Класичні спеціальні функції з матричними змінними
This article focuses on a few of the most commonly used special functions and their key properties and defines an analytical approach to building their matrix-variate counterparts. To achieve this, we refrain from using any numerical approximation algorithms and instead rely on properties of matrice...
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| author | Shutiak, Dmytro Podkolzin, Gleb Bondarenko, Victor Chapovsky, Yury |
| author_facet | Shutiak, Dmytro Podkolzin, Gleb Bondarenko, Victor Chapovsky, Yury |
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| description | This article focuses on a few of the most commonly used special functions and their key properties and defines an analytical approach to building their matrix-variate counterparts. To achieve this, we refrain from using any numerical approximation algorithms and instead rely on properties of matrices, the matrix exponential, and the Jordan normal form for matrix representation. We focus on the following functions: the Gamma function as an example of a univariate function with a large number of properties and applications; the Beta function to highlight the similarities and differences from adding a second variable to a matrix-variate function; and the Jacobi Theta function. We construct explicit function views and prove a few key properties for these functions. In the comparison section, we highlight and contrast other approaches that have been used in the past to tackle this problem. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2024.4.10 |
| first_indexed | 2025-07-17T10:28:41Z |
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Publisher IASA at the Igor Sikorsky Kyiv Polytechnic Institute, 2024
Системні дослідження та інформаційні технології, 2024, № 4 117
UDC 517.58 + 512.643
DOI: 10.20535/SRIT.2308-8893.2024.4.10
CLASSICAL SPECIAL FUNCTIONS OF MATRIX ARGUMENTS
D.O. SHUTIAK, G.B. PODKOLZIN, V.G. BONDARENKO, Y.A. CHAPOVSKY
Abstract. This article focuses on a few of the most commonly used special func-
tions and their key properties and defines an analytical approach to building their
matrix-variate counterparts. To achieve this, we refrain from using any numerical
approximation algorithms and instead rely on properties of matrices, the matrix ex-
ponential, and the Jordan normal form for matrix representation. We focus on the
following functions: the Gamma function as an example of a univariate function
with a large number of properties and applications; the Beta function to highlight the
similarities and differences from adding a second variable to a matrix-variate func-
tion; and the Jacobi Theta function. We construct explicit function views and prove
a few key properties for these functions. In the comparison section, we highlight and
contrast other approaches that have been used in the past to tackle this problem.
Keywords: matrix, special function, matrix function, gamma function, beta func-
tion, Jacobi theta function, Jordan normal form.
INTRODUCTION
Data with matrix responses for each experiment are increasingly common in
modern statistical problems. For example, observations over a time period can be
viewed holistically as a matrix variable, labeling the rows and columns as time
and actual measurements respectively. Temporal and spatial data, multivariate
growth curve data, imaging data, and data from cross-sectional designs also gen-
erate matrix-valued responses. On the other hand, many of these phenomena are
still often built on generalized cases of classical problems, many of which are
solved, or at least interpreted or simplified, by special functions. Therefore, the
motivation of the study was to combine these two parts and to do so as generally
as possible analytically, without relying on a specific problem or purely numerical
methods. There were earlier studies on this topic, but they were aimed at either
generalizing a specific concept (Mitra S. 1970 [1]), or calculating values for cer-
tain classes of matrices needed for further calculations (Kishka Z., Saleem M.
2019 [2]). In this article, based on the theory of matrices, matrix exponents and
using the Jordanian canonical form of matrices, we formulate a basic toolkit of
definitions and key properties of special matrix-variate functions. These proper-
ties are applicable to the widest range of matrices and have an explicit form, that
is, they can be used for further research with minimal changes.
First, the definition and key properties for the matrix Gamma function will
be given, as an example of a univariate special function, followed by a series of
two-variable special functions such as the Beta function and the Jacobi Theta
function. A comparison of the obtained results with the existing methods men-
tioned above will also be made
D.O. Shutiak, G.B. Podkolzin, V.G. Bondarenko, Y.A. Chapovsky
ISSN 1681–6048 System Research & Information Technologies, 2024, № 4 118
UNIVARIATE SPECIAL FUNCTIONS
1 GAMMA FUNCTION
1.1 the definition and the general form of the matrix-variate gamma function
For the Gamma function and all subsequent special functions, the integral defini-
tion of functions was taken as the basis of the study. Specifically for the Gamma
function and the Beta function, the following shift was also made to simplify the
calculations:
Definition. For an arbitrary matrix A , we define
.
replacens calculatio
furthersimplify to
)(
00
dxex
BIA
dxexAГ xBxIA
(2.1)
From this definition using the matrix, we get the following form for the ma-
trix-variate Gamma function:
For an arbitrary matrix ,kkA which has the Jordanian canonical form
AUUJ 1 its’ Gamma function will have the form:
;)()( 1 UJUГAГ
))((
))((
))((
2
1
2
1
lr
r
r
l
JГ
JГ
JГ
JГ — block matrix,
where
))(( jrj
J
)1(0000
!1
)1(
)1(000
!2
))1(
!1
)1(
)1(00
)!2(
)1(
!1
)1(
)1(0
)!1(
)1(
)!2(
)1(
!2
)1(
!1
)1(
)1(
)2(
)1()2(
j
j
j
jj
j
j
j
r
j
j
j
j
r
j
j
r
jj
j
r
rr
j
jj
(2.2)
The blocks of the resulting matrix correspond to the blocks of each of the ei-
genvalues of the Jordan matrix J of the matrix A and have the corresponding
dimensions )( ii rr . It should also be noted that these matrices are upper-
triangular, that is, they have zero-values below the main diagonal. This fact will
also be important for the subsequent special functions.
Classical special functions of matrix arguments
Системні дослідження та інформаційні технології, 2024, № 4 119
1.2 Main functional equation
One of the most important and used properties of the Gamma function is the func-
tional equation, as several other properties of the Gamma function are based on it.
Also, this property allows you to recursively find the values of the function,
thereby significantly simplifying calculations.
For a scalar argument, the identity has the following form:
)(*)1( ГГ . (2.3)
To prove this statement in the matrix case, we first consider the following
auxiliary equality:
). 1(
100
11
00
110
0011
ir
J
(2.4)
Let us now use this and definition (2.2) to generalize identity (2.3):
)2.2(
)1(
)4.2(
))((
using
JГ
using
IJГ irir ii
)2(0000
!1
)2(
)2(000
!2
))2(
!1
)2(
)2(00
)!2(
)1(
!1
)2(
)2(0
)!1(
)2(
)!2(
)1(
!2
)1(
!1
)1(
)2(
)2(
)1()2(
j
j
j
jj
j
j
j
r
j
j
j
j
r
j
j
r
jj
j
r
rr
j
jj
Using the properties of the derivative of the Gamma function and the proper-
ties of the Gamma function itself:
)2(0000
!1
)2(
)2(000
!2
))2(
!1
)2(
)2(00
)!2(
)1(
!1
)2(
)2(0
)!1(
)2(
)!2(
)1(
!2
)1(
!1
)1(
)2(
)2(
)1()2(
j
j
j
jj
j
j
j
r
j
j
j
j
r
j
j
r
jj
j
r
rr
j
jj
D.O. Shutiak, G.B. Podkolzin, V.G. Bondarenko, Y.A. Chapovsky
ISSN 1681–6048 System Research & Information Technologies, 2024, № 4 120
000
)1()1(0
!1
)1()1()1(
)1()1(
ii
iii
ii
)1()1(0
!1
)1()1()1(
!)1(
)1()1()1()1( )2()1(
ii
iii
i
i
r
ii
r
i
r
r ii
Now we split the obtained matrix into two separate matrices, grouping all
terms with the coefficient )1( i into the first, all others into the second:
)1()1(000
!1
)1()1(
)1()1(0
)!1(
)1()1(
!1
)1()1(
)1()1(
)1(
ii
ii
ii
i
i
r
iii
ii
Г
Г
Г
r
ГГ
Г
i
;
0000
!1
))1((
00
!)1(
)1()1(
!1
))1((
0
)2(
i
i
i
r
ii
Г
r
ГrГ i
subtract )1( i from the first term as a matrix and reduce the factorial and coef-
ficient in the second:
)1(000
!1
)1(
)1(0
)!1(
)1(
!1
)1(
)1(
1
1
i
i
i
i
i
r
i
i
i
Г
Г
Г
r
ГГ
Г
i
0000
!1
))1((
0
!2
))1((''
!1
))1((
00
!)2(
)1(
!2
))1((''
!1
))1((
0
)2(
i
ii
i
i
r
ii
Г
ГГ
r
ГГГ i
Classical special functions of matrix arguments
Системні дослідження та інформаційні технології, 2024, № 4 121
.)0())(())(()1(
iii ririri JJГJГ
That is, we get the following identity:
.)0())(())(()1())((
iiii riririir JJГJГIJГ
Generalizing for the matrix :) ( 1 UJUA iri
.) 0( )()()1()( 1 UJUAГAГIAГ
iri (2.5)
As we can see, it was possible to prove a property similar to (2.3), but to
which the correcting term 1) 0( )( UJUAГ
ir
is added. Indeed, if we take the di-
mension of the matrix A as )11( , i.e. return to a scalar variable, then the second
term of the identity will be equal to 0 and we will return to the widely-known
identity (2.3).
1.3 Euler’s reflection formula
Before moving on to the generalization of the reflection formula, we give an addi-
tional auxiliary property:
dxe
e
r
exm
e
dxeemJГ x
xm
i
xmr
xm
xxmJ
ir
i
ii
i
ir
ln
ln1
ln
0
ln)(
0 0
!)1(
)ln(
))((
.
)1(000
!1
)1(
)1(0
!)1(
)1(
!1
)1(
)1(
1
i
i
i
i
i
r
i
i
mГ
mГ
Г
r
mГmГ
mГ
i
(3.9)
Now let's return to Euler's reflection formula:
))(())(( irrir JIГJГ
i
)1(
100
11
00
110
0011
)( ir
i
i
i
i
irr ii
JJI
)1(000
!1
)1(
)1(0
!)1(
)1(
!1
)1(
)1(
))1(())((
)1(
9.3
.
i
i
i
i
i
r
i
i
вл
irir
Г
Г
Г
r
ГГ
Г
JГJГ
i
ii
D.O. Shutiak, G.B. Podkolzin, V.G. Bondarenko, Y.A. Chapovsky
ISSN 1681–6048 System Research & Information Technologies, 2024, № 4 122
)2(000
!1
)2(
)2(0
!1
)2(
!1
)2(
)2(
1
i
i
i
i
i
r
i
i
Г
Г
Г
r
ГГ
Г
i
))1(())(( irir ii
JГJГ
Derivation of Euler's reflection formula:
Consider the following product:
;)(
2
1
)(
2
1
irrirr iiii
JIГJIГ
;
2
1
2
1
00
1
2
1
00
1
2
1
0
001
2
1
)(
2
1
ir
i
i
i
i
irr JJI
i
.
2
1
2
1
00
1
2
1
00
1
2
1
0
001
2
1
)(
2
1
ir
i
i
i
i
irr JJI
i
2
1
2
1
)(
2
1
)(
2
1
iririrrirr JГJГJIГJIГ
iiii
2
3
000
!1
2
3
2
3
0
)!1(
2
3
!1
2
3
2
3
)1(
i
i
i
i
i
r
i
i
Г
Г
Г
r
ГГ
Г
i
Classical special functions of matrix arguments
Системні дослідження та інформаційні технології, 2024, № 4 123
))1(())((
2
1
000
!1
2
1
2
1
0
)!1(
2
1
!1
2
1
2
1
)1(
irir
i
i
i
i
i
r
i
i
ii
i
JГJГ
Г
Г
Г
r
ГГ
Г
2
1
2
3
00
2
1
2
3
)!1(!
2
1
2
3
2
1
2
3
0
1
0
2
1
2
3
)1()(
0
i
i
i
i
i
kir
i
k
ij
k
i
i
i
i
Г
Г
Г
Г
kirk
ГГ
Г
Г
rj
i
Г
Г
2 BETA FUNCTION
2.1 Definition of the matrix-variate beta function
Similarly to the previous subsection, let's start with the definition of the matrix-
variate Beta function, using the integral definition of the Beta function:
.1),( 11
1
0
dtttyxB yx
For two matrices , kkYX with Jordan canonical forms ;1
111
UJUX
1
222
UJUY we consider the function of two matrix variables. We will first per-
form the following calculations for their Jordan matrices, respectively:
dteeJJB rr JtJt
rr
)()1(ln)()(ln
1
0
21
21))(), ((
Now we present and analyze the integral product separately:
D.O. Shutiak, G.B. Podkolzin, V.G. Bondarenko, Y.A. Chapovsky
ISSN 1681–6048 System Research & Information Technologies, 2024, № 4 124
,
0 )1(ln)(ln
,
)1(ln)(ln
)()1(ln)(ln
21
21
21
tt
ml
tt
JtJt
ee
Mee
ee rr
where ),( mlM is an arbitrary product of the l-th row and m-th column of the initial
matrices, and lm :
.00
)!((
)1(ln
00 )1(ln)(ln
),(
21
lm
t
eeM
lm
tt
ml
The number of zeros at the beginning and at the end of the product is l and
lmk – , respectively, so the resulting sum will consist of the middle part of the
vector:
.
)!)((!
))1((ln))((ln )1(ln)(ln
)(
0
),(
21 tt
jlmjlm
j
ml ee
jlmj
tt
M
Then, returning to the integral, we get the following:
; diagonaln the mainelements o)1, 1( 21
)1(ln)(ln
1
0
21 Bdtee tt
)1(ln)(ln
)(
0
1
0
21
)!)((!
))1((ln))((ln tt
jlmjlm
j
ee
jlmj
tt
.
))((0
1
1
,
)!)((!
1
)1, 1(
2
1)(
)(
0
21
jlmB
y
x
yx
yxB
jlmj
B
jlmj
lmlm
j
It should be noted that the resulting matrix, namely an arbitrary element in
the form of the sum ),( mlM and the corresponding resulting sums of derivatives
depend only on the difference of indices ),( ml , and not on each of them separately.
This, in turn, means that these elements are equal to each other for m and l on the
corresponding diagonals, which significantly reduces the number of necessary
calculations for finding the explicit form of the matrix ),(B for specific values.
Now let us return to the initial general definition for arbitrary matrices YX , :
tion decomposi
Jordanusing the
),( )1(ln)(ln
1
0
dteeYXB YtXt
;1
2
)1(ln
2
1
1
)(ln
1
1
0
21 dtUeUUeU JtJt
Classical special functions of matrix arguments
Системні дослідження та інформаційні технології, 2024, № 4 125
Compared to the situation with functions of one variable (e.g. Gamma func-
tion), when we start working with functions of several matrix variables, we have
two different Jordan transformations, and that is, two different matrices ,, 21 UU
which greatly complicates the task and makes it impossible to establish a direct
relationship between ),( YXB and ),( 21 JJB to obtain a clear analytical view of
the resulting matrix. In this regard, it is advisable to further consider the matrices
YX , as a pair of commuting matrices, which will give us the opportunity to find
a common Jordan basis for them, i.e. 21 UU . Also, in several points, it will al-
low the use of properties of the matrix exponent only for commuting matrices.
Therefore, taking this into account, we get the following result:
1)1(ln)ln(
1
0
1)1(ln1)(ln
1
0
),( 2121 UdteeUdtUUeUUeYXB JtJtJtJt
.), ( 1
21
UJJUB
2.2 Certain properties of the Beta function
1) Symmetry: ),(),( xyByxB
Since commuting matrices were chosen for research from the previous point,
symmetry for matrix arguments is also preserved.
2) Partial case of the function ./1),1( xxB
For the matrix-variate function, let's start with )( 1rJ :
dteJIB rJt
r
)()1(ln
1
0
1), (
In this case, we have a single matrix of the form:
;
0
!)1(
)1(ln
)1(ln
)1(ln1
)1ln(
)()1(ln
1
1
1
1
t
tk
t
Jt
e
k
et
e
e r
Then, with the absence of a product, we go directly to the integral:
)1, 1(0
1
1),(
!)1(
1
)1, 1(
,
1
1
1
)1(
1
1
B
y
x
y
yxB
k
B
JIB
k
k
r
)1,1(0
1
),1(
!)1(
1
)1,1(
1
1
1
)1(
1
B
yy
yB
k
B
k
k
D.O. Shutiak, G.B. Podkolzin, V.G. Bondarenko, Y.A. Chapovsky
ISSN 1681–6048 System Research & Information Technologies, 2024, № 4 126
)),1((
rixjordan mat
f theproperty o
1
1
0
)1()!1(
)1(
1
1
1
1
1
1
1
1
Jf
k k
k
.
1
)( where
x
xf
So, we see a complete analogy with the property of the scalar Beta function.
Summarizing:
.), (),( 1 UJIUBXIB
2.3 Pascal’s rule or the Beta function recurrence relation
Pascal's rule is one of the key identities in combinatorics and given the relation-
ship between the Beta function and binomial coefficients, as well as its use for the
recurrent computation of the Beta function, it will be appropriate to try to general-
ize it for two arbitrary commuting matrices.
dteeIYXBYIXB YtIXt )1(ln)()(ln
1
0
), (), (
dteeeedtee IYtXtYtctIYtXt )( )()1(ln)(ln)1(ln)(ln
1
0
)1(ln)(ln
1
0
dteUUeUUeUUeeUUe ItJtJtJtItJt )( )1(ln1)1(ln1)(ln1)1(ln)(ln1)(ln
1
0
2121
tnd exponenmatrices acommuting ies ofto propertaccording
dteUeUeeUeUe ItJtJtItJtJt )( )1(ln1)1(ln)(ln)(ln1)1(ln)(ln
1
0
2121
dteeUeUe ItItJtJt ))(( )1(ln)(ln1)1(ln)(ln
1
0
21 .
According to the property of the matrix exponent, the two terms obtained are
found as exponents of the diagonal matrix:
.
0
0
0
0
)1(ln
)1(ln
)(ln
)(ln
)1(ln)(ln I
e
e
e
e
ee
t
t
t
t
ItIt
So, the end result is as follows:
dtIUeUeIYXBYIXB JtJt )(), (), ( 1)1(ln)(ln
1
0
21
Classical special functions of matrix arguments
Системні дослідження та інформаційні технології, 2024, № 4 127
).,(), ( 1
21
1)1(ln)(ln
1
0
21 YXBUJJUBdtUeeU JtJt
Similarly to the scalar Beta function, Pascal's rule holds and has the same
form, unlike many other properties that have additional constructions when work-
ing with matrix variables.
3 JACOBI THETA FUNCTION
3.1 Definition of the main Jacobi Theta function in the form of an infinite sum
,Η))(), ((
2
21
nn
n
rr QJJ
where .Η, 12 π2)( rr iJiJ eeQ
As in the previous section, let's start with each of the factors separately and
then move on to the general form of the product:
)( 2
22
rJinn eQ
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
000
!1
00
)!2(
)(
0
)!1(
)(
)!2(
)(
!1
2
22
12222
in
in
in
ink
in
inkinkin
in
e
ein
e
k
ein
e
k
ein
k
einein
e
)(π2 1Η rinJn e
2
2
1
2
2
1
2
2
111
1
000
!1
00
)!2(
)(
0
)!1(
)(
)!2(
)(
!1
2
12
in
in
in
ink
in
inkinkin
in
e
ein
e
k
ein
e
k
ein
k
einine
e
.
Then the product of these matrices will be:
,
0
Η
2
2
1
2
2
1
2
2
),(
2
inin
ml
inin
nn
ee
Mee
Q
D.O. Shutiak, G.B. Podkolzin, V.G. Bondarenko, Y.A. Chapovsky
ISSN 1681–6048 System Research & Information Technologies, 2024, № 4 128
where ),( mlM is the product of the l -th row and the m -th column of the initial
matrices for lm .
,00
!0
)2(
)!(
)(
)!(
)2(
!0
)(
00
0202
2
),(
2
2
1
in
lm
in
lm
inin
eeM
lmlm
inin
ml
while the number of zeros at the beginning and end will be equal to l and
1mk respectively. Then as a result we get the sum:
.
)!(!
)2()(
2
2
12
2
0
),(
inin
jlmjlm
j
ml ee
jlmj
inin
M
Thus, similar to the product for the Beta function, we get a matrix element that
depends only on the difference in the indices of the initial row and column, i.e. all the
elements of the resulting matrix will be equal on the corresponding diagonals.
The next step is to return to the initial form of the function, namely to the sum:
;
), (0
), (
)!(!
1
), (
))(), ((
21
2
1
0
21
21
z
z
z
jlmj
JJ
jjlm
lmlm
j
rr
1
21 ), (
commuting
, for
), ( UJJU
TZ
TZ
3.2 The period of the Jacobi theta function
The scalar Jacobi theta function is periodic with a period of 1 in z :
),, (), 1( zz and by completing the square, — quasiperiodic in z :
)., (), ( 2 zez izi
In the case of matrix variables, the 1-periodicity of the first variable is trans-
formed into the periodicity of the unit matrix I :
))(), 1(()1()())(, )(( 211121 11 rrrrrr JJJIJJIJ
), 1(0
1
), (
)!(!
1
), 1(
21
2
1
0
21
z
z
z
jlmj jjlm
lmlm
j
Classical special functions of matrix arguments
Системні дослідження та інформаційні технології, 2024, № 4 129
)).(), ((
), (0
), (
)!(!
1
), (
21
21
2
1
0
21
rr
jjlm
lmlm
j
JJ
z
z
z
jlmj
Then for two arbitrary commuting matrices TZ , :
TinIinZin
n
TinZin
n
eeeeeTIZ
22 π222π2)1(2), (
1π2221π2212 UU 2
2
12
2
1 JinIinJin
n
JinIinJin
n
eeUeUeeUUe
1
21
1π2)(2 ))(, )((U2
2
1 UJIJUeUe rr
JinIJin
n
.), (), ( 1
21 TZUJJU
4 COMPARISON OF THE OBTAINED RESULTS WITH EXISTING METHODS
OF WORKING WITH MATRIX-VARIATE FUNCTIONS
4.1 Comparison of obtaining the matrix Gamma function using the Lanczos
approximation method and the obtained method
Computing the matrix Gamma function by the Lanczos method [3] is performed
on the basis of the following formula:
IAIA
eIAAГ 2
1
2
1
2
1
2
,)())1(()()( ,
1
1
0
AeIkAcIc mk
m
k
where )(kc are the Lanczos coefficients that depend on the parameter .
Typically, pre-logarithmization is used to optimize calculations and avoid
overflow problems during calculations:
IAIAIAAГ
2
1
2
1
ln
2
1
)2(ln
2
1
))((ln
)())1((ln ,
1
1
0 AeIkAccIc mk
m
k
.
D.O. Shutiak, G.B. Podkolzin, V.G. Bondarenko, Y.A. Chapovsky
ISSN 1681–6048 System Research & Information Technologies, 2024, № 4 130
It is also important to note that the set of coefficients )(kc is found empiri-
cally [3] and for the example for the pair 10,9 m the following values are
used:
k ) 9(kc
0 1.000000000000000174663
1 5716.400188274341379136
2 14815.30426768413909044
3 14291.49277657478554025
4 6348.160217641458813289
5 1301.608286058321874105
6 108.1767053514369634679
7 2.605696505611755827729
8 0.742345251020141615 210
9 0.538413643250956406 210
10 0.402353314126823637 210
Now let's compare the actual algorithms for finding matrices by these two
methods:
Algorithm for finding the function
using the Lanczos method
Algorithm for finding the function
using the Jordan form
1. Set 9 ; 10m ; 1
10
AcIcS ; 1. Eigenvalues i of matrix A ;
2. for 10: 2 k 2. Eigen and adjoint vectors ix ;
3. ;))1(( 1 IkAcSS k 3. Jordan form J ;
4. end
4. )(JГ and transitional matrix U
based on ix ;
5.
IAIAIL
2
17
ln
2
1
)2ln(
2
1
)ln(
2
17
SIA
;
5. 1)()( UjUГAГ .
6. .)( LeAГ –
So, as we can see, the proposed algorithm is much more convenient for actu-
ally finding the values of the matrix Gamma function )(AГ in comparison with
some existing numerical methods. It is also in addition to the above that our
method has the advantage of being able to use the obtained function and its prop-
erties in further research. Similar results were obtained for Spouge’s approxima-
tion method [3], since both of them have similar algorithms.
4.2 Research using the Schur decomposition
The Schur decomposition method [4] is based on the decomposition of the input
matrix and its representation through unitary and upper triangular:
.;: , , , **: )(, *
2
*
121 QQRBQQRARRUABBAnnBA
Classical special functions of matrix arguments
Системні дослідження та інформаційні технології, 2024, № 4 131
Thanks to this, in their work L. Jódar, J. CCortés [5] for two commuting ma-
trices proved several properties of the matrix-variate Beta function, namely the
symmetry of the variables and the connection with the matrix Gamma function:
.)(1)()( ), ( QPГQГPГQPB
It should be noted that the last property was proved only for diagonalizable,
commuting matrces P and Q. Compared to the obtained results, we coan sii that
the main advantage of using the Jordan canonical form is the presence of an ex-
plicit form of the resulting matrix. This, in turn, gives us the following advantages
compared to the Shur Schedule:
Ability to derive properties associated with certain partial cases and spe-
cific function values;
From the point of view of computational complexity, although histori-
cally the calculation of the Jordan canonical form was usually considered a very
difficult task, the properties of the matrix function from the Jordan matrix allow
us to bypass this step, and so the need remains only to find the eigenvalues and
the corresponding vectors to form a basis. Then, comparing to the Schur decom-
position, which has a computational complexity of )( 3nO , our method will have
an approximate complexity of 376.2 2),( nO .
4.3 The zonal polynomials method
The method of zonal polynomials [6] is one of the methods for studying such
functions using integrals and the difference in approach will be illustrated on its
example.
In this method, the studied function differs from others, namely, it has the
following form:
.)(det)(det),( 2
)1(
2
)1(
0
dXXIXbaB
m
b
m
m
aI
m
m
Additional results and generalizations of this function were found using
zonal polynomials and evaluating the resulting integral for them. The main use
case of it and its generalized forms is the matrix beta distribution:
For ),,(~ baBU I
p the distribution density of the positive definite square
matrix U :
.)(det)(det
), (
1
)( 2
1
2
1
p
b
p
p
a
p
UIU
baB
Uf
As we can see, this function and similar functions of this type contain only
matrix determinants and, in some cases, trace. This means that these functions are
limited to uses only in problems in which the input signal has a matrix form, and
the output signal is already scalar. This has a number of disadvantages in solving
some statistical problems in which it is important to leave connections between
certain vectors or blocks of vectors, like the problems that were mentioned in the
introductory section.
D.O. Shutiak, G.B. Podkolzin, V.G. Bondarenko, Y.A. Chapovsky
ISSN 1681–6048 System Research & Information Technologies, 2024, № 4 132
REFERENCES
1. Sujit Kumar Mitra, “A Density-Free Approach to the Matrix Variate Beta Distribu-
tion,” Sankhyā: The Indian Journal of Statistics, Series A (1961-2002), vol. 32,
no. 1, pp. 81–88, 1970. Available: http://www.jstor.org/stable/25049638
2. Amr Elrawy, Mohammed Saleem, and Z. Kishka, “The Matrix of Matrices Exponen-
tial and Application,” International Journal of Mathematical Analysis, 13, pp. 81–97,
2019. doi: 10.12988/ijma.2019.9210
3. João Cardoso, Amir Sadeghi, “Computation of matrix gamma function,” BIT Nu-
merical Mathematics, 59, 2019. doi: 10.1007/s10543-018-00744-1
4. Maya Neytcheva, “On element-by-element Schur complement approximations,”
Linear Algebra and its Applications, vol. 434, issue 11, pp. 2308–2324, 2011. Avail-
able: https://doi.org/10.1016/j.laa.2010.03.031
5. L Jódar, J.C Cortés, “Some properties of Gamma and Beta matrix functions,” Ap-
plied Mathematics Letters, vol.11, issue 1, pp. 89–93, 1998. Available:
https://doi.org/10.1016/S0893-9659(97)00139-0
6. Daya Nagar, Sergio Gómez-Noguera, and Arjun Gupta, “Generalized Extended Ma-
trix Variate Beta and Gamma Functions and Their Applications,” Ingeniería y Cien-
cia, 12, pp. 51–82, 2016. doi: 10.17230/ingciencia.12.24.3
Received 02.11.2023
INFORMATION ON THE ARTICLE
Dmytro O. Shutiak, ORCID: 0009-0008-6480-3706, World Data Center for
Geoinformatics and Sustainable Development of the National Technical University of Ukraine
“Igor Sikorsky Kyiv Polytechnic Institute”, Ukraine, e-mail: dima.shutyak@gmail.com
Gleb B. Podkolzin, ORCID: 0000-0002-7120-2772, Educational and Research Institute
for Applied System Analysis of the National Technical University of Ukraine “Igor Sikor-
sky Kyiv Polytechnic Institute”, Ukraine, e-mail: podkolzin.gleb@lll.kpi.ua
Victor G. Bondarenko, ORCID: 0000-0003-1663-4799, Educational and Research Insti-
tute for Applied System Analysis of the National Technical University of Ukraine “Igor
Sikorsky Kyiv Polytechnic Institute”, Ukraine, e-mail: bondarenvg@gmail.com
Yury A. Chapovsky, ORCID: 0009-0001-8981-4742, Educational and Research Institute
for Applied System Analysis of the National Technical University of Ukraine “Igor Sikor-
sky Kyiv Polytechnic Institute”, Ukraine
КЛАСИЧНІ СПЕЦІАЛЬНІ ФУНКЦІЇ З МАТРИЧНИМИ ЗМІННИМИ /
Д.О. Шутяк, Г.Б. Подколзін, В.Г. Бондаренко, Ю.А. Чаповський
Анотація. Розглянуто декілька найбільш часто використовуваних спеціальних
функцій та їх ключові властивості, а також запропоновано аналітичний підхід
до побудови їх аналогів із метричними змінними. Щоб досягти цього, ми уни-
кали використання будь-яких алгоритмів чисельного наближення та натомість
покладались на властивості матриць, матричної експоненти та Жорданову но-
рмальну форму для представлення матриць. Ми зосередились на таких функ-
ціях: гамма-функція як приклад функції однієї змінної з великою кількістю
властивостей і застосувань; бета-функція, щоб підкреслити подібності та від-
мінності від додавання другої змінної до функції матричної змінної; тета-
функція Якобі. Побудовано явні представлення функцій і доведено декілька
ключових властивостей для цих функцій; висвітлено та порівняно інші підхо-
ди, які використовувалися в минулому для вирішення цих задач.
Ключові слова: матриця, спеціальна функція, гамма-функція, бета-функція,
тета-функції Якобі, Жорданова нормальна форма.
|
| id | journaliasakpiua-article-322530 |
| 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/ff/19bcc815917d92ddbf6229d3e0912cff.pdf |
| spelling | journaliasakpiua-article-3225302025-02-09T21:55:38Z Classical special functions of matrix arguments Класичні спеціальні функції з матричними змінними Shutiak, Dmytro Podkolzin, Gleb Bondarenko, Victor Chapovsky, Yury matrix special function matrix function gamma function beta function Jacobi theta function Jordan normal form матриця спеціальна функція гамма-функція бета-функція тета-функції Якобі Жорданова нормальна форма This article focuses on a few of the most commonly used special functions and their key properties and defines an analytical approach to building their matrix-variate counterparts. To achieve this, we refrain from using any numerical approximation algorithms and instead rely on properties of matrices, the matrix exponential, and the Jordan normal form for matrix representation. We focus on the following functions: the Gamma function as an example of a univariate function with a large number of properties and applications; the Beta function to highlight the similarities and differences from adding a second variable to a matrix-variate function; and the Jacobi Theta function. We construct explicit function views and prove a few key properties for these functions. In the comparison section, we highlight and contrast other approaches that have been used in the past to tackle this problem. Розглянуто декілька найбільш часто використовуваних спеціальних функцій та їх ключові властивості, а також запропоновано аналітичний підхід до побудови їх аналогів із метричними змінними. Щоб досягти цього, ми уникали використання будь-яких алгоритмів чисельного наближення та натомість покладались на властивості матриць, матричної експоненти та Жорданову нормальну форму для представлення матриць. Ми зосередились на таких функціях: гамма-функція як приклад функції однієї змінної з великою кількістю властивостей і застосувань; бета-функція, щоб підкреслити подібності та відмінності від додавання другої змінної до функції матричної змінної; тета-функція Якобі. Побудовано явні представлення функцій і доведено декілька ключових властивостей для цих функцій; висвітлено та порівняно інші підходи, які використовувалися в минулому для вирішення цих задач. 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/322530 10.20535/SRIT.2308-8893.2024.4.10 System research and information technologies; No. 4 (2024); 117-132 Системные исследования и информационные технологии; № 4 (2024); 117-132 Системні дослідження та інформаційні технології; № 4 (2024); 117-132 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/322530/312910 |
| spellingShingle | матриця спеціальна функція гамма-функція бета-функція тета-функції Якобі Жорданова нормальна форма Shutiak, Dmytro Podkolzin, Gleb Bondarenko, Victor Chapovsky, Yury Класичні спеціальні функції з матричними змінними |
| title | Класичні спеціальні функції з матричними змінними |
| title_alt | Classical special functions of matrix arguments |
| title_full | Класичні спеціальні функції з матричними змінними |
| title_fullStr | Класичні спеціальні функції з матричними змінними |
| title_full_unstemmed | Класичні спеціальні функції з матричними змінними |
| title_short | Класичні спеціальні функції з матричними змінними |
| title_sort | класичні спеціальні функції з матричними змінними |
| topic | матриця спеціальна функція гамма-функція бета-функція тета-функції Якобі Жорданова нормальна форма |
| topic_facet | matrix special function matrix function gamma function beta function Jacobi theta function Jordan normal form матриця спеціальна функція гамма-функція бета-функція тета-функції Якобі Жорданова нормальна форма |
| url | https://journal.iasa.kpi.ua/article/view/322530 |
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