Generalized eigenvector
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In linear algebra, a generalized eigenvector of an n × n matrix is a vector which satisfies certain criteria which are more relaxed than those for an (ordinary) eigenvector.[1]
Let be an n-dimensional vector space; let
be a linear map in L(V), the set of all linear maps from
into itself; and let
be the matrix representation of
with respect to some ordered basis.
There may not always exist a full set of n linearly independent eigenvectors of that form a complete basis for
. That is, the matrix
may not be diagonalizable.[2][3] This happens when the algebraic multiplicity of at least one eigenvalue
is greater than its geometric multiplicity (the nullity of the matrix
, or the dimension of its nullspace). In this case,
is called a defective eigenvalue and
is called a defective matrix.[4]
A generalized eigenvector corresponding to
, together with the matrix
generate a Jordan chain of linearly independent generalized eigenvectors which form a basis for an invariant subspace of
.[5][6][7]
Using generalized eigenvectors, a set of linearly independent eigenvectors of can be extended, if necessary, to a complete basis for
.[8] This basis can be used to determine an "almost diagonal matrix"
in Jordan normal form, similar to
, which is useful in computing certain matrix functions of
.[9] The matrix
is also useful in solving the system of linear differential equations
where
need not be diagonalizable.[10][11]
Contents
Overview and definition
There are several equivalent ways to define an ordinary eigenvector.[12][13][14][15][16][17][18][19] For our purposes, an eigenvector associated with an eigenvalue
of an
×
matrix
is a nonzero vector for which
, where
is the
×
identity matrix and
is the zero vector of length
.[20] That is,
is in the kernel of the transformation
. If
has
linearly independent eigenvectors, then
is similar to a diagonal matrix
. That is, there exists an invertible matrix
such that
is diagonalizable through the similarity transformation
.[21][22] The matrix
is called a spectral matrix for
. The matrix
is called a modal matrix for
.[23] Diagonalizable matrices are of particular interest since matrix functions of them can be computed easily.[24]
On the other hand, if does not have
linearly independent eigenvectors associated with it, then
is not diagonalizable.[25][26]
Definition: A vector is a generalized eigenvector of rank m of the matrix
and corresponding to the eigenvalue
if
but
Clearly, a generalized eigenvector of rank 1 is an ordinary eigenvector.[28] Every ×
matrix
has
linearly independent generalized eigenvectors associated with it and can be shown to be similar to an "almost diagonal" matrix
in Jordan normal form.[29] That is, there exists an invertible matrix
such that
.[30] The matrix
in this case is called a generalized modal matrix for
.[31] If
is an eigenvalue of algebraic multiplicity
, then
will have
linearly independent generalized eigenvectors corresponding to
.[32] These results, in turn, provide a straightforward method for computing certain matrix functions of
.[33]
The set spanned by all generalized eigenvectors for a given , forms the generalized eigenspace for
.[34]
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.[35][36][37] Suppose
Then there is only one eigenvalue, , and its algebraic multiplicity is m = 2.
There are several ways to see that there will be only one generalized eigenvector. The easiest is to 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 (or one could note that the vector space is of dimension 2, so there can be only one generalized eigenvector). Alternatively, one could compute the dimension of the nullspace of
to be p = 1, and thus there are m – p = 1 generalized eigenvectors. (See the nullspace page.)
Computing the ordinary eigenvector is left to the reader. (See the eigenvector page for examples.) Using this eigenvector, we compute the generalized eigenvector
by solving
Writing out the values:
This simplifies to
The element has no restrictions. The generalized eigenvector is then
, where a can have any scalar value. The choice of a = 0 is usually the simplest.
Note that
so that is a generalized eigenvector,
so that is an ordinary eigenvector, and that
and
are linearly independent and hence constitute a basis for the vector space
.
Example 2
This example is more complex than Example 1. Unfortunately, it is a little difficult to construct an interesting example of low order.[38] The matrix
has eigenvalues and
with algebraic multiplicities
and
, but geometric multiplicities
and
.
The generalized eigenspaces of are calculated below.
is the ordinary eigenvector associated with
.
is a generalized eigenvector associated with
.
is the ordinary eigenvector associated with
.
and
are generalized eigenvectors associated with
.
This results in a basis for each of the generalized eigenspaces of . Together the two chains of generalized eigenvectors span the space of all 5-dimensional column vectors.
An "almost diagonal" matrix in Jordan normal form, similar to
is obtained as follows:
where is a generalized modal matrix for
, the columns of
are a canonical basis for
, and
.[39]
Jordan chains
Definition: Let be a generalized eigenvector of rank m corresponding to the matrix
and the eigenvalue
. The chain generated by
is a set of vectors
given by
-
(1)
-
Thus, in general,
-
(2)
The vector , given by (2), is a generalized eigenvector of rank j corresponding to the eigenvalue
. A chain is a linearly independent set of vectors.[40]
Canonical basis
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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 that are in the Jordan chain generated by
are also in the canonical basis.[41]
Let be an eigenvalue of
of algebraic multiplicity
. First, find the ranks (matrix ranks) of the matrices
. The integer
is determined to be the first integer for which
has rank
(n being the number of rows or columns of
, that is,
is n × n).
Now define
The variable designates the number of linearly independent generalized eigenvectors of rank k corresponding to the eigenvalue
that will appear in a canonical basis for
. Note that
.[42]
Computation of generalized eigenvectors
In the preceding sections we have seen techniques for obtaining the n linearly independent generalized eigenvectors of a canonical basis for the vector space associated with an n × n matrix
. These techniques can be combined into a procedure:
- Solve the characteristic equation of
for eigenvalues
and their algebraic multiplicities
;
- For each
- Determine
;
- Determine
;
- Determine
for
;
- Determine each Jordan chain for
;
- Determine
Example 3
The matrix
has an eigenvalue of algebraic multiplicity
and an eigenvalue
of algebraic multiplicity
. We also have n = 4. For
we have
.
The first integer for which
has rank
is
.
We now define
Consequently, there will be three linearly independent generalized eigenvectors; one each of ranks 3, 2 and 1. Since corresponds to a single chain of three linearly independent generalized eigenvectors, we know that there is a generalized eigenvector
of rank 3 corresponding to
such that
-
(3)
but
-
(4)
Equations (3) and (4) represent linear systems that can be solved for . Let
Then
and
Thus, in order to satisfy the conditions (3) and (4), we must have and
. No restrictions are placed on
and
. By choosing
, we obtain
as a generalized eigenvector of rank 3 corresponding to . Note that it is possible to obtain infinitely many other generalized eigenvectors of rank 3 by choosing different values of
,
and
, with
. Our first choice, however, is the simplest.[43]
Now using equations (1), we obtain and
as generalized eigenvectors of rank 2 and 1 respectively, where
and
The simple eigenvalue can be dealt with using standard techniques and has an ordinary eigenvector
A canonical basis for is
and
are generalized eigenvectors associated with
.
is the ordinary eigenvector associated with
.
It should be noted that this is a fairly simple example. In general, the numbers of linearly independent generalized eigenvectors of rank k will not always be equal. That is, there may be several chains of different lengths corresponding to a particular eigenvalue.[44]
Generalized modal matrix
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Let be an n × n matrix. A generalized modal matrix
for
is an n × n matrix whose columns, considered as vectors, form a canonical basis for
and appear in
according to the following rules:
- All Jordan chains consisting of one vector (that is, one vector in length) appear in the first columns of
.
- All vectors of one chain appear together in adjacent columns of
.
- Each chain appears in
in order of increasing rank (that is, the generalized eigenvector of rank 1 appears before the generalized eigenvector of rank 2 of the same chain, which appears before the generalized eigenvector of rank 3 of the same chain, etc.).[45]
Jordan Normal Form
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Let be an n-dimensional vector space; let
be a linear map in L(V), the set of all linear maps from
into itself; and let
be the matrix representation of
with respect to some ordered basis. It can be shown that if the characteristic polynomial
of
factors into linear factors, so that
has the form
where are the distinct eigenvalues of
, then each
is the algebraic multiplicity of its corresponding eigenvalue
and
is similar to a matrix
in Jordan normal form, where each
appears
consecutive times on the diagonal, and each entry above each
(that is, on the superdiagonal) is either 0 or 1; the entry above the first occurrence of each
is always 0. All other entries are zero. The matrix
is as close as one can come to a diagonalization of
. If
is diagonalizable, then all entries above the diagonal are zero.[46] 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.[47][48]
Every n × n matrix is similar to a matrix
in Jordan normal form, obtained through the similarity transformation
, where
is a generalized modal matrix for
.[49]
Example 4
Find a matrix in Jordan normal form that is similar to
Solution: The characteristic equation of is
, hence,
is an eigenvalue of algebraic multiplicity three. Following the procedures of the previous sections, we find that
and
Thus, and
, which implies that a canonical basis for
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
and one chain of one vector
. Designating
, we find that
and
where is a generalized modal matrix for
, the columns of
are a canonical basis for
, and
.[50] Note that since generalized eigenvectors themselves are not unique, and since some of the columns of both
and
may be interchanged, it follows that both
and
are not unique.[51]
Example 5
In Example 3, we found a canonical basis of linearly independent generalized eigenvectors for a matrix . A generalized modal matrix for
is
A matrix in Jordan normal form, similar to is
so that .
Applications
Matrix functions
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Three of the most fundamental operations which can be performed on square matrices are matrix addition, multiplication by a scalar, and matrix multiplication.[52] These are exactly those operations necessary for defining a polynomial function of an n × n matrix .[53] 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.[54] If
is diagonalizable, that is
with
then
and the evaluation of the Maclaurin series for functions of is greatly simplified.[55] For example, to obtain any power k of
, we need only compute
, premultiply
by
, and postmultiply the result by
.[56]
Using generalized eigenvectors, we can obtain the Jordan normal form for and these results can be generalized to a straightforward method for computing functions of nondiagonalizable matrices.[57] (See Matrix function#Jordan decomposition.)
Differential equations
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Consider the problem of solving the system of linear ordinary differential equations
-
(5)
where
and
If the matrix is a diagonal matrix so that
for
, then the system (5) reduces to a system of n equations which take the form
-
(6)
In this case, the general solution is given by
In the general case, we try to diagonalize and reduce the system (5) to a system like (6) as follows. If
is diagonalizable, we have
, where
is a modal matrix for
. Substituting
, equation (5) takes the form
, or
-
(7)
where
-
(8)
The solution of (7) is
The solution of (5) is then obtained using the relation (8).[58]
On the other hand, if is not diagonalizable, we choose
to be a generalized modal matrix for
, such that
is the Jordan normal form of
. The system
has the form
-
(9)
where the are the eigenvalues from the main diagonal of
and the
are the ones and zeros from the superdiagonal of
. The system (9) is often more easily solved than (5). We may solve the last equation in (9) for
, obtaining
. We then substitute this solution for
into the next to last equation in (9) and solve for
. Continuing this procedure, we work through (9) from the last equation to the first, solving the entire system for
. The solution
is then obtained using the relation (8).[59]
Notes
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References
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- ↑ Kreyszig (1972, pp. 273,684)
- ↑ Nering (1970, p. 104)
- ↑ Burden & Faires (1993, p. 401)
- ↑ Beauregard & Fraleigh (1973, pp. 270-274)
- ↑ Bronson (1970, pp. 179-183)
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- ↑ Nering (1970, p. 118)
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- ↑ Bronson (1970, p. 185)
- ↑ Bronson (1970, pp. 209-218)
- ↑ Beauregard & Fraleigh (1973, pp. 274-275)
- ↑ Beauregard & Fraleigh (1973, p. 317)