University of Tennessee, Knoxville
Trace: Tennessee Research and Creative
Exchange
Masters Theses
Graduate School
8-1953
The Characteristic Roots of Certain Real
Symmetric Matrices
Joseph Frederick Elliott
University of Tennessee - Knoxville
Recommended Citation
Elliott, Joseph Frederick, "The Characteristic Roots of Certain Real Symmetric Matrices. " Master's Thesis, University of Tennessee,
1953.
http://trace.tennessee.edu/utk_gradthes/2384
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To the Graduate Council:
I am submitting herewith a thesis written by Joseph Frederick Elliott entitled "The Characteristic Roots
of Certain Real Symmetric Matrices." I have examined the final electronic copy of this thesis for form and
content and recommend that it be accepted in partial fulfillment of the requirements for the degree of
Master of Science, with a major in Mathematics.
Wallace Givens, Major Professor
We have read this thesis and recommend its acceptance:
Edgar D. Graves, O. G. Harrold
Accepted for the Council:
Dixie L. Thompson
Vice Provost and Dean of the Graduate School
(Original signatures are on file with official student records.)
August 13, 1953
To the Graduate Council:
I am submitting herewith a thesis written by Joseph FrederiCk.
Elliott entitled "The Characteristic Roots of Certain Real Symmetric
Matrices." I recommend that it be accepted for � quarter hours of
credit in partial fulfillment of the requirements for the degree of
Master of Science, with a major in Mathematics.
'
Major Professor
·-x.#�a.
We have read this thesis
and recommend its acceptance:
(�! t;( 2··�·),
Accepted for the Council:
Dean of the Graduate School
THE CHARACTERISTIC ROOTS OF CERTAIN REAL SYMMETRI C MATRICES
A THESIS
Submitted to
The Graduate Council
of
The University of Tennessee
in
Partial Fulfillment of the Requirements
for the degree of
Master of Science
by
Joseph Frederick Elliott
August 1953
ACKNOWLEDGEMENT
This thesis was completed while the author was an Oak Ridge
Institute of Nuclear Studies Fellow at the Oak Ridge National Laboratory.
The author wishes to express his sincere appreciation to
Professor Wallace Givens of the University of Tennessee for his
encouraging and patient supervision of the writing of this thesis
and to Dr. A. S. Householder and the Mathematics Panel of the Oak
Ridge National Laboratory for their assistance in the final preparation
of this paper.
TABLE OF CONTENTS
SECTION
I.
II.
III.
rv.
PAGE
INTRODUCTION
•
•
•
•
•
•
•
•
•
•
1
•
PHYSICAL SOURCE OF THE PROBLEM
3
BACKGROUND
8
•
•
GENERAL RESULTS
•
•
16
•
1.
An oscillation matrix
16
2.
The Fibonacci numbers
25
3.
Variations of the oscillation matrix
26
4.
Alternating values on the diagonal
31
5.
The inverse of a certain matrix
33
6.
Symmetric circulants
36
1·
An extension by using the direct product
38
8.
Bounds for characteristic roots
41
9·
The general case
10 .
BIBLIOGRAPHY
Summary
•
•
•
•
•
•
•
•
43
45
50
IN!RODUCTION
I.
The main purpose of this thesis is to collect and coordinate
some known results in the stuAy of characteristic roots of certain
real symmetric matrices.
have their elements
a
ij
Specifically, most of the matrices considered
=
known as Jacobi matrices.
papers by
D. E.
0
except for
Ii
-
J
I <
1
and are
The thesis fills in the details of two
Rutherford
[12; 13 J 1
, with some translation of his
results for determinants to the problem of finding characteristic
roots of matrices.
Some of the results obtained are briefly discussed
relative to certain theorems on bounds for characteristic roots.
The first part of the paper is an attempt to show some of the
physical sources of the problem and indicate some applications of the
results.
The aim here is to exhibit matrices of the type considered
in the problem and little effort is made to give a detailed theory of
the physical laws involved .
A background
A rather
for the study is the purpose of the next section.
general treatment of theorems concerning characteristic roots
is given, especially with respect to real symmetric matrices.
More
recent discoveries concerning bounds of characteristic roots are given.
Proofs are either sketched briefly or omitted entirely since most of
the theorems are well known.
The main problem is introduced by solving one of the physical
problems discussed in the section on applications.
�
eferences in brackets,
end of the thesis.
[ J,
The results obtained
are to the bibliography at the
2
are used as a basis for a more generalized treatment With the aim of
finding the characteristic roots of several clas s es of
n
by
n
matrices.
The matrices to be c onsidered are at best s omewhat restricted
in s cope.
This s eems to limit the us efulnes s of the results.
However,
it is feas ible from a practical viewpoint to reduce any real symmetric
matrix to a Jacobi matrix in s uch a way as to preserve the characteristic
roots.
[6 J
This fact,
,
gives added value to the results obtained.
Aside from the actual solution of c ertain physical problems,
it is
hoped that results here obtained might be of some importance for at
first,
least two reasons:
matrices of order
n
it makes available s everal clas s es of
for which the proper values can be obtained
from tables of trigonometric functions and, secondly, the methods used
here might s uggest an approach to the s olution of the problem for more
general real s ymmetric matrices .
special case of the Jacobi matrix.
Much remains to be done, even in the
II. PHYSICAL SOURCE OF
TBE
PROBLEM
The task of finding the characteristic roots of a matrix is
very often the crux of a physical p�oblem. The example below, which
deals with a real symmetric matrix, is taken from the theory of
oscillations
[
11,
p.
184]
'J:.
�
Consider
n
+
2
I
-
�
L-
-
-
ttl
x... ,
particles, each of mass v , distributed
equally along a stretched string. Let the distance between two
consecutive particles at equilibrium be a so that the length of the
string is
(n
+
l) a
•
Let x represent the displacement of the i-th
i
particle along a line perpendicular to the equilibrium ltne, with
positive displacement on one side and negative displacement on the
other side, subject to the condition that only plane transverse motion
is to be considered. Suppose the tension in the string is
K , that
gravity is not to be considered and that the end points of the string
a.re fixed; that is,
x0
xn+l 0
With these hypotheses, it is
desired to study the motion of the system after a small displacement
=
=
•
from equilibrium, assuming that each particle moves in a line perpendicula.r to the equilibrium line at the equilibrium position of the
particle.
A fter a small displacement, the tension in the string is still
approximately K
•
Then the resultant force acting on the i-th particle
4
to restore it to equilibrium is given by
..
=
v x
i
-K
sin g
-
K
sin � ,
where
and
i
1, 2,
=
•
•
•
, n .
For small displacement, sin
tan Q
equal to
tan �
and
Q
sin �
and
respectively.
are approximately
Hence the equation of
motion becomes
-
-
v
.
-
a
-
v
.
a
'
i
==
1, 2,
. . .
2
In matrix notation , this gives
r-
1
,1!
I
=
-2K
K
va
va
� _:
K
va
K
va
-2K
va
K
va
K
va
-2K
va
K
va
K
va
-2K
va
=
K
va
A
K
va
2
For purposes of this thesis, zero elements of displayed
matrices Will generally be omitted.
, n
•
5
It is immediately obvious thAt the i-th particle does not
exeC'U'te simple banaonic motion unless verr special initial. coDditions
are
imposed.
Cr
aDd
x
However, by a suitable linear transformation
column vectors), an equation
obtailled vhere
T A T-1
y=
T A T-l y
is a diagonal matrix (Th. 3-3).
y = Tx
can be
In this
equation the motion is simple harmonic motion and the normal modes of'
vibration
are
the relation
treqency.
l
obtained from the characteristic roots of' T A T-
TAT-1 y
2
-v
=
where
y
v = f/2'Tf
1
X=T- y
It will then follow tba't
•
by
the angular
gives the motion of
the original system.
'.f A T-l
The characteristic root s o:f
are
the same
eba'raeteristic roots o:f A , by similarity properties.
as
the
So the solution
ot the probl.ea basically depel.lds on the characteristic roots of'
A
or, in general, on the characteristic roots of' a matrix of' the form
ra
i
b
'b
l
.
r
t
I
.
.
L
a
b
b
a
.
.
.
b
.
.
.
b
a
b
b
a
Various matrices of' this general type J'111J:3' be obtained by
'ftr'yiD& tl!le initial coDditions.
:ror example, if' the distances
between the :first tvo particles and the 1ut tvo particles
are
changed
( -K K )
--
v a.1
xn
••
K
--x +
va. 1
-
va.
x
2
- K)
(_:!
..!._X
va. n -1 + va.
=
6
i
=
2, 3,
. . . , n-1
va.n xn
+l
Now the solution of the problem depends upon the characteristic
roots of a. matrix of the form
a.+c
b
b
a
b
b
.
.
.
.
.
b
a.
.
.
.
.
b
a
b
b
a.+d
In general, if the distance between the i-th and (i -l)st
particle is a.1 , the equation of motion in matrix notation is
rI
l
I
I
I
l
L
va.2
va.2
l
K
va.2
va.2
-
va.3
.
I
n:
J
-
-
,
IIX..
!
a.l
,
•
I •
I
�-K Kv K
K �K K�
x
l
va.3
.
.
.
.
.
.
.
.
.
�/-K
va.n
_
K�
va.
n+
...J
7
One is therefore interested essentially in the characteristic
roots of a matrix of the form
�
b
b
l
a
b
l
b
2
2
b
3
�
2
.
.
.
.
.
b
n-1
.
.
.
a
n-1
b
n-1
b
n-1
a
n
The oscillation problem requires that the
the conditions
b
i
>
0
,
i
b + a
+ b
i
i+1
i+l
•
11
2,
• • •
=
0 ,
i
=
1, 2,
a
•
and
i
•
•
, n-2
b
1
satisfy
and
, n-1 , but these restrictions will be
ignored in the following mathematical investigations.
III.
BA CKGROUND
The matrix obtained from a matrix A by changing rows to
columns is called the transpose of A , denoted in this paper by AT .
If the matrix A is such that A AT then A is said to be
symmetric. One says that A is a real symmetric matrix if A=AT
and all the elements of A are real numbers. It should be noted that
a symmetric matrix is necessarily square; that is, the number of rows
is the sane as the number of columns. Hence one is justified i.n
speaking of a symmetric matrix of order n where n indj_cates the
number of rows or columns.
If, for a square matrix A of order n with elements in the
field of complex numbers, there exists a non-zero column vector
(an n by 1 matrix; that is, n rows and 1 column) such that
A x l x for some complex number A , then A is said to be
characteristic root (proper value, latent root, eigenvalue) of A
and x is said to be a characteristic ve�tor (proper vector, eigen�
vector) of A corresponding to A . It is clear that (A :A. In)x = 0
has a non-zero solution if and only if the determinant of the system
vanishes; that is, if and only if lA A In I 0 This is called the
characteristic equation of the matrix A . From it one obtains the
characteristic roots of A .
Theorem 3.1. The characteristic roots of a real symmetric
matrix are real.
Proof: Let A be a characteristic root of real symmetric
matrix A Then there exists a non-zero column vector x for which
=
,
x
a
=
-·
-
=
•
a.
•
9
AX
(1)
A
=
X '
where one does not yet know that A is real and so must allow the
x to be complex. Multiplying both sides of (1) by
components of
the transposed conjugate of x one obtains
T
AX
T
(2)
x Ax =
x
Taking the transposed conjugate of both sides of (2) it follows
that
T
X AX
which implies that A
=
X
by an exercise in Birkhoff
Definition 3. 1.
w:1. th
Hence
.
l
and
If x
MacLane
and
real elements, then x and
T
only if x y
0
l/2 .
be (x xT)
=
•
l
y
-T
A
=
is real.
[ 1,
X
This proof is suggested
p. 309
are two
X
1
J .
by n row vectors
y are said to be orthogonal if
The length of the row vector
and
x is defined to
Definition 3. 2. A square matrix P is called orthogonal if and
T
only if P P
I ; that is, each row of P has length one and any
=
two rows are orthogonal.
Definition 3·3·
AT = AC
A square matrix A is called hermitian if
C
where A = (a1 )
j
It should be noted that hermitian matrices are important for
•
our purposes because a real symmetric matrix is an hermitian matrix.
10
Hence results may often be obtained for real symmetric matrices as
special hermitian matrices.
A
Definition ;;.4.
·
ATC
=
A square matrix A is called unitary if
I .
Characteristic vectors corresponding to distinct
Theorem 3.2.
roots of a real symmetric matrix A are orthogonal.
Proof :
A.
Let >.. and ).. be distinct characteristic roots of
i
j
Then there are non -zero column vectors x and y {characteristic
vectors of A ) such that
From A X
=
>..1
A
x
=
li x and
A
y
it follows that XT A
X
=
>..
j
y
T
i x
)..
=
•
Multiplying
both sides by y , one obtains
T
X
Since
Ay
=
)..i
T
)..i � >.. , then x y
j
=
T
X
0 .
y
=
)..j
T
X
Hence
y
x and
y are
orthogonal.
Definition 3 . 5.
a field F
,
If
A
and B are n by n
then A is said to be similar to B
matrices over
over
F
if there
is a non singular matrix P with elements in F such that A P
l
T
P , then A is said to be orthogonally similar to B .
If P=
-1
=
If P is a unitary matrix, then A is said to be unitarily similar
to B
•
Many
of the properties of orthogonal similarity are implied
by the properties of unitary similarity.
Theorem
.3·.3·
Every real symmetric matrix is orthogonally
B P.
ll
similar to a diagonal matrix.
Proof: One proof of the theorem [10, p. 186] shows the·
existence of orthonormal basis of the vector space Vn (R) ,
consisting of characteristic vectors of the real symmetric matrix A
If the matrix P is formed by letting these characteristic vectors of
A be the columns of P , then it is easily seen that P is orthogonal
and PT A P diag. (k1, k2, , kn)
An important property of orthogonal similarity is that two
matrices which are orthogonally similar are both similar and congruent.
Hence they have the same characteristic roots and if one is symmetric,
so is the other.
A similar theorem is true for hermitian matrices, with the
substitution of unitarily similar for orthogonally similar and of
conjunctive for congruent.
A trivial corollary to Theorem 3.3 is that a real symmetric
matrix is orthogonally similar to a Jacobi matrix. This has
significance because the reduct.ion to a Jacobi matrix is much more
feasible in actual practice than a reduction to diagonal form. Since
the characteristic roots of a Jacobi matrix can be calculated without
too much difficulty, one has ·a practical method for obtaining the
characteristic roots of real symmetric matrices [6 J
Theorem 3.4. Let A be a square matrix of order n over
the field F of complex numbers and denote the characteristic roots
If g(x) is a polynomial in F [x J,
then the characteristic roots of g(A) are g(11), g(12), , g(1n )
an
•
=
•
•
•
•
.
•
•
12
Proof: The proof is indicated as an exercise in Perlis [10,
p. 138] A more concise presentation of the same method is given by
:zm.miihl [14, p. 228] . The latter proof is as follows:
Let the polynomial g(x) - y be written as
.
Then
g(A)-
y I
•
(A - x1
a
I) ( A- x2I)
.
.
•
(A-� I)
from which it follows that
I g(A)
- Y
I
I
•
a n I A - xl I I j A
.
-
x2 I
I
/ A.
. . .
-
l1t
I
j
I t is clear that
Hence the characteristic roots of g(A)
Theorem
'·t·
If
are
g(l ), g(�),
i
•
•
•
,
g(ln)
the characteristic roots of a real symmetric
•
13
< An) and the characteristic
striking out the last row and the
< 8
<
n-l) and if
<
8 1 < A2 < 8 2
<
<
<
matrix A are (11
1 <
2
roots of the matrix B obtained by
last column of A are (51 < 82
li � 8 j tor every i, j, then ll
8n-1 < An
Proof: The matrix A has the tom.
A
row vector, and
a
.� )
yT
Y
nn
is
a
ann
is
orthogonal
and
of A Since B is a
is an orthogonal matrix P such that
single element
real symmetric matrix there
The matrix
where y is a l by (n-1)
•
14
where
w
i
is a polynomial in the elements of
y
and P
•
Since Q
is orthogonally similar to A one knows that the characteristic
polynomial of A is given by
f(l.)
where
that
g(l.)
w
i
�
•
•
,
5
'
n-l
0
for if'
+
e><:>
,
w1 = 0
then 51
If one lets
l.
It is then clear
•
is a zero of
take on values
the alternation in sign of
< 5n-1 < l. n .
n
n-1
- a ) g(l.) - �
nn
i= l
is the characteristic polynomial of B
to the hypothesis.
.
(l.
=
=
f(l.) , contrary
-
f(l.)
C><J
, 51, 52,
implies
11
<
It should be noted that the cases where
is an odd number and where
n
is
an
even number must be treated
separately.
Removing the restriction that the characteristic roots of
and
B
A
be distinct, the result is not a strict separation of roots,
since some of the characteristic roots of B
characteristic roots of A .
to be a diagonal matrix.
As
an
might be equal to
extreme example one might take A
For a detailed discussion of these aspects
of the problem, the reader is referred to Browne [2].
Theorem
3.6.
The characteristic roots of A 0 B
characteristic roots of A
Proof:
together with those of B
•
The symbol ® indicates the direct sum.
defined to be the block matrix
are the
A(±) B
is
15
The characteristic polynomial of A® B
and the theorem follows immediately.
[Parker,
Theorem 3. 7.
J
p. 105 .
IA
is then
If
A.
-
:>..
I
I IB
·
-
:>..
I
I
is a characteristic
root of the matrix A and R is the greatest sum obtained for the
absolute values of the elements of a row and T
is the greatest sum
obtained for the absolute values of the elements of a column, then
I >..I
.(
min (R, T)
Theorem 3.8.
•
[Parker,
J
p. 107 .
matrix with characteristic roots :>.. -(
�
1
I I
[
( An - Al) ) max aij , i � j
Theorem 3. 9.
Brauer, p 390
•
.
•
J.
If A
>..2 <.
( ai
=
·
·
·
j
)
-(
is
an
hermitia.n
A , then
n
The absolute value of one of
the characteristic roots of A equals R ( see notation in Theorem
3.7) only if R
� and
�l' �2,
=
1
•
•
R2
R and arg. ( a )
� + �i - �j where
ij
i�
is
. , �n are arbitrary real numb era. Then R e
=
• • •
=
=
a characteristic root of A .
The proofs of the last three theorems are found in the references
indicated.
Many other theorems concerning characteristic roots of matrices
might be included in this section.
It was thought best, however, to
include only those which are used most in the thesis.
The reader is
referred to the bibliography for references to more complete treatments.
IV.
1.
GENERAL
RESULTS
An Oscillation Matrix
It has been briefly indicated that the matrices to be studied
arise naturally in certain physical problems.
On the other hand, an
exact physical application is not always available when one begins to
generalize the first basic examples.
It is hoped that the extensions
Will at least be of some mathematical interest.
Consider first, then, the triple diagonal matrix
A
which arose
during the consideration of the modes of vibration of a system consisting
The solution of the
of certain masses attached to a stretched string.
problem depends upon the characteristic roots of the following
n
n
by
matrix:
A
where
a
and
b
a
b
b
a
b
b
a
b
.
=
.
.
b
.
a
b
b
a
are certain real numbers determined by the conditions
of the physical problem.
The task of finding the proper values of
using the theorem that if
A
=
g(B)
then the characteristic roots of
n , where
�
A
where
A
g(x)
are given by
can be reduced by
is a rational function,
g(�) , k
is a characteristic root of the matrix
B
•
=
11 2,
Applying
• • • ,
17
this theorem (Th.
B
3.4)
here, one sees that A
0
1
1
0
1
1
0
=
a I + b B
n
where
1
=
.
.
.
.
1
0
1
1
0
It is then sufficient to know the characteristic roots of
B .
These can be obtained by considering the characteristic equation of
Let us denote the characteristic polynomial of
x,
-1,
� (x)= j xi-B I =
n
x,
-1
-1,
.
I
-1
x,
-1
X
B
by
1
1 X
.
-1,
x,
-1
-1,
X
so that
1
1 X
=
� (x)
n
B
1
1 X
1
1 X
The last equality follows from elementary properties of
It is also clear from determinant theory that
determinants.
x �
(x)
n+1
+
� (x) = 0
n
polynomials of
respectively.
B
where
when
B
(x)
�
n+2
and
is of order
�
(x)
n+ 1
(n + 2)
and
From the theory of finite differences
then knows that
�
(x) n+2
are characteristic
(n + 1)
[7,
p.
543] one
•
18
where
and
c
l
and
r
1
c
are
2
The roots of
r
2
are
r
2
to
roots of the equation
determined from
be
1
- :x r +
2
two known
1
- x r +
values
of
- 4)
2
i9
and
r
2
= e
-i9
and the value for
cp (:x) = x = e
1
It is easily seen that
x
2
- 1 = e
2i9
+ 1 + e
-2i9
e
19
and
cp (x)
k
•
and
These are simplified if one makes the sUbstitution
r = e
1
0
1
2
2
x + .{x
Then
=
are
0
•
r
•
i9
cp (x)
n
-i9
+ e
x = 2 cos 9 .
becomes
and
cp2 (x) =
Using these values, one finds that
and
-i9
- e
c
2
=-
e
iQ
e
-19
- e
-iQ
Hence
e
=
-( l)i9 =
- e n+
i9
-i9
e
- e
(n+l)i9
sin (n + 1)9
sin 9
which is a result that will be referred to throughout the paper.
should note that it assumes the indeterminate form
However, if
one
writes
cp (x)
n
n
r.
r=O
g
when
I xI
One
= 2
as
e
(n-2r)i9
=
e
-(n+l)i9
- e
i9
-i9
e
- e
{n+l)i9
•
then it is clear that
�
(2)
=
n
These are never zero so that
1
n+
all
and
�
(-2)
n
the zeroes of
�
=
( -l) n ( n + 1)
n( x)
19
•
can be obtained
from the expression
=
where x
=
2
sin �{n + �1)9
-si"'"n---,9
,
cos 9 .
So the characteristic roots of the matrix B , where
0
1
B
=
1
0
1
1
0
l
l
0
1
l
0
are the roots of its characteristic equation
s in ( n + 1) 9 = 0
sin 9
=
�
.
k'T(
n+1 , k 1, 2,
k'TT'
. , n
Hence the characteristic roots of B are 2 cos n+
1 ,
k
l, 2,
n
It is then immediate that the characteristic roots
It is easily seen that
.
.
=
•
n {x)
is zero for 9
=
=
---
---
. . . '
•
of the matrix A , where
n .
are � = a + 2 b cos k'/1
n+l , k l, 2,
This result was obtained by Gantmakher and Kre�n 4, p. 134
=
.
.
•
,
[
]
who also found an explicit formula for the characteristic vectors of
A
.
If
vj
vj
l
=
vj
2
.
vnj
is the characteristic vector corresponding to the characteristic root
l ,
j
then it is shown that vkj
cos
=
Q
C
sin k
Q
j
a
l
-
2b
j
j
where
'
and
C
is some constant . When one substitutes the value already found
for
l
, it follows that
j
Vj
k
=
C
sin knj +-rrl
Actually, this is shown in Rutherford's paper
[13,
]
p. 241
21
•
Before considering this section, however, it is necessary to discuss
some concepts of power series of matrices.
It is well known that the series
COS
X = 1
-
If A is a square matrix, it is
converges for every number x
cos
natural to define
A
4
2
X
X
+
7=\T
liT
2.
as
cos A =
I
+
- .
..
One would then ask what is meant by the convergence of such a series
p
r
as cos A
In general, the answer is that a series S = L. ar A ,
P
r=O
where A is a square matrix and ar is a real number, is said to
converge as p �
=
if every element of S
p
converges as p-�
00
.
It can then be shown that the series cos A converges for every
, a )
square matrix A
Furthermore, since A = diag . (a1, a2,
n
k k
k
k
implies A = diag . (a1, a ,
, a ) , it follows that if A =
2
n
• • •
•
• • .
.
. ., a )
n
cos A = diag . (cos a1, cos a2, . . . , cos an )
One can now develop a formula for the characteristic vectors of
the matrix
then
•
22
a
b
A
=
b
a
b
b
a
b
b
a
b
b
a
0
b
b
0
Let
A(O)
0
b
b
0
b
b
0
=
b
b
Then A
=
a I + A(O)
A
(3)
Q
=
.
It is asserted that
2 b cos Qn
a I +
B
in which B
n Bn t n n
n is a matrix whose
1/2
sin r
element is (2/n+l)
and
n +
where
=
8�
'11
n + 1
1
0
0
0
2
0
0
0
n
2
The proof of (3) begins by proving Bn
=
I .
r , s-th
From the definition
of Bn
it follows that if
i, j-th element of
c1
is the
j
C
=
23
2
Bn
then
n
2
n+l
=
j x1/
sin inx;r' 1 sin
n + 1
x =l
�
By a trigonometric substitution one obtains
n
1
c.. = n+l � cos (i n- +j ) x1 'JT
l.J
X=l
-
n
1
) x11
cos (i +n j
1
+
X=l
�
n+l
One now needs the fact (which is easily proved) that
n
"t... cos x r'1(
n + 1
x:::l
if r is an odd number
if r is
an
even number
Using this it is immediate that
cj_j
=
cij
=
1
n
-+n+l n+l
0
0
1
=
0
=
if i
if (i
=
-
(i
=
2
It ha.s now been shown that B
n
=
I
j
j)
-
is an odd number .
j)
is
an
even number.
From this it follows that
24
cos gn
= cos (Bn V n B n)
'
which is obtained by examining the cosine series for
g .
n
The proof
of assertion (3) will be completed by showing that A(O) = Bn(2b cos Vn)Bn·
If the
p , q-th element of Bn(2b cos
(4)
4b
d nq = n+l
.r
v
)B
n n
is
d
pq
then
n
x'l/ . xq1Y
"
u
sin PX1T
1 cos -1 s1n
n
+
n+
n + 1
x=l
--
which is obtained in a straightforward manner and makes use of the fact
that
2b cos
2b
:1
2'1/
2b cos n+l
l
31T
2b cos n+ 1
-
cos ... n =
2b cos
By a trigonometric substitution,
t
n_.,.,....,
-1-1
n+l
_j
dpq can be written as
b �
(:p-q+l)xil + cos (p-q-l)x1T - cos (p+q+l)XJ/
dpq =n+l u cos n + 1
n + 1
n + 1
x=l
-
cos
(:p+q-l)x11
n + 1
l
_j
From this it follows tha.t
d
pq
=
0
if p
=
q
25
b
d q = n+1(n + 1) = b if IP- qj = 1
p
dpq =
o
if 1 p - q l > 1
Thus A(O) = Bn(2b cos V n)Bn = 2b cos Qn
Hence A = a I + A(O)=
+ 2b cos Qn and the proof of assertion (3) is complete. It follows
•
�
I
that
A Bn = Bn ( a
I
+ 2b cos Vn)
I
2b cos '�'n)
are exactly the characteristic roots of A , then the columns of Bn
Since the diagonal elements of the diagonal matrix ( a
+
�-
But the elements of the j -th
are the characteristic vectors of A
k j f( ,
. n+I
column of Bn are vkj = ( 2/n+l) 1/2 s1n
k 1, 2,
, n ,
which is the result obtained by Gantmakher and Krein 4, p. 134
=
[
.
.
•
]
2. The Fibonacci Numbers
As a curiosity, the result that
X
cp
n(x)=
1
..
1
1
X
. 1 X. 1
1
.
X
1
1
X
n
k--rr)
= x cpn_1 ( x)-cpn_2 ( x) = lJI (x-2cos n+
l
k=l
I
can be related to Fibonacci numbers. Consider the numbers
5,
8,
13, . . . where the i-th number (i > 3)
o,
1, 2, 3 ,
is the sum of the two
26
numbers immediately preceding it and the first three numbers are 0,
It is interesting to note that the (n + l)st number in
the sequence is in �n (-i) . For if one lets x -i and multiplies
n
�n (x) x �n-1(x) - �n-2( x) through by i , it follows that
1 and 2
•
=
=
.n � (-i) = ('1) n-1 � ( ') - 1.n � ( • ) =
n-1 -l
n-2 -1.
n
1
facts that i �l( -1. ) = -1.2 = 1 and 1.2 �2 ( -1. ) = J.. 4 - 1 2 = 2
complete the inductive process. Hence the n-th Fibonacci number could
The
.
be written as
n l n-1
(i) n-l �n-1(-i) = (i ) - -r/ (-i - 2 cos kn1()
k=l
n-1
iT (1
k=l
-
=
k11' )
2.l cos n
). Variations of the Oscillation Matrix
The next question might be a consideration of matrices which are
slight variations of the matrix A where
A =
a b
b a b
b a b
b a b
b a
.
I
L
.
.
-----,
I
I
I
i
.
J
27
If some of the elements are changed, could th e characteristic roots
still be found? These changes might be due to changes in initial
conditions in the physical problem or to merely an arbitrary extension
to more general cases. In any case one might suspect that variations
in the elements of A would lead to quite different results .
First, suppose the first and last elements on the main diagonal
of A are changed, thus obtaining
a+d
b
b
a
b
b
a
b
a
b
Proceeding
before, we note that Rn = a I + b
as
d
1
1
0
b
c
where q>k ( x)
0
1
-
c!
where
1
1
C
c,
1
=
The proper values of
I xI
b
0
1
1
c
b
are obtained from the zeroes of
-
c+ d
• b
cd
( x)
q>n-1 ( x) + �
b q>n-2
has the same meaning as before. Substituting the values
28
already found for q>k(x) it follows that
cd sin (n - l)Q
C+d sin n Q + �
sin (n+l)Q - �
b
lx I C I =
sin Q
-
In particular, consider the following cases as examples:
1) If c = -d b , then
=
I
IX
-
cI
- sin (n - l)Q = 2 cos n Q
= sin (n + l)Q
sin Q
n . The proper
which vanishes if Q =(2k-1}7f/2n , k =1, 2,
values of the matrix C are therefore 7k =2 cos (2��l)7r , k =1, 2,
n . This implies in turn that the characteristic roots of Rn ,
)'71
subject to the restrictions noted, are � =a + 2b cos (2k-1
2n ,
n.
k =1, 2,
2) If c =0 and. d = -b , then
.
.
.
•
.
•
,
'
I
IX
-
c
I=
sin (n + 1)Q sin nQ =
sin Q
+
which is zero if Q =2k/(/2n + 1) , k =1, 2, ... , n For this
example, the matrix C has characteristic roots 7k =2 cos 2k-1T
2n+l ,
k =1, 2, , n and the corresponding Rn has characteristic roots
k""/l , k =1, 2, , n .
�=a + 2b cos 22n+l
3) If c = d =b , then
•
.
.
•
.
•
.
29
lx
I
-
C
= sin ( n+l) Q
I
2
=
which vanishes if Q
that Q
=
-1"(
•
sin
2
-
sin nQ + sin ( n-l) Q
sin Q
nQ
( cos Q
sin Q
k1Y/n, k
2,
1,
=
1)
-
•
.
, n.
•
gives an indeterminate form but x
I
verified directly to be a zero of I :x:
-
C
=
It should be noted
-2
is easily
I . Renee the characteristic
roots of
a.+b
b
b
a
b
Rn =
are � = a +
2b
b
a
.
k'71 k = 1,
cos -n-,
.
.
2,
b
b
•
.
.
a
b
•
,
b
a+b
n.
One special combination of results already obtained may be
mentioned. Suppose
X
1
cpn ( x ) =
1
X
1
1
X
1
1
X
1
and
1
X
30
y
1
1
y
1
y
1
.
where
q>
(x)
n
1
.
.
1
.
.
y
1
1
y
the determinant of a matrix Pn(y)
(z-1),
2a ,
2a,
2a,
z,
2a
1,
2a,
n(y)
= x y + 2 and 2a = (x +
l
y) .
(z-1)
Since it has already been
shown that
q> n(r)
n
k-'lf
= /( (r - 2 cos n+I) ,
k=1
then it follows that
n
n
/1
= IT (x - 2 cos k ) 1T (y - 2 cos k
n+
D+1 k=1
k=1
-
n
is
1
1,
z
q>
Then
•
z, 2a, 1
2a,
z, 2a,
l ,
where
and
is the determinant of a matrix P (x)
n
'71'1 )
=
-� r�z - (2 + a) 2 + (a - 2 cos k--n' ) 2
D+l
L
k=l
]
31
This implies that the characteristic roots of
-2a,
-2a,
o,
-2a,
-1,
-2a,
o,
.
.
.
-
n
l
l,
-l
.
.
-l
-2a,
.
.
.
.
.
.
.
.
.
-1,
are
2
k-rT 2
2 + a - (a - 2 cos ---1) , k
n+
1, 2,
=
. • •
.
-2a,
-1,
, n
.
o,
-2a,
-2a
l
•
4 . Alternating values on the diagonal
At least one case in which the terms on the main diagonal
r
alternate in value can be solved by the results already obtained.
82
n
a
l
l
b
l
=
l
a
.
l
.
.
.
.
.
a
l
l
.
n by 2n
.
.
l
b
__j
Then
-l
(>..-a),
-1,
(>..-b),
-1,
(>..-a),
-1
-1,
-l
(>..-a)'
-1, (>..-b)
Let
by n
32
It is easily seen that A = a or A = b are not zeroes of [AI 2n -s2n[
Hence one is permitted to multiply each row containing ( A - a) by
( A - b) 1/ 2 , multiply each row containing ( A - b) by ( A - a) 1/2 ,
·
by ( A - a) 1/2 and divide
One thus obtains
each column containing (l - b) by (l - b) 1/ 2
divide each column containing ( A - a)
•
the result that
z,
-1,
-1
z,
-1,
-1
z,
-1
=
where z
r,( A
- a) ( A - b )
] 112 = 2 cos
a + b .:!: [( a - b) 2 +
k
=
1, 2,
. . .
,
n
z
sin ( 2n + 1)9
sin Q
L
the characteristic roots of s2n are
=
-1
z,
-1,
-1,
Q
16
•
It follows easily that
cos2 k-rr
2n+l
J
1/ 2
,
.
By a similar argument one finds that the characteristic
the matrix
roots
of
33
82n+l
a
1
1
b
1
1
a
1
...
.
1
.
b
.
.
.
1
a
1
are
a + b + I.
L(a - b)
k
•
2
2
+ 16 cos
k --rr
2n+2
J
1/2
'
A
1, 2, ... , n , together with the characteristic root
•
a
which
may be verified directly.
5.
The Inverse of a Certain Matrix
Another possible means of extending the results thus far obtained
is to find inverses for some of the matrices for which characteristic
roots have already been found.
manner in at least one case.
Consider the matrix
[x+b
II
R
n
(x, a, b)
=
Rn (x,
X
1
.
.
a, b)
l
1
1
L
The inverse of
This can be done in a straightforw-ard
.
1
X
II
1
.
.
.
1
. .
X
1
.
1
x+a
I
I
_j
is found by malting use of the relationship
I Rn(x,
a, b)
\
=
�n(x) + (a + b) �n- 1(x) + a b� 2{x)
n-
where the value of � {x)
k
is the same as before .
If the r, s-th element of
[Rn(x,
a, b)
] -l
is denoted by
hrs , it is not too difficult to see that
hrs
This is obtained by considering the cofactors of the elements of
R (x, a, b) divided by the determinant of the matrix . If one
n
substitutes the value found previously for � (x) , then
n
[sin r9+b sin(r-l)Q] [sin(n-s+l)Q+a sin(n-s)QJ
sin Q [sin{n+l)9+(a+b) sin nQ+a b sin(n -l)QJ
sin t3 Q
sin(t3+l)Q and
By certain trigonometric manipulations one gets
.
This expression can be simplified by lett�ng
b
=
�!{�lJQ
- s
.
hrs
=
a
=
-
2
sin Q sin(r + a)Q sin(n+t3 -s+l)Q
sin3Q sin(n + t3 + a + l) Q
l) Q
=
(-l)
=
r+s
� r-l(x) �n+ s(x)
��
�n+t3+a(x)
It should be noted that the last form of
hrs
is not valid
35
extend our definition of �n ( x ) so that even when n is
not an integer, we have
unless
we
and �n ( 2)
=
n
(2 cos 9)
1 , � ( -2)
n
�n
+
sin ( n + 1)9
sin 9
n
( - 1 ) ( n + 1) . This must be done
=
=
'
because, in general, a and 13 would not be j_ntegers.
As a simple example to illustrate the mett.od suppose one wishes
R3(2,
to find the inverse of
hrs
0: 0)
hsr
=
[R3(2,
o,
I
0) i
J
-1
=
11
1
2
,o
1
L..
0
Then
(-1 ) r+s
=
from which it follows that
l2
.
-1
l
�J
=
=
1,
2, 3,
4
r-
I
I
l_
R3(2,
Since the characteristic roots of
k
1
4
(r) (4
-
-2
3
1
0, 0)
s)
-2
4
r
/
�
s
1
3
are 2 2 cos k-7/
-2
-2
--'
+
then the characteristic roots of
3 -2 1
: -2 4 -2
IL 1 -2 3
2 k1T k 1, 2, 3·
sec T'
I
I
Q
II
I
=
k1[
are qk 2/( 1 + cos T )
Another case where the matrj_x is of order n gives the result
=
that
=
=
4" '
36
-2
[
R (-2,1, -1
n
.
r
l
1
1
1
1
3
3
1
3
5
1
3
5
-
=
3
1
5
H
2n-l
H
It follows that the characteristic roots of
1
�
6.
k
2 (2k-l)
�
csc
'
4n
=
1, 2 '
•
.
•
,
n
are
•
Symmetric Circulants
Another class of matrices for which it is possible to find
exact solutions for characteristic roots are the matrices which are
called circulants
a
E
A circulant matrix is a matrix of the tonn
•
a
o
a
a
n-1
=
.
a
l
a
l
.
a2
a
�
o
.
a
2
n-1
n-2
.
a
a,
o
where the elements of the i-th row are the elements of the (i - l)st
row permuted in the cyclic manner indicated.
K
If
0
1
0
0
0
0
0
1
0
0
0
0
0
=
.
.
1
.
.
0
0
0
0
0
1
1
0
0
0
0
37
then
E
•
E
Hence the characteristic roots of
•
•
n-1
an-1 K
+
can be obtained if the charac-
The characteristic equation of K
teristic roots of K are known.
is easily seen to be
X
-1
0
0
0
0
X
-1
0
0
0
0
.
.
.
.
.
X
.
.
.
-1
.
0
0
0
0
-1
0
0
0
.
.
.
X
0
.
n
X
=
- 1
-1
X
2k1/i
n
e
' k
It follows that the characteristic roots of K are
•
.
•
, n .
=
1, 2,
E
Hence the characteristic roots of the circulant
2k'1T'i
n
k
0
=
•
•
•
, n .
If E
4k-11i
n
•
•
+
•
1, 2,
are
2(n-l)k'T(i
n
a -1 e
n
--
+
=
is symmetric so that ar
'
an-r , then the
=
proper values are
+
Hence
+
+
L
'
k
=
1, 2,
4k1/
2a2 cos n +
•
•
•
.
+
.
•
L
, n
,
k
=
1, 2,
. . .,
n
38
where L
n
=
2s
2sk-rl
n
is 2 as cos
if
-­
n
+
1
n-l
=
=
2s
=
a
and
L
i � 1 , n-1
k
a
s
if
1
1
and
ai
=
0 for
This gives a matrix of the form
•
0
G
1
1
1
=
0
1
1
0
1
1
0
1
It is immediate that the characteristic roots of G
=
( -1)
•
As a special case, choose a
k
=
2k'tr
are 2 cos n- - ,
1, 2, ..., n . Actually, one could just as easily obtain the more
general result that the characteristic roots of a matrix
H
.
=
.
7.
+
b
.
.
a
.
.
b
. . .
a
b
2 b Cos 2k-rr'
n , k
I
b
b
are a
iI
b
b
a
a
b
1 ' 2'
=
• • •
, n
•
An Extension by using the Direct Product
A further extension to other matrices for which the characteristic
roots can be found requires another theorem.
Theorem 4.1. If G
n
and
G ® Im
and H
are real symmetric matrices of order
m respectively, then the characteristic roots of G(H)
+
In® H
are
11
+
a
j
where
1
1,
.
.
•
, A.n
=
are the characteristic
roots of
G
and
The symbol
�'
am
,
• • •
is used to iDd.ic&te the direct product.
®
H
the characteristic roots of
are
G
Since
and
H
are real. symmetric
possible to find ortbogoDSl. matrices
diag.
wbere
(l:t,
. ...
T
P2 1:! P
Dp
2
is
am
, 1.0)
T
P2 11 P2
""
are denoted by D
1
P1
diag.
and
aDd
(�,
D
2
P2
•
.
,
The DBtrix
D
P
If
am).
respectively, then
is of the follov1ng fo1'111.:
wbere
B
m by m
i
=
and
� by
m
=
diag.
(c;_, ��
p.
�
pl G pl.
T
P
l G pl
a diagonal. matrix With the characteristic roots of
aJ.ong its d1agonal...
[s,
matrices it is
such tbat
.
•
For &
discussion of direct product tbe reader is referred to .MacDuf'fee
Proof:
39
• • •
,
am)
•
G(H)
•
]
82 .
4o
Hence
where
and the proof is complete.
As an example of an application of this theorem, suppose
I
Gn by
n
a
b
=
b
a
b
b
a
b
b
a
b
and
1fl by m
d
c
=
c
d
c
c
d
l
·:j
c
c
d
c
c
d
Then G(H) is the matrix in which the diagonal elements of G are
replaced by ( a I m + H) and the elements b are replaced by b I m
We know
with zeroes elsewhere to complete a matrix of order m n
•
that the characteristic roots of G are
,
41
�
and
=
k1/
a + 2 b cos n+l
H
the characteristic roots of
a
j
=
k
'
are
Jff
m+l
d + 2c cos
1, 2, ... , n
=
j
=
1, 2,
It follows that the characteristic roots of G(H)
.
.
•
, m .
are
k� + 2 c cos j11
m+l
a + d + 2 b cos
n+l
where k
1, 2,
=
•
.
•
,
n
and
j
=
1, 2,
.
.
,
•
m ,
With all possible
combinations being taken.
8. Bounds for Characteristic Roots
Let us now consider the results obtained for at least one case
with reference to bounds for characteristic roots. Specifically, take
a
b
A
=
b
a
b
b
a
b
b
a
b
b
a
for which the characteristic roots are �
k
=
1, 2,
•
.
•
,
n and
=
k--r/ ,
a + 2 b cos i:i+I
42
r
B
a+b
b
=
l..
.
b
a
b
.
with characteristic roots �
For matrix
A
b
a
b
.
b
=
a
b
b
a + 2
b
a+b
;' ,
cos k
k
=
l, 2,
•
.
•
,
n .
we have that
Here, then, we have a strict inequality for Theorem 3-7 which
concluded that
j � \-< j a I + 2jb j.
That strict inequality does not
hold in general is immediately seen in the case of matrix B
matrix B we have
with the equality holding when a >
0
.•
b < 0
and k
=
.
For
n . General
conditions necessary for the equality to hold are given in Theorem 3.9.
Let us examine the spread of the characteristic roots of
A
relative to Theorem 3.8, where the spread is indicated as
and 2l (an 0]_)
-
. .. -( an
!o..
2 n
Then
-
1
1
>
=
b(2 cos ..:Z:C.)
�
n+l "'
if b > 0
b
and
1
)
2 n - a..
-J.
-(a
=
(-b) (1
+
cos�)
> 1 b l if b <
n
One then notes that the equality holds in the first case for n
=
o .
2
but that strict inequality holds for the characteristic roots of B
Much work is currently being done in this area of bounds for
characteristic roots. Results that are sharper than those listed in
this thesis are contained in the papers of Brauer
[3] together with
references to other writers in this field. However, the notation and
background that must be built up limits us here to this brief mention
of a topic that seems to offer opportunities for much further exporation.
9.
The General Case
It would clearly be desirable to consider the matrix
al
bl
bl
a2
b2
b2
a3
b3
bn-2
an-1
bn-1
bn-1
an
J
44
One can at least prove the following theorem:
Theorem 4.2.
Givens
[5].
If A has a characteristic root
of multiplicity k , then at least ( k
-
zero.
1)
of the b i 's must be
Proof: The theorem is shown true for k
=
2 and extended by
induction. If Aj is a double character:i.stic root of A , then
(A - Aj I) has rank at most (n - 2) . By striking out the first row
one obtains immediately an ( n
and last column of (A - A. I)
J
{n
-
(A -
submatrix whose determinant is
1)
A.
1)
by
If the ran.'k of
equals zero. Hence
I) is at most (n - 2) , then
J
-
some b i is zero. One can then use the theorem that the characteristic
roots of the direct sum of two matrices B and C are the characteristic roots of B together with those of C
For if
is a
characteristic root of A of multiplicity k , then certainly at
least one b1 is zero. Then A can be represented as a direct sum
of two matrices which we may call B and C . If k > 2 then
either B or C
(or both) must have at least a double characteristic
root; hence, another
that if
b.
1
is zero. Continuing in this way it follows
is of multiplicity
k
,
then at least (k - 1)
b.1's are zero.
Another result applicable to the general case
[
]
is
of the
contained in
Gantmakher and Kref'n 4, p. 134 and concerns bounds for the largest
and smallest characteristic roots of A .
�
A
If the characteristic roots
n , then the results obtained are
A general method for calculating the characteristic roots of A
for arbitrary ai and b i is given in
10.
[6] .
Summary
C haracteristic roots were found for several classes of real
symmetric matrices.
It was thought desirable to tabulate these
results in order to make them more readily accessable .
Matrix
a b
b a
b
b
�
-1t
a b
b
a-b b
b
a
b
�b
cos
2b
2k7(
cos 2n+l ,
c:;
k 1(
n+l
b
a
b
� = a +
a
b
b
a
1
k =
1,
'
2
'
·
•
•
'
n
2,
•
.
•
,
n
n by n
a b
b
k =
n by n
a-b b
.
'
a
b
.
= a +
a b
b
b
Characteristic Roots
n by n
..
a b
b
.
� = a + 2b cos
.
a
b
b
a+b
(2k-l):U:
, k =
2n
1,
2,
•
.
•
,
n
.
��
b b
a b
b a b
l· .
-1
2a
1
.
.
.
b a b
b a+b
2a 1
0 2a
2a 2a
46
n by n
.
:>..
-k
= a+
2b cos
k 'i( k = 1 2
n '
'
'
n
· · · '
•
n by n
1
1
a 1
1 b 1
1 a 1
� = ( a-2
2a 0
1 2a
1 b 1
1 a
k'l( 2 - ( a2+2)
n;r)
k = 1, 2,
2a
-1
.
.
•
,
, n
•
2n by 2n
A =
1/ 2
k
11
2
2
a+b+ L( a-b ) + 16 c os 2ii+I
-----�---------���-----
]
�
�
1 a 1
1 b
a 1
1 b 1
1 a 1
cos
'
k = 1, 2, . . . , n
( 2n+1) by ( 2n+1 )
A =
r, a-b ) 2
a+ b + �
+
"71
16 cos 2 k20+2
J
1/ 2
--------��-
k = 1, 2,
•
.
•
'
, n and A=a.
a
n-1
n
by n
If'
a
n-2
�
L
wbere
=
2a8 cos
1
1 1
1
3
1
3
3
1
3
5
(a I
+
•
2sk71'
n
if'
b i•
=
l..
-k
5
R)
n
= a
tben
n-r
= a0
2s
cos
+
2B:J_
6k-ff
--n
+ 1 and
cos
+
L
2k--rr
--n
• • •
+
L
( -1)
=
�
+
k
cos
lfk-rr +
--n
, k = 11 2,
i:f
as
n
• • •
=
2s
D
,
•
n b;y n
3
. . •
r
�
1
5
a
=
2 (2k-1) 17'
1
esc
lin
2
1
k = 1, 2,
• • •
, n
(2n-1)
b Im
( a r_ + R)
n b;y n
(bl.oeks}
b i
m
( a I + B)
b Im
b I
m
m
b
d
c
c
=
.
c
d
c
.
d
.
.
c
c
d
c
c
d
I
a
{a I
m
+ H)
48
Im is the unit matrix of order m , then
a
r,k
=
a + d +
k
=
1,
2b
1
r'IT
cos kn+11 + 2c cos m+1
2, . . . , n ; r
=
1, 2 ,
'
.
.
•
, m
BIBLIOGRAPHY
1.
Birkhoff, Garrett and MacLane, Saunders . � Survey of Modern
Algebra . New York : MacMillan, 1947 .
2 . Browne, E.
T.
"On the Separation Property of the Roots of the
Secular Equation" . Amer . J. Math. 52, 843-850 ( 1930) .
3.
Brauer, Alfred. "Limits for the Characteristic Roots of a Matrix" .
Duke Math. J. 13, 387-395 ( 1946 ) . ( First of a series in this
Journal. )
4.
Gantmakher, F. and Kre�n, M. G.
iadra. (Russian) .
5·
Givens, Wallace. Private communication .
6.
Ostsillla.ts innye matritsy .!_
"Numerical Computation of Characteristic Values of a
Real Symmetric Matrix". Forthcoming report at the Oak Ridge
National Laboratory.
7.
Jordan, Charles . Calculus of Finite Differences . New York :
Chelsea, 1950 .
8.
MacDuffee,
9.
10 .
11 .
c.
C. The Theory of Matrices .
New York :
Chelsea,
1946 .
Parker, w . v . "Characteristic Roots and Field of Values of a
Matrix" . Bull . Amer . Math . Soc . ( 2) 57, 103-108 ( 1951) .
Perlis, Sam. Theory of Matrices .
Wesley, 1952 .
Cambridge
Rutherford, D . E . Classical Mechanics.
1951 .
42,
London :
Mass. : AddisonOliver and Boyd,
12.
"Some Continuant Determinants arising in Physics and
Chemistryn . Proc . Roy. Soc . Edinburgh . ( A ) 62, 229-236, 1945 .
13 .
"Some Continuant Determinants arising in Physics and
Chemistry !In . Proc . Roy. Soc . Edinburgh . ( A ) 63 , 232-241,
14 .
Zurmiihl ,
R . Matrizen . Berlin :
1950 .
1950 .