if there exists a nonzero vector x such that Ax=λx. The vector x is called an eigenvector.

In matrix theory, an element in the underlying ring R of a square matrix A is called a right eigenvalue if there exists a nonzero column vector x such that Ax=λx, or a left eigenvalue if there exists a nonzero row vector y such that yA=yλ. If R is commutative, the left eigenvalues of A are exactly the right eigenvalues of A and are just called eigenvalues. If R is not commutative, e.g. quaternions, they may be different.

In graph theory, an eigenvalue of a graph is simply an eigenvalue of the graph's adjacency matrix.

The word eigenvalue comes from the German Eigenwert which means "proper or characteristic value."

Multiplicity

Suppose A is a square matrix over a commutative ring. The algebraic multiplicity (or simply multiplicity) of an eigenvalue λ of A is the number of factors t-λ of the characteristic polynomial of A. The geometric multiplicity of λ is the number of factors t-λ of the minimal polynomial of A or equivalently the nullity of (λI-A).

An eigenvalue of algebraic multiplicity 1 is called a simple eigenvalue.

Spectrum

In functional analysis, the spectrum of a bounded linear operator A on a Banach space is the set of scalars ν such that νI-A does not have a bounded two-sided inverse. Note that by the closed graph theorem, if a bounded operator has an inverse, the inverse is necessarily bounded.

If the underlying Banach space is finite dimensional, then the spectrum of A is the same of the set of eigenvalues of A. This follows from the fact that on finite dimensional spaces injectivity of a linear operator A is equivalent to surjectivity of A.

Multiset of eigenvalues

Occasionally, in an article on matrix theory, one may read a statement like:

The eigenvalues of a matrix A are 4,4,3,3,3,2,2,1.

It means the algebraic multiplicity of 4 is two, of 3 is three, of 2 is two and of 1 is one.

This style is used because algebraic multiplicity is the key to many mathematical proofs in matrix theory.

Trace and determinant

Suppose the eigenvalues of a matrix A are λ₁,λ₂,...,λ_n. Then the trace of A is λ₁+λ₂+...+λ_n and the determinant of A is λ₁λ₂...λ_n. These two are very important concepts in matrix theory.

See also

Please refer to eigenvector for some other properties of eigenvalues.

• wikibooks:Algebra:Eigenvalues and eigenvectors