The Eigenvalue reference article from the English Wikipedia on 24-Jul-2004 (provided by Fixed Reference: snapshots of Wikipedia from wikipedia.org)

Eigenvalue

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 Topics in mathematics related to linear algebra Edit Vectors | Vector spaces | Linear span | Linear transformation | Linear independence | Linear combination | Basis | Column space | Row space | Dual space | Orthogonality | Eigenvector | Eigenvalue | Least squares regressions | Outer product | Cross product | Dot product | Transpose | Matrix decomposition
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 Axx. 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 Axx, 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.

Multiplicity

Suppose A is a square matrix over 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 factor 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 scalar 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 λ12,...,λn. Then the trace of A is λ12+...+λn and the determinant of A is λ1λ2...λn. These two are very important concepts in matrix theory.