eigenvalue (of a matrix)


Let A be an complex n×n matrix. A number λ is said to be an eigenvalueMathworldPlanetmathPlanetmathPlanetmathPlanetmath of A if there is a nonzero n×1 column vectorMathworldPlanetmath x for which


Computation of Eigenvalues

The computation of the eigenvalues of a given matrix A is relatively easy from a theoretical point of view, though often computationally infeasible, or at least difficult. The basic procedure is to note that the eigenvalues of a matrix are precisely the solutions to the equation


where I denotes the n×n identity matrixMathworldPlanetmath and det is the determinantMathworldPlanetmath function. As the above determinant is simply a polynomialMathworldPlanetmathPlanetmathPlanetmath (of degree n, called the characteristic polynomialMathworldPlanetmathPlanetmath of A) in λ with coefficients in , its roots can be calculated or approximated accordingly to give the eigenvalues of the matrix. Following this train of thought, we also note that this polynomial has degree at least 1, so since is algebraically closedMathworldPlanetmath, it is thus guaranteed that any A has at least one eigenvalue (and at most n). If λ is a multiple root (say, of multiplicityMathworldPlanetmath k) of the defining polynomial, we say that λ is an eigenvalue of multiplicity k.

If one is given a n×n matrix A of real numbers, the above argument implies that A has at least one complex eigenvalue; the question of whether or not A has real eigenvalues is more subtle since there is no real-numbers analogue of the fundamental theorem of algebraMathworldPlanetmath. It should not be a surprise then that some real matrices do not have real eigenvalues. For example, let


In this case det(λI-A)=λ2+1; clearly no real number λ λ2+1=0; hence A has no real eigenvalues (although A has complex eigenvalues i and -i).

If one converts the above into an algorithm for calculating the eigenvalues of a matrix A, one is led to a two-step procedure:

  • Compute the polynomial det(λI-A).

  • Solve det(λI-A)=0.

Unfortunately, computing n×n determinants and finding roots of polynomials of degree n are both computationally messy procedures for even moderately large n, so for most practical purposes variations on this naive are needed. See the eigenvalue problem for more .


Title eigenvalue (of a matrix)
Canonical name EigenvalueofAMatrix
Date of creation 2013-03-22 13:42:57
Last modified on 2013-03-22 13:42:57
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 11
Author mathcam (2727)
Entry type Definition
Classification msc 65-00
Classification msc 15A18
Classification msc 15-00
Classification msc 65F15