# Eigenvalue

Topics in mathematics related to linear algebra
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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 |

**eigenvalue**(in some older texts, a

**characteristic value**) of a linear mapping

*A*if there exists a nonzero vector

*x*such that

*Ax*=λ

*x*. 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*=λ*x*, or a **left eigenvalue** if there exists a nonzero row vector *y* such that *yA*=*y*λ. 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.

Table of contents |

2 Spectrum 3 Multiset of eigenvalues 4 Trace and Determinant 5 See also |

## 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.

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 λ

_{1},λ

_{2},...,λ

_{n}. Then the trace of

*A*is λ

_{1}+λ

_{2}+...+λ

_{n}and the determinant of

*A*is λ

_{1}λ

_{2}...λ

_{n}. These two are very important concepts in matrix theory.

## See also

Please refer to eigenvector for some other properties of eigenvalues.