Generalized eigenvector

In linear algebra, a generalized eigenvector of an formula_1 matrix formula_2 is a vector which satisfies certain criteria which are more relaxed than those for an (ordinary) eigenvector.
Let formula_3 be an formula_4-dimensional vector space; let formula_5 be a linear map in , the set of all linear maps from formula_3 into itself; and let formula_2 be the matrix representation of formula_5 with respect to some ordered basis.
There may not always exist a full set of formula_4 linearly independent eigenvectors of formula_2 that form a complete basis for formula_3. That is, the matrix formula_2 may not be diagonalizable. This happens when the algebraic multiplicity of at least one eigenvalue formula_13 is greater than its geometric multiplicity (the nullity of the matrix formula_14, or the dimension of its nullspace). In this case, formula_13 is called a defective eigenvalue and formula_2 is called a defective matrix.
A generalized eigenvector formula_17 corresponding to formula_13, together with the matrix formula_14 generate a Jordan chain of linearly independent generalized eigenvectors which form a basis for an invariant subspace of formula_3.
Using generalized eigenvectors, a set of linearly independent eigenvectors of formula_2 can be extended, if necessary, to a complete basis for formula_3. This basis can be used to determine an "almost diagonal matrix" formula_23 in Jordan normal form, similar to formula_2, which is useful in computing certain matrix functions of formula_2. The matrix formula_23 is also useful in solving the system of linear differential equations formula_27 where formula_2 need not be diagonalizable.
The dimension of the generalized eigenspace corresponding to a given eigenvalue formula_29 is the algebraic multiplicity of formula_29.
Overview and definition.
There are several equivalent ways to define an ordinary eigenvector. For our purposes, an eigenvector formula_31 associated with an eigenvalue formula_29 of an formula_4 × formula_4 matrix formula_2 is a nonzero vector for which formula_36, where formula_37 is the formula_4 × formula_4 identity matrix and formula_40 is the zero vector of length formula_4. That is, formula_31 is in the kernel of the transformation formula_43. If formula_2 has formula_4 linearly independent eigenvectors, then formula_2 is similar to a diagonal matrix formula_47. That is, there exists an invertible matrix formula_48 such that formula_2 is diagonalizable through the similarity transformation formula_50. The matrix formula_47 is called a spectral matrix for formula_2. The matrix formula_48 is called a modal matrix for formula_2. Diagonalizable matrices are of particular interest since matrix functions of them can be computed easily.
On the other hand, if formula_2 does not have formula_4 linearly independent eigenvectors associated with it, then formula_2 is not diagonalizable.
Definition: A vector formula_58 is a generalized eigenvector of rank "m" of the matrix formula_2 and corresponding to the eigenvalue formula_29 if
but
Clearly, a generalized eigenvector of rank 1 is an ordinary eigenvector. Every formula_4 × formula_4 matrix formula_2 has formula_4 linearly independent generalized eigenvectors associated with it and can be shown to be similar to an "almost diagonal" matrix formula_23 in Jordan normal form. That is, there exists an invertible matrix formula_48 such that formula_69. The matrix formula_48 in this case is called a generalized modal matrix for formula_2. If formula_29 is an eigenvalue of algebraic multiplicity formula_73, then formula_2 will have formula_73 linearly independent generalized eigenvectors corresponding to formula_29. These results, in turn, provide a straightforward method for computing certain matrix functions of formula_2.
Note: For an formula_78 matrix formula_2 over a field formula_80 to be expressed in Jordan normal form, all eigenvalues of formula_2 must be in formula_80. That is, the characteristic polynomial formula_83 must factor completely into linear factors. For example, if formula_2 has real-valued elements, then it may be necessary for the eigenvalues and the components of the eigenvectors to have complex values.
The set spanned by all generalized eigenvectors for a given formula_85, forms the generalized eigenspace for formula_85.
Examples.
Here are some examples to illustrate the concept of generalized eigenvectors. Some of the details will be described later.
Example 1.
This example is simple but clearly illustrates the point. This type of matrix is used frequently in textbooks.
Suppose
Then there is only one eigenvalue, formula_88, and its algebraic multiplicity is "m" = 2.
Notice that this matrix is in Jordan normal form but is not diagonal. Hence, this matrix is not diagonalizable. Since there is one superdiagonal entry, there will be one generalized eigenvector of rank greater than 1 (or one could note that the vector space formula_89 is of dimension 2, so there can be at most one generalized eigenvector of rank greater than 1). Alternatively, one could compute the dimension of the nullspace of formula_90 to be "p" = 1, and thus there are "m" – "p" = 1 generalized eigenvectors of rank greater than 1.
The ordinary eigenvector formula_91 is computed as usual (see the eigenvector page for examples). Using this eigenvector, we compute the generalized eigenvector formula_92 by solving
Writing out the values:
This simplifies to
The element formula_96 has no restrictions. The generalized eigenvector of rank 2 is then formula_97, where "a" can have any scalar value. The choice of "a" = 0 is usually the simplest.
Note that
so that formula_92 is a generalized eigenvector,
so that formula_101 is an ordinary eigenvector, and that formula_102 and formula_103 are linearly independent and hence constitute a basis for the vector space formula_89.
Example 2.
This example is more complex than Example 1. Unfortunately, it is a little difficult to construct an interesting example of low order.
The matrix
has "eigenvalues" formula_106 and formula_107 with "algebraic multiplicities" formula_108 and formula_109, but "geometric multiplicities" formula_110 and formula_111.
The "generalized eigenspaces" of formula_2 are calculated below.
formula_113 is the ordinary eigenvector associated with formula_114.
formula_115 is a generalized eigenvector associated with formula_114.
formula_117 is the ordinary eigenvector associated with formula_118.
formula_119 and formula_120 are generalized eigenvectors associated with formula_118.
This results in a basis for each of the "generalized eigenspaces" of formula_2.
Together the two "chains" of generalized eigenvectors span the space of all 5-dimensional column vectors.
An "almost diagonal" matrix formula_23 in "Jordan normal form", similar to formula_2 is obtained as follows:
where formula_48 is a generalized modal matrix for formula_2, the columns of formula_48 are a canonical basis for formula_2, and formula_137.
Jordan chains.
Definition: Let formula_58 be a generalized eigenvector of rank "m" corresponding to the matrix formula_2 and the eigenvalue formula_29. The chain generated by formula_58 is a set of vectors formula_142 given by
Thus, in general,
The vector formula_143, given by (), is a generalized eigenvector of rank "j" corresponding to the eigenvalue formula_29. A chain is a linearly independent set of vectors.
Canonical basis.
Definition: A set of "n" linearly independent generalized eigenvectors is a canonical basis if it is composed entirely of Jordan chains.
Thus, once we have determined that a generalized eigenvector of rank "m" is in a canonical basis, it follows that the "m" − 1 vectors formula_145 that are in the Jordan chain generated by formula_146 are also in the canonical basis.
Let formula_147 be an eigenvalue of formula_2 of algebraic multiplicity formula_149. First, find the ranks (matrix ranks) of the matrices formula_150. The integer formula_151 is determined to be the "first integer" for which formula_152 has rank formula_153 ("n" being the number of rows or columns of formula_2, that is, formula_2 is "n" × "n").
Now define
The variable formula_157 designates the number of linearly independent generalized eigenvectors of rank "k" corresponding to the eigenvalue formula_147 that will appear in a canonical basis for formula_2. Note that
Computation of generalized eigenvectors.
In the preceding sections we have seen techniques for obtaining the formula_4 linearly independent generalized eigenvectors of a canonical basis for the vector space formula_3 associated with an formula_78 matrix formula_2. These techniques can be combined into a procedure:
Example 3.
The matrix
has an eigenvalue formula_175 of algebraic multiplicity formula_176 and an eigenvalue formula_177 of algebraic multiplicity formula_178. We also have formula_179. For formula_180 we have formula_181.
The first integer formula_185 for which formula_186 has rank formula_187 is formula_188.
We now define
Consequently, there will be three linearly independent generalized eigenvectors; one each of ranks 3, 2 and 1. Since formula_180 corresponds to a single chain of three linearly independent generalized eigenvectors, we know that there is a generalized eigenvector formula_193 of rank 3 corresponding to formula_180 such that
but
Equations () and () represent linear systems that can be solved for formula_193. Let
Then
and
Thus, in order to satisfy the conditions () and (), we must have formula_199 and formula_200. No restrictions are placed on formula_201 and formula_202. By choosing formula_203, we obtain
as a generalized eigenvector of rank 3 corresponding to formula_205. Note that it is possible to obtain infinitely many other generalized eigenvectors of rank 3 by choosing different values of formula_201, formula_202 and formula_208, with formula_200. Our first choice, however, is the simplest.
Now using equations (), we obtain formula_115 and formula_113 as generalized eigenvectors of rank 2 and 1, respectively, where
and
The simple eigenvalue formula_177 can be dealt with using standard techniques and has an ordinary eigenvector
A canonical basis for formula_2 is
formula_218 and formula_193 are generalized eigenvectors associated with formula_114, while formula_117 is the ordinary eigenvector associated with formula_118.
This is a fairly simple example. In general, the numbers formula_171 of linearly independent generalized eigenvectors of rank formula_224 will not always be equal. That is, there may be several chains of different lengths corresponding to a particular eigenvalue.
Generalized modal matrix.
Let formula_2 be an "n" × "n" matrix. A generalized modal matrix formula_48 for formula_2 is an "n" × "n" matrix whose columns, considered as vectors, form a canonical basis for formula_2 and appear in formula_48 according to the following rules:
Jordan normal form.
Let formula_3 be an "n"-dimensional vector space; let formula_5 be a linear map in , the set of all linear maps from formula_3 into itself; and let formula_2 be the matrix representation of formula_5 with respect to some ordered basis. It can be shown that if the characteristic polynomial formula_238 of formula_2 factors into linear factors, so that formula_238 has the form
where formula_242 are the distinct eigenvalues of formula_2, then each formula_244 is the algebraic multiplicity of its corresponding eigenvalue formula_13 and formula_2 is similar to a matrix formula_23 in Jordan normal form, where each formula_13 appears formula_244 consecutive times on the diagonal, and the entry directly above each formula_13 (that is, on the superdiagonal) is either 0 or 1: the entry above the first occurrence of each formula_13 is always 0; all other entries on the superdiagonal are 1. All other entries (that is, off the diagonal and superdiagonal) are 0. The matrix formula_23 is as close as one can come to a diagonalization of formula_2. If formula_2 is diagonalizable, then all entries above the diagonal are zero. Note that some textbooks have the ones on the subdiagonal, that is, immediately below the main diagonal instead of on the superdiagonal. The eigenvalues are still on the main diagonal.
Every "n" × "n" matrix formula_2 is similar to a matrix formula_23 in Jordan normal form, obtained through the similarity transformation formula_257, where formula_48 is a generalized modal matrix for formula_2. (See Note above.)
Example 4.
Find a matrix in Jordan normal form that is similar to
Solution: The characteristic equation of formula_2 is formula_262, hence, formula_263 is an eigenvalue of algebraic multiplicity three. Following the procedures of the previous sections, we find that
and
Thus, formula_266 and formula_267, which implies that a canonical basis for formula_2 will contain one linearly independent generalized eigenvector of rank 2 and two linearly independent generalized eigenvectors of rank 1, or equivalently, one chain of two vectors formula_269 and one chain of one vector formula_270. Designating formula_271, we find that
and
where formula_48 is a generalized modal matrix for formula_2, the columns of formula_48 are a canonical basis for formula_2, and formula_137. Note that since generalized eigenvectors themselves are not unique, and since some of the columns of both formula_48 and formula_23 may be interchanged, it follows that both formula_48 and formula_23 are not unique.
Example 5.
In Example 3, we found a canonical basis of linearly independent generalized eigenvectors for a matrix formula_2. A generalized modal matrix for formula_2 is
A matrix in Jordan normal form, similar to formula_2 is
so that formula_137.
Applications.
Matrix functions.
Three of the most fundamental operations which can be performed on square matrices are matrix addition, multiplication by a scalar, and matrix multiplication. These are exactly those operations necessary for defining a polynomial function of an "n" × "n" matrix formula_2. If we recall from basic calculus that many functions can be written as a Maclaurin series, then we can define more general functions of matrices quite easily. If formula_2 is diagonalizable, that is
with
then
and the evaluation of the Maclaurin series for functions of formula_2 is greatly simplified. For example, to obtain any power "k" of formula_2, we need only compute formula_296, premultiply formula_296 by formula_48, and postmultiply the result by formula_299.
Using generalized eigenvectors, we can obtain the Jordan normal form for formula_2 and these results can be generalized to a straightforward method for computing functions of nondiagonalizable matrices. (See Matrix function#Jordan decomposition.)
Differential equations.
Consider the problem of solving the system of linear ordinary differential equations
where
If the matrix formula_2 is a diagonal matrix so that formula_304 for formula_305, then the system () reduces to a system of "n" equations which take the form
In this case, the general solution is given by
In the general case, we try to diagonalize formula_2 and reduce the system () to a system like () as follows. If formula_2 is diagonalizable, we have formula_312, where formula_48 is a modal matrix for formula_2. Substituting formula_315, equation () takes the form formula_316, or
where
The solution of () is
The solution formula_321 of () is then obtained using the relation ().
On the other hand, if formula_2 is not diagonalizable, we choose formula_48 to be a generalized modal matrix for formula_2, such that formula_257 is the Jordan normal form of formula_2. The system formula_327 has the form
where the formula_147 are the eigenvalues from the main diagonal of formula_23 and the formula_330 are the ones and zeros from the superdiagonal of formula_23. The system () is often more easily solved than (). We may solve the last equation in () for formula_332, obtaining formula_333. We then substitute this solution for formula_332 into the next to last equation in () and solve for formula_335. Continuing this procedure, we work through () from the last equation to the first, solving the entire system for formula_336. The solution formula_321 is then obtained using the relation ().