Generalized eigenvector

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In linear algebra, a generalized eigenvector of an matrix is a vector which satisfies certain criteria which are more relaxed than those for an (ordinary) eigenvector.[1]

Let be an -dimensional vector space and let be the matrix representation of a linear map from to with respect to some ordered basis.

There may not always exist a full set of 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]

The dimension of the generalized eigenspace corresponding to a given eigenvalue is the algebraic multiplicity of .[12]

Overview and definition

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There are several equivalent ways to define an ordinary eigenvector.[13][14][15][16][17][18][19][20] 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  .[21] 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  .[22][23] The matrix   is called a spectral matrix for  . The matrix   is called a modal matrix for  .[24] Diagonalizable matrices are of particular interest since matrix functions of them can be computed easily.[25]

On the other hand, if   does not have   linearly independent eigenvectors associated with it, then   is not diagonalizable.[26][27]

Definition: A vector   is a generalized eigenvector of rank m of the matrix   and corresponding to the eigenvalue   if

 

but

  [28]

Clearly, a generalized eigenvector of rank 1 is an ordinary eigenvector.[29] 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.[30] That is, there exists an invertible matrix   such that  .[31] The matrix   in this case is called a generalized modal matrix for  .[32] If   is an eigenvalue of algebraic multiplicity  , then   will have   linearly independent generalized eigenvectors corresponding to  .[33] These results, in turn, provide a straightforward method for computing certain matrix functions of  .[34]

Note: For an   matrix   over a field   to be expressed in Jordan normal form, all eigenvalues of   must be in  . That is, the characteristic polynomial   must factor completely into linear factors;   must be an algebraically closed field. For example, if   has real-valued elements, then it may be necessary for the eigenvalues and the components of the eigenvectors to have complex values.[35][36][37]

The set spanned by all generalized eigenvectors for a given   forms the generalized eigenspace for  .[38]

Examples

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Here are some examples to illustrate the concept of generalized eigenvectors. Some of the details will be described later.

Example 1

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This example is simple but clearly illustrates the point. This type of matrix is used frequently in textbooks.[39][40][41] Suppose

 

Then there is only one eigenvalue,  , and its algebraic multiplicity is  .

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   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   to be  , and thus there are   generalized eigenvectors of rank greater than 1.

The ordinary eigenvector   is computed as usual (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 of rank 2 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, because

 

so that   is an ordinary eigenvector, and that   and   are linearly independent and hence constitute a basis for the vector space  .

Example 2

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This example is more complex than Example 1. Unfortunately, it is a little difficult to construct an interesting example of low order.[42] 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  .[43]

Jordan chains

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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)

where   is always an ordinary eigenvector with a given eigenvalue  . 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.[44]

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.[45]

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

 .[46]

Computation of generalized eigenvectors

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In the preceding sections we have seen techniques for obtaining the   linearly independent generalized eigenvectors of a canonical basis for the vector space   associated with an   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  ;

Example 3

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The matrix

 

has an eigenvalue   of algebraic multiplicity   and an eigenvalue   of algebraic multiplicity  . We also have  . 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.[47]

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  , while   is the ordinary eigenvector associated with  .

This is a fairly simple example. In general, the numbers   of linearly independent generalized eigenvectors of rank   will not always be equal. That is, there may be several chains of different lengths corresponding to a particular eigenvalue.[48]

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.).[49]

Jordan normal form

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An example of a matrix in Jordan normal form. The grey blocks are called Jordan blocks.

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 the entry directly above each   (that is, on the superdiagonal) is either 0 or 1: in each block the entry above the first occurrence of each   is always 0 (except in the first block); all other entries on the superdiagonal are 1. All other entries (that is, off the diagonal and superdiagonal) are 0. (But no ordering is imposed among the eigenvalues, or among the blocks for a given eigenvalue.) The matrix   is as close as one can come to a diagonalization of  . If   is diagonalizable, then all entries above the diagonal are zero.[50] 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.[51][52]

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  .[53] (See Note above.)

Example 4

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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  .[54] 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.[55]

Example 5

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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

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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.[56] These are exactly those operations necessary for defining a polynomial function of an n × n matrix  .[57] 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.[58] If   is diagonalizable, that is

 

with

 

then

 

and the evaluation of the Maclaurin series for functions of   is greatly simplified.[59] For example, to obtain any power k of  , we need only compute  , premultiply   by  , and postmultiply the result by  .[60]

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.[61] (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).[62]

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).[63]

Lemma:

Given the following chain of generalized eigenvectors of length  

 
 
 
 
 ,

these functions solve the system of equations,

 

Proof:

Define

 
 

Then, as   and  ,

 .

On the other hand we have,   and so

 
 
 
 
 

as required.

Notes

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  1. ^ Bronson (1970, p. 189)
  2. ^ Beauregard & Fraleigh (1973, p. 310)
  3. ^ Nering (1970, p. 118)
  4. ^ Golub & Van Loan (1996, p. 316)
  5. ^ Beauregard & Fraleigh (1973, p. 319)
  6. ^ Bronson (1970, pp. 194–195)
  7. ^ Golub & Van Loan (1996, p. 311)
  8. ^ Bronson (1970, p. 196)
  9. ^ Bronson (1970, p. 189)
  10. ^ Beauregard & Fraleigh (1973, pp. 316–318)
  11. ^ Nering (1970, p. 118)
  12. ^ Bronson (1970, p. 196)
  13. ^ Anton (1987, pp. 301–302)
  14. ^ Beauregard & Fraleigh (1973, p. 266)
  15. ^ Burden & Faires (1993, p. 401)
  16. ^ Golub & Van Loan (1996, pp. 310–311)
  17. ^ Harper (1976, p. 58)
  18. ^ Herstein (1964, p. 225)
  19. ^ Kreyszig (1972, pp. 273, 684)
  20. ^ Nering (1970, p. 104)
  21. ^ Burden & Faires (1993, p. 401)
  22. ^ Beauregard & Fraleigh (1973, pp. 270–274)
  23. ^ Bronson (1970, pp. 179–183)
  24. ^ Bronson (1970, p. 181)
  25. ^ Bronson (1970, p. 179)
  26. ^ Beauregard & Fraleigh (1973, pp. 270–274)
  27. ^ Bronson (1970, pp. 179–183)
  28. ^ Bronson (1970, p. 189)
  29. ^ Bronson (1970, pp. 190, 202)
  30. ^ Bronson (1970, pp. 189, 203)
  31. ^ Bronson (1970, pp. 206–207)
  32. ^ Bronson (1970, p. 205)
  33. ^ Bronson (1970, p. 196)
  34. ^ Bronson (1970, pp. 189, 209–215)
  35. ^ Golub & Van Loan (1996, p. 316)
  36. ^ Herstein (1964, p. 259)
  37. ^ Nering (1970, p. 118)
  38. ^ Nering (1970, p. 118)
  39. ^ Nering (1970, p. 118)
  40. ^ Herstein (1964, p. 261)
  41. ^ Beauregard & Fraleigh (1973, p. 310)
  42. ^ Nering (1970, pp. 122, 123)
  43. ^ Bronson (1970, pp. 189–209)
  44. ^ Bronson (1970, pp. 194–195)
  45. ^ Bronson (1970, pp. 196, 197)
  46. ^ Bronson (1970, pp. 197, 198)
  47. ^ Bronson (1970, pp. 190–191)
  48. ^ Bronson (1970, pp. 197–198)
  49. ^ Bronson (1970, p. 205)
  50. ^ Beauregard & Fraleigh (1973, p. 311)
  51. ^ Cullen (1966, p. 114)
  52. ^ Franklin (1968, p. 122)
  53. ^ Bronson (1970, p. 207)
  54. ^ Bronson (1970, pp. 208)
  55. ^ Bronson (1970, p. 206)
  56. ^ Beauregard & Fraleigh (1973, pp. 57–61)
  57. ^ Bronson (1970, p. 104)
  58. ^ Bronson (1970, p. 105)
  59. ^ Bronson (1970, p. 184)
  60. ^ Bronson (1970, p. 185)
  61. ^ Bronson (1970, pp. 209–218)
  62. ^ Beauregard & Fraleigh (1973, pp. 274–275)
  63. ^ Beauregard & Fraleigh (1973, p. 317)

References

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  • Anton, Howard (1987), Elementary Linear Algebra (5th ed.), New York: Wiley, ISBN 0-471-84819-0
  • Axler, Sheldon (1997). Linear Algebra Done Right (2nd ed.). Springer. ISBN 978-0-387-98258-8.
  • Beauregard, Raymond A.; Fraleigh, John B. (1973), A First Course In Linear Algebra: with Optional Introduction to Groups, Rings, and Fields, Boston: Houghton Mifflin Co., ISBN 0-395-14017-X
  • Bronson, Richard (1970), Matrix Methods: An Introduction, New York: Academic Press, LCCN 70097490
  • Burden, Richard L.; Faires, J. Douglas (1993), Numerical Analysis (5th ed.), Boston: Prindle, Weber and Schmidt, ISBN 0-534-93219-3
  • Cullen, Charles G. (1966), Matrices and Linear Transformations, Reading: Addison-Wesley, LCCN 66021267
  • Franklin, Joel N. (1968), Matrix Theory, Englewood Cliffs: Prentice-Hall, LCCN 68016345
  • Golub, Gene H.; Van Loan, Charles F. (1996), Matrix Computations (3rd ed.), Baltimore: Johns Hopkins University Press, ISBN 0-8018-5414-8
  • Harper, Charlie (1976), Introduction to Mathematical Physics, New Jersey: Prentice-Hall, ISBN 0-13-487538-9
  • Herstein, I. N. (1964), Topics In Algebra, Waltham: Blaisdell Publishing Company, ISBN 978-1114541016
  • Kreyszig, Erwin (1972), Advanced Engineering Mathematics (3rd ed.), New York: Wiley, ISBN 0-471-50728-8
  • Nering, Evar D. (1970), Linear Algebra and Matrix Theory (2nd ed.), New York: Wiley, LCCN 76091646