Новий підхід до пошуку власних векторів для кратних власних чисел матриці
An efficient method of calculating eigenvectors for multiple eigenvalues of a matrix is proposed. This method is based on a formalized transformation of the problem of solving degenerate systems of equations into a regular problem by “repairing” their matrices and correspondingly correcting the righ...
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System research and information technologies| _version_ | 1867334449600921600 |
|---|---|
| author | Petrenko, Anatolii |
| author_facet | Petrenko, Anatolii |
| author_institution_txt_mv | [
{
"author": "Anatolii Petrenko",
"institution": "Educational and Research Institute for Applied System Analysis of the National Technical University of Ukraine \"Igor Sikorsky Kyiv Polytechnic Institute\", Kyiv"
}
] |
| author_sort | Petrenko, Anatolii |
| baseUrl_str | http://journal.iasa.kpi.ua/oai |
| collection | OJS |
| datestamp_date | 2025-02-09T21:55:38Z |
| description | An efficient method of calculating eigenvectors for multiple eigenvalues of a matrix is proposed. This method is based on a formalized transformation of the problem of solving degenerate systems of equations into a regular problem by “repairing” their matrices and correspondingly correcting the right-hand sides of the equations, as well as “exclusion” during calculations from the spectrum eigenvalues of the matrix of one of the multiple values. In the case of non-defective multiples of the matrix, orthogonal eigenvectors are formed in contrast to the results obtained using the Mathematica program. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2024.4.09 |
| first_indexed | 2025-07-17T10:28:41Z |
| format | Article |
| fulltext |
Publisher IASA at the Igor Sikorsky Kyiv Polytechnic Institute, 2024
Системні дослідження та інформаційні технології, 2024, № 4 107
TIДC
НОВІ МЕТОДИ В СИСТЕМНОМУ АНАЛІЗІ,
ІНФОРМАТИЦІ ТА ТЕОРІЇ ПРИЙНЯТТЯ РІШЕНЬ
UDC 621.372.061:391.266
DOI: 10.20535/SRIT.2308-8893.2024.4.09
NEW APPROACH TO FINDING EIGENVECTORS
FOR REPEATED EIGENVALUES OF A MATRIX
A.I. PETRENKO
Abstract. An efficient method of calculating eigenvectors for multiple eigenvalues
of a matrix is proposed. This method is based on a formalized transformation of the
problem of solving degenerate systems of equations into a regular problem by “re-
pairing” their matrices and correspondingly correcting the right-hand sides of the
equations, as well as “exclusion” during calculations from the spectrum eigenvalues
of the matrix of one of the multiple values. In the case of non-defective multiples of
the matrix, orthogonal eigenvectors are formed in contrast to the results obtained us-
ing the Mathematica program.
Keywords: eigenvectors, multiples of eigenvalues, algebraic and geometric multi-
plicity, solutions of degenerate systems, change of spectrum of a matrix, defective
and non-defective multiples of a matrix.
INTRODUCTION
Finding the eigenvectors ix for multiple eigenvalues i of the matrix A is the
least formalized task of modern Linear Algebra, as it is related to the solution of
homogeneous (degenerate) systems of equations that have an infinite number of
solutions:
0)( iiii xEAxB for ni ,,1 , (1)
since by definition the eigenvectors cannot be zero even for zero eigenvalues. An
eigenvector corresponding to an eigenvalue creates an eigenspace associated with
it. The set of all eigenvectors for different eigenvalues forms the vector space of
the matrix.
It is well known that equation (1) has nonzero solutions for the vector ix if
and only if the matrix )( EA i has a zero determinant, which determines the
characteristic polynomial of the matrix [1–3]. It can be used to find the eigenval-
ues of the matrix (for small order tasks) together with a more powerful QR algo-
rithm with orthogonal Householder or Givens rotation matrices, which reduce the
original matrix to a triangular matrix, on the diagonal of which there are real ei-
genvalues or 22 blocks of eigenvalues (for large-scale tasks).
A.I. Petrenko
ISSN 1681–6048 System Research & Information Technologies, 2024, № 4 108
The article attempts to formalize the procedure for solving equation (1), ex-
cluding the traditional manual selection of individual components of the solution
vector xi in order to eliminate the degeneracy of the problem. Special attention is
paid to the case of multiple eigenvalues, for which equation (1) has the same
form, but there can be both different and coincident solutions. In the latter case,
the rank of the matrix, which is determined by the number of independent eigen-
vectors, is lower than its order, so such multiples of eigenvalues are called “defec-
tive” [4–6].
The analysis of publications showed that there is great uncertainty in the is-
sue of finding eigenvectors for multiples of the matrix, the Internet is full of re-
quests from specialists of different countries for help and consultations [7–11] and
educational materials [12–14]. This motivated the conduct of own research, the
results of which formed the basis of this article.
STATE OF AFFAIRS
The existing method of solving the problem of finding eigenvectors ix for multi-
ple eigenvalues i of the matrix A is best considered using some examples, say,
the matrix
222
254
245
, (2)
which has a spectrum of eigenvalues 101 and 132 .
When choosing an eigenvalue 12 , the degenerate system of equations (1)
is reduced to the form that demonstrates the relationship of all three equations:
0
0
0
122
244
244
3
2
1
22
v
v
v
xB . (3)
The system of equations (2) is transformed into the following form by the
“row-reduction” procedure (i.e., reduction to a normal trapezoidal form):
0
0
0
000
000
5.011
3
2
1
v
v
v
, (4)
from which the following expression is formed
1
0
5.0
0
1
15.0
32
3
2
32
3
2
1
vv
v
v
vv
v
v
v
.
Then the eigenvectors corresponding to the eigenvalues 132 have the
form:
0,
1
0
5.0
0
1
1
2
x . (5)
New approach to finding eigenvectors for repeated eigenvalues of a matrix
Системні дослідження та інформаційні технології, 2024, № 4 109
The condition 0 excludes the selection of a zero eigenvector. At
the same time, and can take any values, since the degenerate equations (3)
have many solutions. By choosing different and , using (5), different values
of 2x are obtained and checked whether they satisfy the basic equation
iii xAx . (6)
For example, choosing, say, 0 and 1 , we get an eigenvector
}1 ,0 ,5.0{ 2 x , (7)
which satisfies (6). On the contrary, choosing 1 and 0 , we get a solution
0} ,1 ,1{ 3 x , (8)
which also satisfies the basic equation (6).
It is interesting that when choosing 1 and 1 , we get according to (5),
an eigenvector
}1 ,1 ,5.1{4 x ,
which also satisfies equation (6) and is independent with respect to vectors 2x
and 3x . But they are not all orthogonal because 5.2,5.0 4332 xxxx and
75.142 xx .
In the case of multiple eigenvalues of the matrix, two of them are chosen
from the set of independent solutions obtainable from (5) (eg, 2x and 3x ) as the
corresponding eigenvectors for the multiple eigenvalues. It is clear that when the
selected eigenvectors are multiplied by an arbitrary scale coefficient m, the basic
equation (6) continues to be fulfilled.
For the completeness of the picture, one more method should be mentioned,
which recommends, in the case of repeated eigenvalues, to use instead of equation
(1) its modification [1, 2]
0)( i
k
ii
k
i xEAxB ,
where k is an indicator of the algebraic multiplicity of an eigenvalue, and the set
of solutions ix for 1k corresponds to the so-called root eigenvectors. But,
based on equation (3), it can be shown that this is rather a delusion. Indeed, for
2k we get instead of (3) the expression:
0
0
0
91818
183636
183636
3
2
1
2
2
2
v
v
v
xB ,
which is transformed by the “row-reduction” procedure to the already known
equation (4)
0
0
0
000
000
5.011
3
2
1
v
v
v
,
what indicates that the eigenvectors for multiple eigenvalues can be chosen from
the set of solutions of equation (4), as was done above. A similar result is main-
tained if the multiplicity index k is increased.
A.I. Petrenko
ISSN 1681–6048 System Research & Information Technologies, 2024, № 4 110
Most likely, the considered selection of solutions is implemented in the well-
known Mathematica program (the algorithmic support of which, unfortunately, is
not described in detail in its documentation), because with its help (Fig. 1) for
multiples of 132 of the matrix (2) eigenvectors 2} ,0 ,1{ 2 x ,
}0 ,1 ,1{3 x can be obtained, which coincide with the accuracy of the coeffi-
cient with the values (7) and (8), which were previously manually selected when
solving the system (4).
A={{5,4,2},{4,5,2},{2,2,2}};
{vals, vecs} = Eigensystem[A]
{{10, 1, 1}, {{2, 2, 1}, {1, 0, 2}, {1, 1, 0}}
Fig. 1. A fragment of the Mathematica code
It is interesting to compare the results of calculations obtained with the help
of the Mathematica for two matrices that have the same spectrum of eigenvalues,
but the multiples of the second one are defective (Fig. 2).
Fig. 2. Eigenvectors of two matrices with the same eigenvalues
The defect of a multiple eigenvalues matrix A2 is reflected in the Mathe-
matica results by generation of a zero eigenvector (Fig. 2, b), what can mislead
beginners who suspect an error in the program’s operation.
But the Mathematica, unfortunately, sometimes contradicts itself, because it
is enough to use the another its operator JordanDecomposition[A], related also to
the calculation of eigenvectors and eigenvalues, and to find out with surprise that
the same matrix A2 now has different eigenvectors 0} ,0 ,1{ 2 x and
0} ,1 ,0{ 3 x for the same multiple 332 (Fig. 3).
A2={{3,1,1},{0 ,3,2}, {0,0,1}};
JordanDecomposition [A2]
{{{0, 1, 0}, {-1, 0, 1}, {1, 0, 0}}, {{1, 0, 0}, {0, 3, 1}, {0, 0, 3}}}
/MatrixForm =
010
001
110
310
130
001
Fig. 3. The Jordanian normal form of the matrix A2
But the obtained value of 2x does not satisfy the basic equation (6). In addi-
tion, the eigenvector 0} ,1 ,0{ 1 x for 11 differs from its value 1} ,1- ,0{ 1 x
shown before in Fig. 2, and also does not satisfy equation (6).
A1={{3,0,1},{0,3,2},{0,0,1}};
{vals, vecs} = Eigensystem[A1]
{{3, 3, 1}, {{0, 1, 0}, {1, 0, 0}, {-1, -2, 2}}}
A2={{3,1,1},{0 ,3,2}, {0,0,1}};
{vals, vecs} = Eigensystem[A2]
{{3, 3, 1}, {{1, 0, 0}, {0, 0, 0}, {0, -1, 1}}}
a b
New approach to finding eigenvectors for repeated eigenvalues of a matrix
Системні дослідження та інформаційні технології, 2024, № 4 111
Since the algorithmic core of the Mathematica is also used in other well-
known calculation programs (Matlab, Mathcad, Maple), the results of their appli-
cation to calculating eigenvectors for multiple eigenvalues will be similar.
THE PROPOSED METHOD
The paper contains a procedure for generating orthogonal vectors for multiple
non-defective eigenvalues, what does not interfere with Mathematica, and two
depended vectors for defective multiples. In this case, the solution of degenerate
systems of type (3) is formalized by diagonal correction of the systems matrix
after “row-reduction” (4) with a simultaneous correction of the zero vector of the
right side of the system.
The system’s degeneration (4) is manifested by zero k-th diagonal elements
in its matrix. Similar to the method of diagonal correction [3], this matrix is “re-
paired” (so that degeneracy is eliminated) by replacing zero diagonal elements
with a number equal to one or by some constant g, which is chosen to be equal to
the middle value of the elements of the matrix B row. Then the solution is ongo-
ing with the already ingenerated matrix and the new right-hand side
0} ,...1 ,...0{ 1 b , represented by a transposed vector of dimension n , such that all
elements are zero and only k-th elements are equal to one.
However, if at the same time there is a zero column and row in the matrix of
equations (4), then only the position of the diagonal element of the column is ad-
justed in the vector of the right part.
Let us illustrate what has been said with the example of the degenerate sys-
tem of equations (4):
1
1
0
100
010
5.011
0
0
0
000
000
5.011
3
2
1
3
2
1
v
v
v
v
v
v
. (9)
The zero second and third diagonal elements of the zero rows of the original
matrix are corrected by introducing the constant 132 gg and an additional
vector of the right part 2b is formed. From the solution of the adjusted system, we
get
}1 ,0 ,5.0{ 2 x (10)
or normalized value
}894427.0 .,0 ,447214.0{ ]][N[ 2 2 xNormalizeX . (11)
The eigenvector 1x of the matrix for 101 is found quite similarly. In this
case, the system of equations (1) looks like:
0
0
0
822
254
245
3
2
1
11
v
v
v
xB
and by the “row-reduction” procedure it is transformed into the following form:
A.I. Petrenko
ISSN 1681–6048 System Research & Information Technologies, 2024, № 4 112
0
0
0
000
210
201
3
2
1
v
v
v
. (12)
Unlike the matrix of the system of equations (4), the matrix of the system
(12) has only one zero row, so its correction is performed differently:
1
0
0
100
210
201
0
0
0
000
210
201
3
2
1
3
2
1
v
v
v
v
v
v
. (13)
As a result, we obtain the solution of the adjusted system (13)
}1 ,2 ,2{1 x (14)
or normalized value
.}333333.0 ,666667.0 ,666667.0{]][[ 1 1 xNNormalizeX (15)
Let us now consider an innovative procedure for finding the eigenvector for
the second multiple of the eigenvalue 12 of the matrix. For this purpose, it is
proposed to apply the following transformation of the matrix A, in which one of
its multiple roots is excluded (zeroed), and then the problem is reduced to the pre-
vious one, when all the eigenvalues of the new matrix A1 are different. Such a
transformation is performed according to the formula [3]:
222222 ,roductKroneckerP1A XXAXXA , (16)
where vector multiplication according to Kronecker, which results in a matrix,
and the normalized eigenvector 2X (11) are used.
According to the formula (16) taking into account (11), we build the matrix
2.124.2
254
4.248.4
1A ,
for which the spectrum of eigenvalues }8613110.3.,1.,10{ 16 does not con-
tain multiples.
According to (1), we obtain a homogeneous system of equations
0
0
0
2.024.2
244
4.248.3
3
2
1
33
v
v
v
xB . (17)
Using the “row-reduction” procedure, the system of equations (17) is trans-
formed and then is being “repaired” taking into account the fact that after the
“row-reduction” procedure, only one zero row is formed in the matrix:
33xB
.
1
0
0
100
5.210
201
0
0
0
000
5.210
201
0
0
0
2.024.2
244
4.248.3
3
2
1
3
2
1
3
2
1
v
v
v
v
v
v
v
v
v
(18)
New approach to finding eigenvectors for repeated eigenvalues of a matrix
Системні дослідження та інформаційні технології, 2024, № 4 113
As a result of the solution (18), we obtain the value of the second eigenvector
1} ,2.5- ,2{ 3 x (19)
or in normalized form
}298142.0 ,745356.0 ,596285.0{ ]][[ 3 3 xNNormalizeX (20)
Thus, for the matrix (2) the following Eigensystem[A] is obtained by the
proposed method
}} ,.,{},1,0,5.0{},1,2,2{{ },1,1,10{{} ,{ 1522 vecsvals (21)
which differs from the results of Mathematica
}} , ,{,}2 ,0 ,1{ },1 ,2 ,2{{ },1 ,1 ,10{{ } ,{ 011vecsvals (22)
presented in Fig. 1, by the value of the eigenvector for the second multiple eigen-
value 13 .
The obtained eigenvectors given (10), (14) and (19) are orthogonal, since
0,0 2131 xxxx and 032 xx .
If in the Mathematica’s solution (22) we denote different components as
1} ,2 ,2{ 1 y , }2 ,0 ,1{2 y and 0} ,1 ,1{ 3 y , then we can make sure, what
0,0 2131 yyyy , but 132 yy .
This means that these vectors yi although they satisfy the corresponding ba-
sic equations (6), are not orthogonal and therefore, unlike the set of eigenvectors
ix from (21), cannot ensure unmistakably the canonical JordanDecomposition[A]
operation for the matrix A, when
tTDTA , (23)
where T is the orthogonal matrix of eigenvectors, and D is the diagonal matrix of
all eigenvalues, including multiples. Indeed, using the normalized values of the
obtained eigenvectors X1, X2 and X3 from the corresponding formulas (15), (11)
and (20), it is possible to build
298142.0 894427.0 333333.0
745356.0 0 666667.0
596285.0 447214.0 666667.0
T ,
100
010
0010
D
and make sure that according to (23)
ATDT t }}.2 .,2 .,2{ ,}.2 .,5 .,4{ .},2 .,4 ,00001.5{{ .
For comparison, if you normalize the eigenvectors of the matrix A obtained
with the help of Mathematica (22), you can get:
}333333.0 ,666667.0 ,666667.0{ ]][[1 1 yNNormalizeY ,
,}894427.0 .,0 ,447214.0{ ]][[2 2 yNNormalizeY
}.0 ,707107.0 ,707107.0{ ]][[3 3 yNNormalizeY .
A.I. Petrenko
ISSN 1681–6048 System Research & Information Technologies, 2024, № 4 114
and instead of the orthogonal matrix T construct another matrix
0 894427.0 333333.0
707107.0 0 666667.0
0.707107 447214.0 666667.0
1T ,
with which we can check whether equation (23) is satisfied:
,
1.91111 2.22222 1.82222
2.22222 4.94445 3.94445
1.82222 3.94445 5.14445
* 11 ADTTA t
.
while 948684.0][Det][Det 11 tTT and 00001.9*][Det A instead of 10.
The same erroneous Mathematica’s result maу be obtained by applying the
standard JordanDecomposition[A] operator.
The obtained result calls into question the existing lemma that orthogonal ei-
genvectors correspond only to different simple eigenvalues [14], which was for-
mulated, most likely, on the basis of practical results obtained with the help of the
traditional selection of solutions of a homogeneous system of equations, consid-
ered above using the example of the system (3). A new approach with the exclu-
sion of multiples and consideration of two homogeneous systems of equations
provides new opportunities.
It seems interesting, using the method described above, to find the eigenvec-
tors of the matrix A2 for its eigenvalue’s spectrum }1,3,3{ for which the
Mathematica generates a solution with a zero eigenvector (Fig. 2).
For the first multiple eigenvalue 31 , by analogy with the above example,
instead of equation (9), we obtain the following expression
0
0
1
100
100
011
0
0
0
000
100
010
3
2
1
3
1
1
v
v
v
v
v
v
,
from which we find 0} ,0 ,1{ 1 x .
Using the obtained value of 1x , which coincides with its normalized value of
X1, to exclude, according to (16), one of the multiples of the matrix A2, we find
the matrix A3
100
230
110
3A ,
which has a modified spectrum of eigenvalues }1,3,0{ and for which, by
analogy with (18), we construct an equation for finding the eigenvector of the
second multiple 32 of the matrix A2:
0
0
1
100
110
03/11
0
0
0
000
100
03/11
3
2
1
3
2
1
v
v
v
v
v
v
,
New approach to finding eigenvectors for repeated eigenvalues of a matrix
Системні дослідження та інформаційні технології, 2024, № 4 115
from which we get }0 ,0 ,1{2 x .
As you can see, the values of 1x and 2x are linearly dependent (they just co-
incide), which indicates a defect in the multiple eigenvalues of the matrix.
CONCLUSIONS
One of the most important tasks of computational mathematics is the creation of
effective and stable algorithms for finding the eigenvalues and vectors of a matrix
[1]. They are a powerful tool that provides a deep understanding of matrix proper-
ties and opens wide perspectives for its application. Possession of this tool opens
up opportunities for research and innovation in various fields of science and tech-
nology (for example, for identifying the main components and clustering of data
during their analysis, for filtering signals and extracting a useful signal, for clus-
tering and pattern recognition, etc.).
The state of affairs with the formalization of finding the eigenvectors of a
matrix in general and for multiple eigenvalues in particular requires better. The
article takes a certain step in this direction and proposes an innovative method of
calculating eigenvectors for multiple eigenvalues of a matrix, which is based on
the formalized transformation of the problem of solving degenerate systems of
equations into a regular problem by “repairing” their matrices and by correspond-
ingly correcting the right-hand sides of the equations, as well as “exclusion” of
one of the multiple values from the spectrum of eigenvalues of the matrix during
calculations of eigenvectors for multiples eigenvalues. In the case of non-
defective multiples eigenvalues of the matrix, this method allows you to form or-
thogonal eigenvectors in contrast to the results obtained using the Mathematica.
REFERENCES
1. Mathematics: Finding Eigenvectors with repeated Eigenvalues. Available:
https://math.stackexchange.com/questions/144798/finding-eigenvectors-with-
repeated-eigenvalues
2. Repeated Eigenvalues. Available: https://ocw.mit.edu/courses/18-03sc-differential-
equations-fall-2011/051316d5fa93f560934d3e410f8d153d_MIT18_03SCF11_s33_
8text.pdf
3. L.P. Feldman, A.I. Petrenko, and O.A. Dmitrieva, Numerical Methods in Computer
Science: Textbook (in Ukrainian). Kyiv: BHV Publishing Group, 2006, 480 p.
Available: https://library.kre.dp.ua/Books/2-
4%20kurs/%D0%90%D0%BB%D0%B3%D0%BE%D1%80%D0%B8%D1%82%D
0%BC%D0%B8%20%D1%96%20%D0%BC%D0%B5%D1%82%D0%BE%D0%
B4%D0%B8%20%D0%BE%D0%B1%D1%87%D0%B8%D1%81%D0%BB%D0
%B5%D0%BD%D1%8C/%D0%A4%D0%B5%D0%BB%D1%8C%D0%B4%D0%
BC%D0%B0%D0%BD_%D0%A7%D0%B8%D1%81%D0%B5%D0%BB%D1%8
C%D0%BD%D1%96_%D0%BC%D0%B5%D1%82%D0%BE%D0%B4%D0%B8
_%D0%B2_%D1%96%D0%BD%D1%84%D0%BE%D1%80%D0%BC%D0%B0
%D1%82%D0%B8%D1%86%D1%96_2007.pdf
4. Real, Repeated Eigenvalues (Sect. 5.9) Review. Michigan State University. Available:
https://users.math.msu.edu/users/gnagy/teaching/13-summer/mth235/l31-235.pdf
5. Repeated Eigenvalues. Available: https://ocw.mit.edu/courses/18-03sc-differential-
equations-fall-2011/051316d5fa93f560934d3e410f8d153d_MIT18_03SCF11_s33_
8text.pdf
A.I. Petrenko
ISSN 1681–6048 System Research & Information Technologies, 2024, № 4 116
6. Jiry Lebl, Multiple Eigenvalues. Available: https://math.libretexts.org/Bookshelves
/Differential_Equations/Differential_Equations_for_Engineers_(Lebl)/3%3A_Syste
ms_of_ODEs/3.7%3A_Multiple_Eigenvalues
7. How do I find an eigenvector matrix when eigenvalues are repeated? Available:
https://www.quora.com/How-do-I-find-an-eigenvector-matrix-when-eigenvalues-
are-repeated
8. Eigenvalues and Eigenvectors. Available: https://sites.calvin.edu/scofield/courses
/m256/materials/eigenstuff.pdf
9. Marco Taboga, “Linear independence of eigenvectors,” Lectures on matrix algebra,
2021. Available: https://www.statlect.com/matrix-algebra/linear-independence-of-
eigenvectors
10. MATLAB Answers: How to identify repeated eigenvalues of a matrix? Available:
https://www.mathworks.com/matlabcentral/answers/395353-how-to-identify-
repeated-eigenvalues-of-a-matrix
11. Repeated eigenvalues -> crazy eigenvectors? Available: https://www.reddit.com/r/
matlab/comments/lsk3v7/repeated_eigenvalues_crazy_ eigenvectors/?rdt=37293
12. Eigenvalues and Eigenvectors. Available: https://personal.math.ubc.ca/~
tbjw/ila/eigenvectors.html
13. Eigenvalues and Eigenvectors. Available: https://www.math.hkust.edu.hk/~
mabfchen/Math111/Week11-12.pdf
14. Eigenvectors, Eigenvalues, and Diagonalization (solutions). Available:
https://math.berkeley.edu/~mcivor/math54s11/worksheet2.28soln.pdf
Received 08.01.2024
INFORMATION ON THE ARTICLE
Anatolii I. Petrenko, ORCID: 0000-0001-6712-7792, Educational and Research Institute
for Applied System Analysis of the National Technical University of Ukraine “Igor
Sikorsky Kyiv Polytechnic Institute”, Ukraine, e-mail: tolja.petrenko@gmail.com
НОВИЙ ПІДХІД ДО ПОШУКУ ВЛАСНИХ ВЕКТОРІВ ДЛЯ КРАТНИХ
ВЛАСНИХ ЧИСЕЛ МАТРИЦІ / А.І. Петренко
Анотація. Запропоновано ефективний метод обчислення власних векторів для
кратних власних чисел матриці, який базується на формалізованому перетво-
ренню задачі розв’язання вироджених систем рівнянь у звичайну задачу шля-
хом «ремонту» їх матриць і відповідного корегування правих частин рівнянь,
а також «вилучення» під час обчислень зі спектра власних чисел матриці одно-
го з кратних значень. У випадку недефектних кратних чисел матриці форму-
ються ортогональні власні вектори на відміну від результатів, отриманих за
допомогою програми Mathematica.
Ключові слова: власні вектори, кратні власні числа, алгебрична і геометрична
кратність, розв’язання вироджених систем, зміна спектра матриці, дефектні і
недефектні кратні числа матриці.
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| id | journaliasakpiua-article-322526 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-07-17T10:28:41Z |
| publishDate | 2024 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/66/1531c08a7f29040c0197d786a879d366.pdf |
| spelling | journaliasakpiua-article-3225262025-02-09T21:55:38Z New approach to finding eigenvectors for repeated eigenvalues of a matrix Новий підхід до пошуку власних векторів для кратних власних чисел матриці Petrenko, Anatolii власні вектори кратні власні числа алгебрична і геометрична кратність розв’язання вироджених систем зміна спектра матриці дефектні і недефектні кратні числа матриці eigenvectors multiples of eigenvalues algebraic and geometric multiplicity solutions of degenerate systems change of spectrum of a matrix defective and non-defective multiples of a matrix An efficient method of calculating eigenvectors for multiple eigenvalues of a matrix is proposed. This method is based on a formalized transformation of the problem of solving degenerate systems of equations into a regular problem by “repairing” their matrices and correspondingly correcting the right-hand sides of the equations, as well as “exclusion” during calculations from the spectrum eigenvalues of the matrix of one of the multiple values. In the case of non-defective multiples of the matrix, orthogonal eigenvectors are formed in contrast to the results obtained using the Mathematica program. Запропоновано ефективний метод обчислення власних векторів для кратних власних чисел матриці, який базується на формалізованому перетворенню задачі розв’язання вироджених систем рівнянь у звичайну задачу шляхом «ремонту» їх матриць і відповідного корегування правих частин рівнянь, а також «вилучення» під час обчислень зі спектра власних чисел матриці одного з кратних значень. У випадку недефектних кратних чисел матриці формуються ортогональні власні вектори на відміну від результатів, отриманих за допомогою програми Mathematica. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2024-12-25 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/322526 10.20535/SRIT.2308-8893.2024.4.09 System research and information technologies; No. 4 (2024); 107-116 Системные исследования и информационные технологии; № 4 (2024); 107-116 Системні дослідження та інформаційні технології; № 4 (2024); 107-116 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/322526/312908 |
| spellingShingle | власні вектори кратні власні числа алгебрична і геометрична кратність розв’язання вироджених систем зміна спектра матриці дефектні і недефектні кратні числа матриці Petrenko, Anatolii Новий підхід до пошуку власних векторів для кратних власних чисел матриці |
| title | Новий підхід до пошуку власних векторів для кратних власних чисел матриці |
| title_alt | New approach to finding eigenvectors for repeated eigenvalues of a matrix |
| title_full | Новий підхід до пошуку власних векторів для кратних власних чисел матриці |
| title_fullStr | Новий підхід до пошуку власних векторів для кратних власних чисел матриці |
| title_full_unstemmed | Новий підхід до пошуку власних векторів для кратних власних чисел матриці |
| title_short | Новий підхід до пошуку власних векторів для кратних власних чисел матриці |
| title_sort | новий підхід до пошуку власних векторів для кратних власних чисел матриці |
| topic | власні вектори кратні власні числа алгебрична і геометрична кратність розв’язання вироджених систем зміна спектра матриці дефектні і недефектні кратні числа матриці |
| topic_facet | власні вектори кратні власні числа алгебрична і геометрична кратність розв’язання вироджених систем зміна спектра матриці дефектні і недефектні кратні числа матриці eigenvectors multiples of eigenvalues algebraic and geometric multiplicity solutions of degenerate systems change of spectrum of a matrix defective and non-defective multiples of a matrix |
| url | https://journal.iasa.kpi.ua/article/view/322526 |
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