In linear algebra, a generalized eigenvector matrix is a vector which satisfies certain criteria which are more relaxed than those for an (ordinary) eigenvector.

-dimensional vector space be the matrix representation with respect to some ordered basis.

linearly independent may not be diagonalizable. This happens when the algebraic multiplicity of at least one eigenvalue is greater than its geometric multiplicity (the nullity, or the dimension of its nullspace is called a defective eigenvalue is called a defective matrix.

generate a Jordan chain of linearly independent generalized eigenvectors which form a basis for an invariant subspace.

. in Jordan normal form, similar, which is useful in computing certain matrix functions. is also useful in solving the system of linear differential equations need not be diagonalizable.

.

Overview and definition

There are several equivalent ways to define an ordinary eigenvector. identity matrix is the zero vector. is in the kernel of the transformation is similar to a diagonal matrix. That is, there exists an invertible matrix. is called a spectral matrix is called a modal matrix. Diagonalizable matrices are of particular interest since matrix functions of them can be computed easily.

is not diagonalizable.

Definition: is a generalized eigenvector of rank m if

but

Clearly, a generalized eigenvector of rank 1 is an ordinary eigenvector. in Jordan normal form.. in this case is called a generalized modal matrix...

over a field. That is, the characteristic polynomial 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 forms the generalized eigenspace.

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

.

Notice that this matrix is in Jordan normal form but is not diagonal. Hence, this matrix is not diagonalizable. Since there is one superdiagonal 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 generalized eigenvectors of rank greater than 1.

is computed as usual (see the eigenvector by solving

Writing out the values:

This simplifies to

, where a can have any scalar value. The choice of a = 0 is usually the simplest.

Note that

is a generalized eigenvector, because

.

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 with algebraic multiplicities, but geometric multiplicities.

The generalized eigenspaces.

This results in a basis for each of the generalized eigenspaces. Together the two chains of generalized eigenvectors span the space of all 5-dimensional column vectors.

in Jordan normal form is obtained as follows:

is a generalized modal matrix are a canonical basis.

Jordan chains

Definition: be a generalized eigenvector of rank m. The chain generated by given by

. Thus, in general,

, given by (), is a generalized eigenvector of rank j. 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 are also in the canonical basis.

. First, find the ranks is determined to be the first integer (n is n × n).

Now define

designates the number of linearly independent generalized eigenvectors of rank k. Note that

1. .

Computation of generalized eigenvectors

. These techniques can be combined into a procedure:

1. Solve the characteristic equation;

   1. ;

   2. ;

   3. ;

   4. ;

Example 3

The matrix

.

.

We now define

such that

but

Equations () and () represent linear systems. Let

Then

and

Thus, in order to satisfy the conditions () and (, we obtain

. Our first choice, however, is the simplest.

Now using equations ( 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

is

.

will not always be equal. That is, there may be several chains of different lengths corresponding to a particular eigenvalue.

Generalized modal matrix

be an n × n matrix. A generalized modal matrix is an n × n according to the following rules:

• .

• .

• 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.).

Jordan normal form

An example of a matrix in Jordan normal form.The red blocks are called Jordan blocks. be an n be a linear map in with respect to some ordered basis. It can be shown that if the characteristic polynomial has the form

in Jordan normal form (that is, on the superdiagonal 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. (See Note above.)

Example 4

Find a matrix in Jordan normal form that is similar to

Solution: is an eigenvalue of algebraic multiplicity three. Following the procedures of the previous sections, we find that

and

, we find that

and

. are not unique.

Example 5

In Example 3 is

is

.

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. 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. is diagonalizable, that is

with

then

is greatly simplified. For example, to obtain any power k.

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

1. and

, then the system () reduces to a system of n equations which take the form

In this case, the general solution is given by

and reduce the system () to a system like (, equation (, or

where

The solution of () is

of () is then obtained using the relation ().

has the form

. The system () is often more easily solved than (). We may solve the last equation in ( into the next to last equation in (. Continuing this procedure, we work through ( is then obtained using the relation ().

Lemma:

1. ,

these functions solve the system of equations,

Proof:

Define

,

1. .

and so

as required.

Notes

References