Applied Mathematics, 2013, 4, 1260-1268
http://dx.doi.org/10.4236/am.2013.49170 Published Online September 2013 (http://www.scirp.org/journal/am)
Matrix Functions of Exponential Order
Mithat Idemen
Engineering Faculty, Okan University, Istanbul, Turkey
Email: midemen@gmail.com
Received May 31, 2013; revised June 30, 2013; accepted July 7, 2013
Copyright © 2013 Mithat Idemen. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
Both the theoretical and practical investigations of various dynamical systems need to extend the definitions of various
functions defined on the real axis to the set of matrices. To this end one uses mainly three methods which are based on
1) the Jordan canonical forms, 2) the polynomial interpolation, and 3) the Cauchy integral formula. All these methods
give the same result, say g(A), when they are applicable to given function g(t) and matrix A. But, unfortunately, each of
them puts certain restrictions on g(t) and/or A, and needs tedious computations to find explicit exact expressions when
the eigen-values of A are not simple. The aim of the present paper is to give an alternate method which is more logical,
simple and applicable to all functions (continuous or discontinuous) of exponential order. It is based on the two-sided
Laplace transform and analytical continuation concepts, and gives the result as a linear combination of certain n matri-
ces determined only through A. Here n stands for the order of A. The coefficients taking place in the combination in
question are given through the analytical continuation of g(t) (and its derivatives if A has multiple eigen-values) to the
set of eigen-values of A (numerical computation of inverse transforms is not needed). Some illustrative examples show
the effectiveness of the method.
Keywords: Matrix; Matrix Functions; Analytical Continuation; Laplace Transform
1. Introduction
For many theoretical and practical applications one needs
to extend the definitions of functions defined on the real
axis to the set of matrices. The history of the subject goes
back to the second half of the nineteenth century when
Cayley, who is the main instigator of the modern nota-
tion and terminology, introduced the concept of the
square-root of a matrix A [1]. Since then a huge work has
been devoted to the definitions, numerical computations
and practical applications of the matrix functions. A
rather detailed history (including a large reference list)
and important results (especially those concerning the
numerical computation techniques) are extensively dis-
cussed in the book by Higham [2]. So, we eschew here of
making a review of the historical development and giv-
ing a large reference list.
To extend the definition of a scalar function g(t), de-
fined for , to the set of matrices, one starts from
an explicit expression of g(t), which can be continued
analytically into the complex plane C, and replaces
there t by A. If the result is meaningful as a matrix, then
it is defined to be g(A). Before going into further detail, it
is worthwhile to clarify the meaning of the word “de-
fined” appearing in the expression of “defined for
t
t
Th
”. It is especially important when g(t) consists of a
multi-valued inverse function. To this end consider, for
example, the square-root function g(t) = t1/2. Its definition
requires, first of all, a cut connecting the branch point t =
0 to the other branch point t = in the complex plane C.
en, by choosing one of its possible values at a given
point, for example g(1), one defines it completely. The
result consists of a well-defined branch of the square-root
function. If one replaces t in this expression by A, then
one gets a (unique) matrix to be denoted by A1/2. This
matrix satisfies the equation X2 = A which may have
many solutions denoted also by A1/2. The above-men-
tioned function g(A), which consists merely of the exten-
sion of the above-mentioned well-defined branch of the
square-root function, can not permit us to find all these
solutions. For example the equation X2 = I, where I de-
notes the unit 2 × 2 matrix, has infinitely many solutions
given by
cos sin
sin cos

X,
where
stands for any complex angle. All these matrices
are defined to be the square-root I. But the above-
mentioned matrix g(A) gives only one of them, namely I
C
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M. IDEMEN 1261
or (I).
The known classical methods used in this context are
grouped as follows (see [2], Sections 1, 2):
1) Methods based on the Jordan canonical formula;
2) Methods based on the Hermite interpolation for-
mula;
3) Methods based on the Cauchy integral formula.
All these methods are applicable when the function
g(t), defined on the real axis, can be analytically contin-
ued into a domain of the complex-plane, which involves
the spectrum of the matrix A (see def. 1.2 and def. 1.4 in
[2]). Consider, for example, the Heaviside unit step func-
tion H(t) defined on the real axis by

1, 0
12, 0
0, 0.
t
Ht t
t
(1a)
It is obvious that the analytical continuation of H(t)
into the complex z-plane, if it is exists, has the point z = 0
as a singular point. To reveal H(z), let us try to find its
Taylor expansion about any point a > 0. This expansion
is valid in the circle with center at the point z = a and
radius equal to r = a. Since all the coefficients except the
first one are equal to naught, one gets H(z) 1 at all
points inside the circle in question. By letting a one
concludes that H(z) is regular in the right half-plane z >
0. If the above-mentioned Taylor expansion were made
about a point a < 0, then one would get H(z) 0 for all z
with z < 0. This shows that H(z) is a sectionally regular
(holomorphic) function (see [3], Section 2.15). On the
basis of the Plemelj-Sokhotiskii formulas (see [3], Sec-
tion 2.17), for the points on the imaginary axis one writes
H = 1/2, which yields

1, 0
12, 0
0, 0
z
Hz z
z.



(1b)
Notice that (1a) and (1b) can also be obtained by com-
puting the improper integral

22
1
d,
2π2
z
Hz z


(1c)
where the bar on the integral sign stands for the Cauchy
principal value.
From (1b) one concludes that the seemingly general
and elegant method 3), which is based on the Cauchy
integral
 
1
1d,
2πC
gi


AIA (2)
where C stands for a closed contour such that the domain
bounded by C involves all the eigen-values of A and the
function g(z) is regular there, can not be applicable to
find H(A) (and other functions expressible through H(t))
when A has eigen-values having both positive and nega-
tive real parts.
As to the methods 1) and 2), they need, in general,
some tedious and cumbersome computations if A has
multiple eigen-values.
In the present note we will consider the case when the
function g(t), defined on the real axis, is of the exponen-
tial order at both t = + and t = , and give a new
method which seems to be more logical and effective
especially when the matrix A has multiple eigen-values.
It gives the result as a linear combination of n matrices
determined only by the matrix A. To this end we consider
the Laplace transforms of g(t) on the right and left halves
of the real axis, namely:
  
0
ˆed,
st
g
sggt
 
 t (3a)
 
0
ˆed
st
g
sggt
 

 t (3b)
and write [4]
  

11
ˆˆ
eded,
2π2π
,.
ts ts
LL
g
tgssgs
ii
t




s
(3c)
If the orders of the function g(t) for t and t
() are c+ and c, respectively, then the function
ˆ
g
s
is a regular function of s in the right-half plane
s
c
and the integration path L+ appearing in (3c)
consists of any vertical straight-line located in this half-
plane (see Figure 1). Similarly,

ˆ
g
s
is regular in the
half-plane
s
c
L
and the integration path L is any
vertical straight-line in this half-plane (if c+ < c, then
one can assume L
). Furthermore, if g(t) as well as
its derivatives up to the order (m-1) are all naught at t = 0,
i.e. when
 


1
00 0
m
gg g
0
 , (4a)
then one has
c
O
c
+
s
s
L
+
L
Figure 1. Regularity domains of

ˆ
g
s
and the integration
lines L when c < c+.
Copyright © 2013 SciRes. AM
M. IDEMEN
1262



ˆ
mm
g
sg s

(4b)
and



ˆ,
mm
g
sg s

(4c)
which yield inversely
 



11
ˆˆ
ed ed,
2π2π
,.
m
mts mts
LL
g
ss sgss sgt
ii
t


 (4d)
It is worthwhile to remark here that the formula (4d)
permits us to compute


m
g
t only at points on the real
axis although g(t) and its derivatives are defined in (or
can be continued analytically into) the complex t-plane.
Therefore, when the point t is replaced by a complex
C
 , in what follows we will replace the left hand side
of (4d) by the analytical continuation of


m
g
t to the
point
and write
 


11
ˆˆ
ed ed,
2π2π
.
m
ms ms
LL
gsss gsssg
ii
C





(4e)
The formulas (3c), (4d) and (4e) will be the basis of
our approach.
Let A be a square matrix of dimensions n n. We will
define g(A) by replacing t in (3c) by A, namely:
  
11
ˆˆ
edsed.
2π2π
ss
LL
g
gsgs s
ii



AA
A (5)
Thus the computation of g(A) becomes reduced to the
computation of exp{At}. As we will see later on, the lat-
ter consists of a linear combination of certain constant
matrices j ( = order of A). Hence g(A) will
also be a linear combination of these j’s for every g(t).
It is important to notice that to compute the coefficients
in the combinations in question we will never need to
compute the transform functions
1, ,jn

ˆ
g
s
as well as the
integrals of the form (5) if the analytical continuation of


m
g
t is known at the eigen-values of A (see the ex-
amples to be given in Section 4). These points constitute
the essential properties of the definition (5):
1) It unifies the definition of g(A) for all functions g(t)
of exponential order;
2) It gives an expression of g(A) in terms of certain
matrices which take place in the expression of exp(At)
and are determined only by A;
3) It reduces the computation of g(A) to the computa-
tion of exp(At) together with some scalar constants to be
determined in terms of g(t) (and its derivatives when A
has multiple eigen-values) at the eigen-values of A.
The details are given in the theorems that follow.
2. Basic Results
In what follows we will denote a square matrix A of en-
tries ajk by A = [ajk]. Here the first and second indices
show, respectively, the row and column where ajk is
placed. The transpose of A will be indicated, as usual, by
a super index T such as . The characteristic
polynomial of A will be denoted by
T
T
jk
a


A
f

, i.e.
detf

A
I.
Theorem-1. Let A = [ajk] be an n n matrix with char-
acteristic polynomial
f
. Then,
1) when all the zeros of

f
, say 1,,
n
, are dis-
tinct, one has


0
1
expeα
nλt
α
t
ΓA (6a)
with

0
Γ given by

 
T
0
1,
jk
f
a
f



Γ
(6b)
2) when
f
has p distinct zeros, say 1,,
p
,
with multiplicities 1,,
p
mm, respectively, one has
 

12
12 0
1
exp
e
p
mm
mm
t
tt t


 



A
 
(6c)
with the matrices


0, ,1
k
km
a
Γ given by



 
T
1
1
1
! 1!
d.
d
k
m
mk
mk
jk
s
km k
sfs
a
fs
s


 












Γ
(6d)
Theorem-2. Let A = [ajk] be an nn matrix while g(z),
defined in the complex z-plane, is regular at all the eigen-
values of A and its restriction to the real axis is of expo-
nential order at both t = + and t = . If all the eigen-
values of A, say 1,,
n
, are distinct, then one has
 

0
1
.
n
gg
ΓA (7)
Here stands for the matrix taking
place in the expression of exp{At}.

0 1,,a
Γ
n
Theorem-3. Let A = [ajk] be an n n regular matrix
which has p distinct eigen-values 1,,
p
with multi-
plicities 1,,
p
mm
1,,; 0ap
, respectively, while g(z), defined in
the complex z-plane, is regular at all the eigen-values of
A and its restriction to the real axis is of exponential or-
der at both t = + and t = . Let the non-zero matrices
taking place in the expression of exp{At} be
. Then one has


, ,k
ΓK
k
C
opyright © 2013 SciRes. AM
M. IDEMEN
Copyright © 2013 SciRes. AM
1263
G










12
12 0
1
,
pmm
N
mm
gGG


 





ΓΓ ΓAA (8)
where N stands for an integer such that

N
Gtt gt
and its derivatives up to the order (K-1) are all naught at t
= 0.
Let the characteristic polynomial of A be
f
s:

11 121
21 222
12
...
...
...... ... ...
...
n
n
nn nn
as aa
aasa
fs
aaa
Before going into detail of proofs of the above-men-
tioned theorems, it is worthwhile to draw the attention to
the fact that theorems 2 and 3 give the matrix function
g(A) as a combination of the n matrices

k
Γ which
appear in the expression of exp(At). They are the same
(invariant) for all g(t).
s
. (10)
Then the entry of the inverse matrix

1
s
AI,
which is placed at the k-th row and j-th column, can be
computed through the polynomial

f
s as follows:
Proof of theorem-1. Our basic matrix function exp{At}
is defined, as usual, through the infinite series
 
1, ,1,2,,.
jk
f
sjk n
a
fs
(11)

22
1
exp1,
2!
atatatt 
Thus (9) yields
by replacing there the scalar constant a by the square
matrix A, namely:
  
T
11
exped.
2π
ts
jk
L
fs
ts
i a
fs

 

A (12)
 
22
1
exp, .
2!
tI tttt AAA X
Here the integration line L is any vertical straight-line
such that all the eigen-values of A are located in the left
side of L.
It is obvious that X(t) defined as above is the unique
solution to the differential equation

tXX'A t un-
der the initial condition X(0) = I. Hence, by applying the
Laplace transform to this equation one gets
 
1
ˆ
ss

X
AI which permits us to write
 
1ts
1
exped.
2πL
tss
i
 
AAI
If the eigen-values are all simple, then the integral in
(12) is computed by residues and gives (6a). When some
of the eigen-values are multiple, as stated in theorem-1b,
the residue method gives
(9)

 
 
T
1
1
1
.
1d
exp e
1! d
m
m
p
st
m
jk
s
s
tf
a
mfs
s
s












A
(13)
Proof of theorem-2. When the eigen-values of A are
all simple, in (5) one replaces exp(As) by its expression
given in (6a) and obtains
It is obvious that the derivatives in (13) yields a poly-
nomial in t of degree . Hence the final expres-
sion of exp(At) can be arranged as what is given in (6c).
1m

 
 
T
1
111
ˆˆ
ed ed
2π2π
ns
jk LL
s
g
fgss gs
fa ii


 s
 





A. (14)
If all the eigen-values are real, then (3c) reduces (14)
to (7). When some or all of the eigen-values are complex,
we replace (3c) by (4e) with m = 0 and arrive again (7).
Proof of theorem-3. Now consider the case when A
has multiple non-zero eigen-values and define
Gt
N
tgt where the integer N will be determined appro-
priately later on. If in (5) one replaces g(t) by G(t) and
exp{As} by (6c), then we get



112 200
1
,
p
N
mmm m
gcc c
 
 
 

ΓΓ ΓAA (15a)
where


11
ˆˆ
de
e
2π2π
ss
kk
k
LL
cGsssGss
ii






ds
(15b)
M. IDEMEN
1264
with 1,,p
e coeffici
and . Remark that some
hents
0, ,1
a
km

k
of t
Γ may be equal
ub-index k be K
nsi
to naught (see
vativ
Let g(t) be the characteristic polynomial of A (i.e. g(t)
)
In order to show the application and effectiveness of the
ider some simple
One can easily check that A has a tple eigen-value
= 2. Therefore the theorems 1b and 3 are applicable di-
re
ties p and m
mention
= 3. On the other hand
ex.-2)rgest s for which one has

0
k
Γ. Then, by codering the requirements in (4a),
we will choose the integer N such that G(t) and its de-
ries up to the order (K-1) are all naught for t = 0. In
this case all terms existing in (15b) are computed through
(4d) or (4e) and give (8).
3. A Corollary (Cayley-Hamilton Theorem)
. Let the la
f(t)). In this case all the terms taking place in (7) or (8
are equal to zero. When the eigen-values are all simple,
from (7) one gets directly f(A) = 0, which is valid for
both regular and singular matrices. In the case of multi-
ple eigen-values, (8) gives f(A) = 0 if A is not singular.
We remark that the Cayley-Hamilton theorem is correct
for all matrices. We will use this theorem to compute the
factor AN taking place in the formula (8) (see ex.-2).
4. Some Illustrative Examples
method, in what follows we will cons
examples.
Ex.-1 As a first example consider the case where A is
as follows:
812 2

341
122

 




A.
ri
ctly for all functions of exponential order. The quanti-
ed in those theorems are: p = 1, m1
from
 
3
8122
34f

 1 2
122
 
 
one computes

T
f
2
2
2
2612204 2
5 310142
24244
jk
a
 


 

 



which gives (see (6c) and (6d))
22
10
exp et
tttΓΓΓA
2
where
01
612 2
, 361
120
I



and
2
240
1120
2000

Γ.
Since 2 0, one has K = 2 which ows that the for-
mula (8) is applicable with 0 N 2. For example, in
or
 
sh
der to find the expressions of A and sin A (with
2), one can choose N = 0 while A, sin A, sin A,
3
sin A, arcsinA etc. needs N = 1. To compute cosA, logA,
cos A, signA and arcosA one has to choose N = 2.
To check the formulas, we would like to compute first
integer 2) through the formula (7) which gives An (n =

1
112421122
nn
nnn nn
 
21 3 2
210
1
12 22211221 1242.
22
nn
nnnnn
nn
nn nnn nnn nn
nn
  
2
2
n
nn
1
2
n
 


  
 

ΓΓΓA
Thus for n = 2 one gets
2
30 528
13224
484


A.
Notice that by a direct multiplication of A with itself one gets the sam result.
Similarly, one gets also
e

 
4343 4323
3
34
tt t

 
ΓΓΓ ΓAA 1343 3
210 21
2
4, 2
93
t

 


ΓI gt t
C
opyright © 2013 SciRes. AM
M. IDEMEN 1265
  









2222
21
2
21
log loglogloglog
2log234log224log2,
t
gtxtxt xtt xtt xt
xxx

 
 



ΓΓ
ΓΓ
AA
I
0
Γ

  




 


22 22
21
2
2
21
cos cos coscoscos
7
2cos 2sin 24cos22sin 24cos2,
22
t
gtxtxt xtt xtt xt
xx
0
x
xxxx
x
 
 









ΓΓ
ΓΓ
AA
I
Γ

  




    
21
2
2
21
sinsin sinsinsin
3cos2sin 2cos2sin22sin 2,
4
42 2
t
gtxtxt xttxtt xt
xx x
xx xx

 
 




 



ΓΓΓ
ΓΓ
AA
I
0
x

2 2
210
30 528
signsign24413224sign.
484
gt t


 




ΓΓΓ
A
AA AI
Remark that for different branches of t, 3t and

lo one gets different expressions for 3Ag
x
t ,

glo
x
A
,

cos
x
A and

sin
x
A (sSection 5 ee
and theo
Finallylet us conse branche
trigonome arcsint and
ut as shown in Figure 2 into the
re
taking place in the formula (8) can be computed rather
easily by using the Cayley-Hamilton theorem as follows:
rem-4).
, ider ths of the inverse
tric functions g(t) = h(t) = arccost
which map the t-plane c
gions in the g- and h-planes shown in Figures 3 and 4
(the so-called principal branches of these functions!).
For the first function we have to choose N = 1 while
the second one needs N = 2. The matrices A1 and A2
1
6204
1514 2
8

244
A,
2
56144 32
1368816 .
64 1632 16


A
Thus, by starting fromd

arcsinGt tt an
H
t
2arccostt one gets from (8)
816 0
2
arcsin4 8



20 4
010 202
8
24 4
0004802 44
4086 20
45 14



 A 




and
3264 064 1283256144 32
83
arccos16 32032 6416368816
6416 16
0001632 0163216
 

 

 
 
 
 
 
A
with
arcsin2,arccos2, 1 3

.
an interesting exercise to check that It is
π
arcsin arccos .
2
 
0
A
AIII
704
838
80 5
A.
In this case one has
which shows again that the theorems 1b and 3 are appli-
Ex.-2 Now consider the case where A is as follows:
 
13f

 ,
2
C
opyright © 2013 SciRes. AM
M. IDEMEN
1266
1 1
B
+
B
C
2
A
+
A
C
1
t
Figure 2. Complex plane-t cut along the lines () < t < 1
and 1 < t < .
g
B
C
2
C
1
A
/2 /2
A
+
B
+
arcsin
Figure 3. Mapping of the t-plane into the g-plane through
the principal branch of the function g = arcsint.
A
A
+
C
1
C
2
O
h
B
+
B
arccos
Figure 4. Mapping of the t-plane into the h-plane through
the principal branch of the function h = arccost.
cable with p = 2,
1 = 1,
2 = 3, m1 = 1 and m2 = 2.
From the expression of f(
) one gets easily
Therefore from (6c) and (8) we write
.
Since K = 0, (8) is applicable with N = 0 and ges, for
example,
2
Γ
and
 
12
00
1012 0 1
202,212



 

ΓΓ
20220 1



and

2
1
000
000




Γ0.
000



12
3
00
ee e
tt t
ΓΓ
A
iv






 
12
10 2120
12
00
signsign
302
414
sign
 
 
 



ΓΓ
ΓΓ
A
sign
40 3



 



12 22
1021200
HH HH
 
 
AΓΓ ΓΓ
201
212.
20 1





By direct multiplication of these matrices by them-
selves one gets
are the basic properties for the original functions
signt and H(t).
Similarly, one gets also
 


2
2
sign, HHAI AA,
which
20
arcsin2 222
22 02

 


 

A
and
20
arccos2222
22 02

 


 

A
with
arcsin1 ,arcsin3,


arcos1 ,arccos3

.
Here the functions and

arcsin tgtarccos t
ht consist of the functions considered
k that
in the e
ove. One can easily chec
xample-
1 ab
and π
arcsin arccos 2

AA,
sin arcsin
A
AI
h are the basic properties for the original functions
t and arccost.
Ex.-3 Finally, we want to give an exame with com-
plex-valued eigen-values. To this end consider the case
where
whic
arcsin
pl
ab
ba
A
C
opyright © 2013 SciRes. AM
M. IDEMEN
Copyright © 2013 SciRes. AM
1267
has with any real a and b. In this case one




 
 
12
00
12
00
12
00
sign signisigni
if 0
,if 0
ab ab
a
a


 
ΓΓ
ΓΓ
ΓΓ
A
I
12
iabi,ab
 ,
I
and

1
0
1i
1
2i1


Γ,


2
0
1i
1.
2i1


Γ ,i
f 0a
0

and
Thus, for the following functions one writes
 

 

 

 

11
arcsiniarcsinii arcsiniarcsini
22
arcsin 11
iarcsin iarcsin iarcsinarcsin i
2
2
ab abab ab
ab abab ab

  




A.
Here arcsint stands for the function defined in ex.-1
above. It is interesting to check that
A
. Inverse Power of a Matrix
m (= B) which cor-
respond to a given matrix A through the relationm = A.
The next lemma and theorem concern this case.
nction


4
66
g
w
,


4
77
g
w
,


5
88
g
w
.
Theorem-4. Let A be an n n regu matrix with p
different eigen-values
lar


2
sign if 0, sinarcsina AIA
1pn
p different
while m is an integer.
Then there exist at least m matrices B such that
Bm = A.
co
5
Proof. Let the eigen-values of A be

1, ,
Consider a Riemann surface which the
function
jjp
.
rresponds to
From the example 1 considered above one observes
that there are many inverse powers A1/
1m
g
tt and denote, one k-th sheet of this
Riemann surface, the values of the function g(t) at
th
j by
Bpoints



1, 1,,
k
mk
jj
wk m

1m
g
tt, Lemma. Let the values of the fu

 . On the basis of
the lemma, we can ange the cut line appropriately such
that the values of tm at the points
defined on a Riemann surface, at given p points
12
,,,
p

be


0
k
jj
gw
, 1, ,km. Then
there is a Riemann surface such that on one of its sheets
g(t) takes previously chosen values at the points
1
,,,
ch
1/

,,,
12
p

be-
co me equal to previously chosen values. For example,

2
p

, namely:


k
j
j
g
w
with arbitrarily chosen

j
k2
Proof. Let at the first r points (1 r < p) o
1, 2,,, m1,,,j. p
ne has


1
j
j
g
w
while at 1r
one wants to have

11rr

q
g
w

, where ber.
ne the ne which
e branch pocirc

2, ,qm
int t = 0, en
is any num
w cut line as a spiral curve
les the points
Then, we defi
starts from th
,,,
12 r

(see
betw
the Fi q-1) times passes gure 4) and (
een the points
r and 1r
. Thus the analytic con-


tinuation of 1
11
g
w
to
j

2, ,jr b ecomes

1
j
w while at 1r
it is e1r. We continue this
process to adjust also the values at the points 2,,
rn
qual to

q
w
.
Nowhen the super index kj + 1 in

1
1
tice that
j
k
j
w
is smaller
than kj, we can replace kj + 1 (kj + 1 + m) because one
has

1
11
j
k
jj
ww

. At the end at a plane cut
along an appropriate (spiral) curve joining the oint 0
to t = such that at the given points
,,
by
mwe arrive
pt =
 
1j
k
12 ,
p

the
function g(t) has desired values

j
k
j
w. xample on For e
the Riemann sheet shown in Figure 5 one has


1
11
g
w
,




11
22 55
,, ,
g
wg w



11
11
mw
,


13
22
mw
,


12
33
mw
,

1
4
m

1
4
w etc. Since these valun be arranged in mp dif-
fe
es ca
r (8)) can hav
rent forms, the right-hand side of (7) (oe
mp different values. This proves thorem-4.
Ex. LetaC
and bC
any numbers which differ
from zero. Then the matrix
2
2
0
0
a
b
A
1
7
6
5
O
t
8
Figure 5. A cut appropriate for the particular case when r =
5 and q = 4.
M. IDEMEN
1268
has four square-roots given as follows:
0
b
REFERENCES
[1] A. Cayley, “A Memoir on the Theory of Matrices,” Phi-
losophical Transactions of the Royal Society of London
Vol. 148, 1858, pp. 17-37. doi:10.1098/rstl.1858.0002
00
0
, , ,
00 0 0
aaa a
bbb

 
 

 .
,
[2] N. J. Higham, “Functions of Matrices: Theory -
putation,” Society for Industrial and Applied Mathe-
matica (SIAM), Philadelphia, 2008.
[3] N. I. Mushkhral Equations,” P.
and Com
elishvili, “Singular Integ
Noordhoff Ltd., Holland, 1958.
[4] E. C. Titchmarsh, “Introduction to the Theory of Fourier
Integrals,” Chelsea Publishing Company, 1986, Chapter
1.3.
C
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