Generalized eigenvector

Short description: Vector satisfying some of the criteria of an eigenvector

Not to be confused with Generalized eigenvalue problem.

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

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

There may not always exist a full set of n linearly independent eigenvectors of A that form a complete basis for V. That is, the matrix A may not be diagonalizable.[2][3] This happens when the algebraic multiplicity of at least one eigenvalue λi is greater than its geometric multiplicity (the nullity of the matrix (A−λiI), or the dimension of its nullspace). In this case, λi is called a defective eigenvalue and A is called a defective matrix.[4]

A generalized eigenvector xi corresponding to λi, together with the matrix (A−λiI) generate a Jordan chain of linearly independent generalized eigenvectors which form a basis for an invariant subspace of V.[5][6][7]

Using generalized eigenvectors, a set of linearly independent eigenvectors of A can be extended, if necessary, to a complete basis for V.[8] This basis can be used to determine an "almost diagonal matrix" J in Jordan normal form, similar to A, which is useful in computing certain matrix functions of A.[9] The matrix J is also useful in solving the system of linear differential equations 𝐱′=A𝐱, where A need not be diagonalizable.[10][11]

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

## Contents

* 1 Overview and definition
* 2 Examples
* 2.1 Example 1
* 2.2 Example 2
* 3 Jordan chains
* 4 Canonical basis
* 5 Computation of generalized eigenvectors
* 5.1 Example 3
* 6 Generalized modal matrix
* 7 Jordan normal form
* 7.1 Example 4
* 7.2 Example 5
* 8 Applications
* 8.1 Matrix functions
* 8.2 Differential equations
* 9 Notes
* 10 References

## Overview and definition

There are several equivalent ways to define an ordinary eigenvector.[13][14][15][16][17][18][19][20] For our purposes, an eigenvector 𝐮 associated with an eigenvalue λ of an n × n matrix A is a nonzero vector for which (A−λI)𝐮=𝟎, where I is the n × n identity matrix and 𝟎 is the zero vector of length n.[21] That is, 𝐮 is in the kernel of the transformation (A−λI). If A has n linearly independent eigenvectors, then A is similar to a diagonal matrix D. That is, there exists an invertible matrix M such that A is diagonalizable through the similarity transformation D=M−1AM.[22][23] The matrix D is called a spectral matrix for A. The matrix M is called a modal matrix for A.[24] Diagonalizable matrices are of particular interest since matrix functions of them can be computed easily.[25]

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

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

(A−λI)m𝐱m=𝟎

but

(A−λI)m−1𝐱m≠𝟎. [28]

Clearly, a generalized eigenvector of rank 1 is an ordinary eigenvector.[29] Every n × n matrix A has n linearly independent generalized eigenvectors associated with it and can be shown to be similar to an "almost diagonal" matrix J in Jordan normal form.[30] That is, there exists an invertible matrix M such that J=M−1AM.[31] The matrix M in this case is called a generalized modal matrix for A.[32] If λ is an eigenvalue of algebraic multiplicity μ, then A will have μ linearly independent generalized eigenvectors corresponding to λ.[33] These results, in turn, provide a straightforward method for computing certain matrix functions of A.[34]

Note: For an n×n matrix A over a field F to be expressed in Jordan normal form, all eigenvalues of A must be in F. That is, the characteristic polynomial f(x) must factor completely into linear factors; F must be an algebraically closed field. For example, if A 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 λ forms the generalized eigenspace for λ.[38]

## 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.[39][40][41] Suppose

A=(1101).

Then there is only one eigenvalue, λ=1, 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 V 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 A−λI to be p=1, and thus there are m−p=1 generalized eigenvectors of rank greater than 1.

The ordinary eigenvector 𝐯1=(10) is computed as usual (see the eigenvector page for examples). Using this eigenvector, we compute the generalized eigenvector 𝐯2 by solving

(A−λI)𝐯2=𝐯1.

Writing out the values:

((1101)−1(1001))(v21v22)=(0100)(v21v22)=(10).

This simplifies to

v22=1.

The element v21 has no restrictions. The generalized eigenvector of rank 2 is then 𝐯2=(a1), where a can have any scalar value. The choice of a = 0 is usually the simplest.

Note that

(A−λI)𝐯2=(0100)(a1)=(10)=𝐯1,

so that 𝐯2 is a generalized eigenvector, because

(A−λI)2𝐯2=(A−λI)[(A−λI)𝐯2]=(A−λI)𝐯1=(0100)(10)=(00)=𝟎,

so that 𝐯1 is an ordinary eigenvector, and that 𝐯1 and 𝐯2 are linearly independent and hence constitute a basis for the vector space V.

### Example 2

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

A=(1000031000632001063201510632)

has eigenvalues λ1=1 and λ2=2 with algebraic multiplicities μ1=2 and μ2=3, but geometric multiplicities γ1=1 and γ2=1.

The generalized eigenspaces of A are calculated below. 𝐱1 is the ordinary eigenvector associated with λ1. 𝐱2 is a generalized eigenvector associated with λ1. 𝐲1 is the ordinary eigenvector associated with λ2. 𝐲2 and 𝐲3 are generalized eigenvectors associated with λ2.

(A−1I)𝐱1=(0000030000631001063101510631)(03−99−3)=(00000)=𝟎,

(A−1I)𝐱2=(0000030000631001063101510631)(1−1530−1−45)=(03−99−3)=𝐱1,

(A−2I)𝐲1=(−100003−1000630001063001510630)(00009)=(00000)=𝟎,

(A−2I)𝐲2=(−100003−1000630001063001510630)(00030)=(00009)=𝐲1,

(A−2I)𝐲3=(−100003−1000630001063001510630)(001−20)=(00030)=𝐲2.

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

{𝐱1,𝐱2}={(03−99−3),(1−1530−1−45)},{𝐲1,𝐲2,𝐲3}={(00009),(00030),(001−20)}.

An "almost diagonal" matrix J in Jordan normal form, similar to A is obtained as follows:

M=(𝐱1𝐱2𝐲1𝐲2𝐲3)=(010003−15000−9300019−103−2−3−45900),
J=(1100001000002100002100002),

where M is a generalized modal matrix for A, the columns of M are a canonical basis for A, and AM=MJ.[43]

## Jordan chains

Definition: Let 𝐱m be a generalized eigenvector of rank m corresponding to the matrix A and the eigenvalue λ. The chain generated by 𝐱m is a set of vectors {𝐱m,𝐱m−1,…,𝐱1} given by

𝐱m−1=(A−λI)𝐱m,
𝐱m−2=(A−λI)2𝐱m=(A−λI)𝐱m−1,
𝐱m−3=(A−λI)3𝐱m=(A−λI)𝐱m−2,

𝐱1=(A−λI)m−1𝐱m=(A−λI)𝐱2.

|
|

(1)

where 𝐱1 is always an ordinary eigenvector with a given eigenvalue λ. Thus, in general,

𝐱j=(A−λI)m−j𝐱m=(A−λI)𝐱j+1(j=1,2,…,m−1).

|
|

(2)

The vector 𝐱j, given by (2), is a generalized eigenvector of rank j corresponding to the eigenvalue λ. A chain is a linearly independent set of vectors.[44]

## 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 𝐱m−1,𝐱m−2,…,𝐱1 that are in the Jordan chain generated by 𝐱m are also in the canonical basis.[45]

Let λi be an eigenvalue of A of algebraic multiplicity μi. First, find the ranks (matrix ranks) of the matrices (A−λiI),(A−λiI)2,…,(A−λiI)mi. The integer mi is determined to be the first integer for which (A−λiI)mi has rank n−μi (n being the number of rows or columns of A, that is, A is n × n).

Now define

ρk=rank⁡(A−λiI)k−1−rank⁡(A−λiI)k(k=1,2,…,mi).

The variable ρk designates the number of linearly independent generalized eigenvectors of rank k corresponding to the eigenvalue λi that will appear in a canonical basis for A. Note that

rank⁡(A−λiI)0=rank⁡(I)=n.[46]

## 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 V associated with an n×n matrix A. These techniques can be combined into a procedure:

Solve the characteristic equation of A for eigenvalues λi and their algebraic multiplicities μi;
For each λi:

Determine n−μi;
Determine mi;
Determine ρk for (k=1,…,mi);
Determine each Jordan chain for λi;

### Example 3

The matrix

A=(51−24052200530004)

has an eigenvalue λ1=5 of algebraic multiplicity μ1=3 and an eigenvalue λ2=4 of algebraic multiplicity μ2=1. We also have n=4. For λ1 we have n−μ1=4−3=1.

(A−5I)=(01−2400220003000−1),rank⁡(A−5I)=3.
(A−5I)2=(002−80004000−30001),rank⁡(A−5I)2=2.
(A−5I)3=(00014000−40003000−1),rank⁡(A−5I)3=1.

The first integer m1 for which (A−5I)m1 has rank n−μ1=1 is m1=3.

We now define

ρ3=rank⁡(A−5I)2−rank⁡(A−5I)3=2−1=1,
ρ2=rank⁡(A−5I)1−rank⁡(A−5I)2=3−2=1,
ρ1=rank⁡(A−5I)0−rank⁡(A−5I)1=4−3=1.

Consequently, there will be three linearly independent generalized eigenvectors; one each of ranks 3, 2 and 1. Since λ1 corresponds to a single chain of three linearly independent generalized eigenvectors, we know that there is a generalized eigenvector 𝐱3 of rank 3 corresponding to λ1 such that

(A−5I)3𝐱3=𝟎

|
|

(3)

but

(A−5I)2𝐱3≠𝟎.

|
|

(4)

Equations (3) and (4) represent linear systems that can be solved for 𝐱3. Let

𝐱3=(x31x32x33x34).

Then

(A−5I)3𝐱3=(00014000−40003000−1)(x31x32x33x34)=(14x34−4x343x34−x34)=(0000)

and

(A−5I)2𝐱3=(002−80004000−30001)(x31x32x33x34)=(2x33−8x344x34−3x34x34)≠(0000).

Thus, in order to satisfy the conditions (3) and (4), we must have x34=0 and x33≠0. No restrictions are placed on x31 and x32. By choosing x31=x32=x34=0,x33=1, we obtain

𝐱3=(0010)

as a generalized eigenvector of rank 3 corresponding to λ1=5. Note that it is possible to obtain infinitely many other generalized eigenvectors of rank 3 by choosing different values of x31, x32 and x33, with x33≠0. Our first choice, however, is the simplest.[47]

Now using equations (1), we obtain 𝐱2 and 𝐱1 as generalized eigenvectors of rank 2 and 1, respectively, where

𝐱2=(A−5I)𝐱3=(−2200),

and

𝐱1=(A−5I)𝐱2=(2000).

The simple eigenvalue λ2=4 can be dealt with using standard techniques and has an ordinary eigenvector

𝐲1=(−144−31).

A canonical basis for A is

{𝐱3,𝐱2,𝐱1,𝐲1}={(0010)(−2200)(2000)(−144−31)}.

𝐱1,𝐱2 and 𝐱3 are generalized eigenvectors associated with λ1, while 𝐲1 is the ordinary eigenvector associated with λ2.

This is a fairly simple example. In general, the numbers ρk 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.[48]

## Generalized modal matrix

Let A be an n × n matrix. A generalized modal matrix M for A is an n × n matrix whose columns, considered as vectors, form a canonical basis for A and appear in M according to the following rules:

* All Jordan chains consisting of one vector (that is, one vector in length) appear in the first columns of M.
* All vectors of one chain appear together in adjacent columns of M.
* Each chain appears in M 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

[⌜λ11λ1λ1⌝⌜λ21λ2⌝[λ3]⋱⌜λn1λn⌝⌜λ11λ11λ1⌝⌜λ21λ2⌝[λ3]⋱⌜λn1λn⌝⌞λ11λ1λ1⌟⌜λ21λ2⌝[λ3]⋱⌜λn1λn⌝⌜λ11λ11λ1⌝⌜λ21n⌝[λ3]⋱⌜λn1λn⌝⌜λ11λ11λ1⌟⌞λ2λ2⌟[λ3]⋱⌜λn1λn⌝⌜λ11λ11λ1⌝⌜λ21λ2⌝[λ3]⋱⌜λn1λn⌝⌜λ11λ11λ1⌝⌜λ21λ2⌝[λ3]⋱⌜λn1λn⌝⌜λ11λ11λ1⌝⌜λ21λ2⌝[λ3]⋱⌜λn1n⌝⌞λ11λ11λ1⌝⌜λ21λ2⌝[λ3]⋱⌞λnλn⌟]

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

Main article: Jordan normal form

Let V be an n-dimensional vector space; let ϕ be a linear map in L(V), the set of all linear maps from V into itself; and let A be the matrix representation of ϕ with respect to some ordered basis. It can be shown that if the characteristic polynomial f(λ) of A factors into linear factors, so that f(λ) has the form

f(λ)=±(λ−λ1)μ1(λ−λ2)μ2⋯(λ−λr)μr,

where λ1,λ2,…,λr are the distinct eigenvalues of A, then each μi is the algebraic multiplicity of its corresponding eigenvalue λi and A is similar to a matrix J in Jordan normal form, where each λi appears μi consecutive times on the diagonal, and the entry directly above each λi (that is, on the superdiagonal) is either 0 or 1: in each block the entry above the first occurrence of each λi 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 J is as close as one can come to a diagonalization of A. If A 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 A is similar to a matrix J in Jordan normal form, obtained through the similarity transformation J=M−1AM, where M is a generalized modal matrix for A.[53] (See Note above.)

### Example 4

Find a matrix in Jordan normal form that is similar to

A=(042−3834−8−2).

Solution: The characteristic equation of A is (λ−2)3=0, hence, λ=2 is an eigenvalue of algebraic multiplicity three. Following the procedures of the previous sections, we find that

rank⁡(A−2I)=1

and

rank⁡(A−2I)2=0=n−μ.

Thus, ρ2=1 and ρ1=2, which implies that a canonical basis for A 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 {𝐱2,𝐱1} and one chain of one vector {𝐲1}. Designating M=(𝐲1𝐱1𝐱2), we find that

M=(2201300−41),

and

J=(200021002),

where M is a generalized modal matrix for A, the columns of M are a canonical basis for A, and AM=MJ.[54] Note that since generalized eigenvectors themselves are not unique, and since some of the columns of both M and J may be interchanged, it follows that both M and J are not unique.[55]

### Example 5

In Example 3, we found a canonical basis of linearly independent generalized eigenvectors for a matrix A. A generalized modal matrix for A is

M=(𝐲1𝐱1𝐱2𝐱3)=(−142−204020−30011000).

A matrix in Jordan normal form, similar to A is

J=(4000051000510005),

so that AM=MJ.

## Applications

### Matrix functions

Main article: Matrix function

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 A.[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 A is diagonalizable, that is

D=M−1AM,

with

D=(λ10⋯00λ2⋯0⋮⋮⋱⋮00⋯λn),

then

Dk=(λ1k0⋯00λ2k⋯0⋮⋮⋱⋮00⋯λnk)

and the evaluation of the Maclaurin series for functions of A is greatly simplified.[59] For example, to obtain any power k of A, we need only compute Dk, premultiply Dk by M, and postmultiply the result by M−1.[60]

Using generalized eigenvectors, we can obtain the Jordan normal form for A and these results can be generalized to a straightforward method for computing functions of nondiagonalizable matrices.[61] (See Matrix function.)

### Differential equations

Main article: Ordinary differential equation

Consider the problem of solving the system of linear ordinary differential equations

𝐱′=A𝐱,

|
|

(5)

where

𝐱=(x1(t)x2(t)⋮xn(t)),𝐱′=(x1′(t)x2′(t)⋮xn′(t)), and  A=(aij).

If the matrix A is a diagonal matrix so that aij=0 for i≠j, then the system (5) reduces to a system of n equations which take the form

x1′=a11x1
x2′=a22x2

xn′=annxn.

|
|

(6)

In this case, the general solution is given by

x1=k1ea11t
x2=k2ea22t

xn=kneannt.

In the general case, we try to diagonalize A and reduce the system (5) to a system like (6) as follows. If A is diagonalizable, we have D=M−1AM, where M is a modal matrix for A. Substituting A=MDM−1, equation (5) takes the form M−1𝐱′=D(M−1𝐱), or

𝐲′=D𝐲,

|
|

(7)

where

𝐱=M𝐲.

|
|

(8)

The solution of (7) is

y1=k1eλ1t
y2=k2eλ2t

yn=kneλnt.

The solution 𝐱 of (5) is then obtained using the relation (8).[62]

On the other hand, if A is not diagonalizable, we choose M to be a generalized modal matrix for A, such that J=M−1AM is the Jordan normal form of A. The system 𝐲′=J𝐲 has the form

y1′=λ1y1+ϵ1y2⋮yn′−1=λn−1yn−1+ϵn−1ynyn′=λnyn,

|
|

(9)

where the λi are the eigenvalues from the main diagonal of J and the ϵi are the ones and zeros from the superdiagonal of J. The system (9) is often more easily solved than (5). We may solve the last equation in (9) for yn, obtaining yn=kneλnt. We then substitute this solution for yn into the next to last equation in (9) and solve for yn−1. Continuing this procedure, we work through (9) from the last equation to the first, solving the entire system for 𝐲. The solution 𝐱 is then obtained using the relation (8).[63]

Lemma:

Given the following chain of generalized eigenvectors of length r,

X1=v1eλt
X2=(tv1+v2)eλt
X3=(t22v1+tv2+v3)eλt

Xr=(tr−1(r−1)!v1+...+t22vr−2+tvr−1+vr)eλt,

these functions solve the system of equations,

X′=AX.

Proof:

Define

v0=0
Xj(t)=eλt∑i=1jtj−i(j−i)!vi.

Then, as t0=1 and 1′=0,

X'j(t)=eλt∑i=1j−1tj−i−1(j−i−1)!vi+eλtλ∑i=1jtj−i(j−i)!vi.

On the other hand we have, v0=0 and so

AXj(t)=eλt∑i=1jtj−i(j−i)!Avi
=eλt∑i=1jtj−i(j−i)!(vi−1+λvi)
=eλt∑i=2jtj−i(j−i)!vi−1+eλtλ∑i=1jtj−i(j−i)!vi
=eλt∑i=1j−1tj−i−1(j−i−1)!vi+eλtλ∑i=1jtj−i(j−i)!vi
=X'j(t)

as required.

## Notes

1. ↑ (Bronson 1970)
2. ↑ (Beauregard Fraleigh)
3. ↑ (Nering 1970)
4. ↑ (Golub Van Loan)
5. ↑ (Beauregard Fraleigh)
6. ↑ (Bronson 1970)
7. ↑ (Golub Van Loan)
8. ↑ (Bronson 1970)
9. ↑ (Bronson 1970)
10. ↑ (Beauregard Fraleigh)
11. ↑ (Nering 1970)
12. ↑ (Bronson 1970)
13. ↑ (Anton 1987)
14. ↑ (Beauregard Fraleigh)
15. ↑ (Burden Faires)
16. ↑ (Golub Van Loan)
17. ↑ (Harper 1976)
18. ↑ (Herstein 1964)
19. ↑ (Kreyszig 1972)
20. ↑ (Nering 1970)
21. ↑ (Burden Faires)
22. ↑ (Beauregard Fraleigh)
23. ↑ (Bronson 1970)
24. ↑ (Bronson 1970)
25. ↑ (Bronson 1970)
26. ↑ (Beauregard Fraleigh)
27. ↑ (Bronson 1970)
28. ↑ (Bronson 1970)
29. ↑ (Bronson 1970)
30. ↑ (Bronson 1970)
31. ↑ (Bronson 1970)
32. ↑ (Bronson 1970)
33. ↑ (Bronson 1970)
34. ↑ (Bronson 1970)
35. ↑ (Golub Van Loan)
36. ↑ (Herstein 1964)
37. ↑ (Nering 1970)
38. ↑ (Nering 1970)
39. ↑ (Nering 1970)
40. ↑ (Herstein 1964)
41. ↑ (Beauregard Fraleigh)
42. ↑ (Nering 1970)
43. ↑ (Bronson 1970)
44. ↑ (Bronson 1970)
45. ↑ (Bronson 1970)
46. ↑ (Bronson 1970)
47. ↑ (Bronson 1970)
48. ↑ (Bronson 1970)
49. ↑ (Bronson 1970)
50. ↑ (Beauregard Fraleigh)
51. ↑ (Cullen 1966)
52. ↑ (Franklin 1968)
53. ↑ (Bronson 1970)
54. ↑ (Bronson 1970)
55. ↑ (Bronson 1970)
56. ↑ (Beauregard Fraleigh)
57. ↑ (Bronson 1970)
58. ↑ (Bronson 1970)
59. ↑ (Bronson 1970)
60. ↑ (Bronson 1970)
61. ↑ (Bronson 1970)
62. ↑ (Beauregard Fraleigh)
63. ↑ (Beauregard Fraleigh)

## References

* 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, https://archive.org/details/firstcourseinlin0000beau
* Bronson, Richard (1970), Matrix Methods: An Introduction, New York: Academic Press
* Burden, Richard L.; Faires, J. Douglas (1993), Numerical Analysis (5th ed.), Boston: Prindle, Weber and Schmidt, ISBN 0-534-93219-3, https://archive.org/details/numericalanalysi00burd
* Cullen, Charles G. (1966), Matrices and Linear Transformations, Reading: Addison-Wesley
* Franklin, Joel N. (1968), Matrix Theory, Englewood Cliffs: Prentice-Hall
* 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, https://archive.org/details/advancedengineer00krey
* Nering, Evar D. (1970), Linear Algebra and Matrix Theory (2nd ed.), New York: Wiley

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Original source: https://en.wikipedia.org/wiki/Generalized eigenvector. Read more

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