eigenvalue (of a matrix)
Computation of Eigenvalues
The computation of the eigenvalues of a given matrix 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 denotes the identity matrix and is the determinant function. As the above determinant is simply a polynomial (of degree , called the characteristic polynomial of ) in 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 closed, it is thus guaranteed that any has at least one eigenvalue (and at most ). If is a multiple root (say, of multiplicity ) of the defining polynomial, we say that is an eigenvalue of multiplicity .
If one is given a matrix of real numbers, the above argument implies that has at least one complex eigenvalue; the question of whether or not has real eigenvalues is more subtle since there is no real-numbers analogue of the fundamental theorem of algebra. It should not be a surprise then that some real matrices do not have real eigenvalues. For example, let
In this case ; clearly no real number ; hence has no real eigenvalues (although has complex eigenvalues and ).
If one converts the above into an algorithm for calculating the eigenvalues of a matrix , one is led to a two-step procedure:
Compute the polynomial .
Unfortunately, computing determinants and finding roots of polynomials of degree are both computationally messy procedures for even moderately large , so for most practical purposes variations on this naive are needed. See the eigenvalue problem for more .
and have the same eigenvalues.
A real matrix is diagonalizable if all of ’s eigenvalues are real and distinct.
If a symmetric matrix has distinct eigenvalues, it is diagonalizable.
|Title||eigenvalue (of a matrix)|
|Date of creation||2013-03-22 13:42:57|
|Last modified on||2013-03-22 13:42:57|
|Last modified by||mathcam (2727)|