Numerical analysis on set elements for positive definite symmetric matrices by parameterizatio

Geiss, G. R.

Rothschild, D.

English

NASA Technical Reports Server

G r u m m a n  R e s e a r c h  D e p a r t m e n t  M e m o r a n d u m  RM- 386 
PARAMETERIZATION OF THE SET OF P O S I T I V E - D E F I N I T E  MATRICES 
- AND AN ALGORITHM FOR ITS GENERATION 
D. Rothschild 
and 
G .  R .  G e i s s  
S y s t e m s  Research Sect ion 
N o v e m b e r  1967 
T h i s  w o r k  w a s  p e r f o r m e d  under C o n t r a c t  NAS 2-4063 f r o m  NASA A m e s  
Research C e n t e r  and C o n t r a c t  NAS 8-20306 f r o m  NASA G e o r g e  C .  Marshall 
Space F l i g h t  C e n t e r .  
- l e  
C h a r l e s  E .  Mack, Jr. 
D i r e c t o r  of R e s e a r c h  
https://ntrs.nasa.gov/search.jsp?R=19680004145 2020-03-12T11:44:25+00:00Z
ABSTRACT 
An algorithm is  presented f o r  generat ing the  s e t  of n x n 
pos i t i ve -de f in i t e  symmetric matrices, based on t h e  choice of 
a r b i t r a r y  parameters. n(n + 1) 2 
T h i s  method was developed as a necessary s t e p  i n  t h e  de- 
velopment of an approach t o  t h e  problem of est imat ing t h e  domain 
of a t t r a c t i o n  of an equilibrium so lu t ion  t o  a system of nonl inear  
autonomous d i f f e r e n t i a l  equations. The technique however, i s  
genera l ,  and may be appl ied t o  any problem requir ing t h i s  type 
of matr ix .  
ii 
TABLE OF CONTENTS 
Page -I t e m  -
Introduction ............................................. . 1  
2 
8 
9 
10 
Development of the Numerical Algorithm ................... 
Computer Program ......................................... 
References ............................................... 
Appendix - Lis t ing  of Computer Program ................... 
iii 
INTRODUCTION 
In  conjunction with research being done on the  formulation 
of numerical techniques f o r  es t imat ing the  domain of a t t r a c t i o n  
of an equi l ibr ium so lu t ion  t o  a nonl inear  dynamical system, t he  
need arose for an e f f i c i e n t  a lgori thm t o  generate  a s e t  of 
n x n pos i t i ve -de f in i t e  symmetric matrices. These matrices can 
be used as candidate  "Q" matrices which i n  turn  give r ise t o  
associated "P" matrices  through the  so lu t ion  of the  Liapunov 
equation, A P + PA = - Q. Having parameterized the  Q set, T 
one can proceed i n  an order ly  fashion i n  pursu i t  of an "opt imal"  
quadrat ic  form Liapunov function t o  resolve the  domain of sta- 
b i l i t y  problem (see Ref. 1). 
Another problem area f o r  which t h i s  matrix generat ion tech-  
nique has p o t e n t i a l  appl icat ion i s  i n  the  c l a s s  of opt imizat ion 
problems where the  so-called conjugate d i r ec t ion  methods are 
employed (Refs. 2 and 3 ) .  In  many of these  methods, the i n i t i a l  
a r b i t r a r y  o r  random choice of a pos i t i ve -de f in i t e  symmetric 
matrix i s  required.  
This memorandum discusses a procedure and associated computer 
program f o r  s a t i s f y i n g  the  above requirements. 
1 
DEVELOPMENT OF THE NUMERICAL ALGORITHM 
The generat ion of t h e  s e t  of pos i t i ve -de f in i t e  n x n sym- 
metric matrices can be car r ied  out by r e s o r t i n g  t o  the  b ru te  force  
approach of forming an a r b i t r a r y  symmetric matrix and then 
applying the  determinantal  test (Ref. 4 )  f o r  pos i t ive-def in i teness .  
n x n 
n(n + 1) 
matrix elements followed by the  The a r b i t r a r y  choice of 2 
evaluat ion of the  determinants of the  n-pr incipal  minors would 
be necessary.  It would be des i rab le  t o  generate  these  matrices 
by a procedure t h a t  guarantees a l l  t o  be pos i t i ve -de f in i t e ,  and 
i n  addi t ion,  t h a t  the  e n t i r e  set of pos i t i ve -de f in i t e  matr ices  be 
spanned. 
It i s  w e l l  known (Ref. 4 )  t h a t  a l l  real symmetric matrices 
are or thogonal ly  s i m i l a r  t o  a diagonal matrix, and t h a t  a l l  posi-  
t i v e - d e f i n i t e  (pd) matrices are then orthogonally s i m i l a r  t o  a 
diagonal matrix with posi t ive-diagonal  elements; i . e . ,  l e t  Q be 
pd, then 
(1) T Q = S A S ,  
where 
Ai > 0 , i = 1, 2, ..., n 
T S S = I .  
2 
Thus, the parameterization of all pd matrices, Q, is reduced 
to the parameterization of the group of orthogonal matrices, S. 
In Ref. 5, Murnaghan proves that the parameterization of the 
group of n x n unitary matrices U is accomplished by the fac- 
torization 
U = D  
where 
11-1 
I I  'n-k 
k= 1 
= D x Un-l x ... x u1 , 
D = diag iE1 , e itj2 , ,.., e "n-1, ei'n} , 
'k r: 
Y =  
n- 1 
(2n - k)(k - 1) + 
2 
(2n - k)(k - 1) + P =  2 l + n -  a ,  
(2n - k - 2 ) ( k  - 1) 
2 + (n - 0 9 v =  
( 3 )  
(4)  
( 5 )  
3 
ii 
Ukk = cos 0 
= COS e a a  
u = O , i # j , i , j # k , 4  I i j  
- i o  uka = - e s i n  8 
uak = + e s i n  t) , 
The f a c t o r i z a t i o n  of the  group of orthogonal matrices is  imme- 
d i a t e l y  obtained by requir ing U t o  be real; i .e.,  
S = D1 In; k= 1 'n-kl ' 
= diag {I, ..., 1, k 1) , Dl 
6 = 0 = 0 ,  
In p a r t i c u l a r ,  
(7) 
(2n - k - 2)(k - 1) 
c L =  2 + n -  4 .  
4 
(n - 1) (n - 2) t he t a s  and n phis ,  2 This f a c t o r i z a t i o n  contains 
o r  a t o t a l  of 
Eq. (2) raise the  number of parameters t o  2 + 1, o r  one 
n(n - 1) 
2 + 1 parameters. The n lambdas i n  
n(n + 1) 
more than required.  Thus, i f  w e  r e s t r i c t  S t o  be a r o t a t i o n  
matrix ( i . e . ,  choose 
- p(n + 1) 
2 
matrix.  The choice 'pn = 0 i s  i n t u i t i v e l y  motivated by the con- 
s i d e r a t i o n  t h a t  w e  wish t o  r o t a t e  and sca l e  the  e l l i p s o i d  associated 
'pn = 0) ,  t he  number of parameters w i l l  be 
, the  number required t o  represent  an a r b i t r a r y  symmetric 
with the  quadrat ic  form formed from the  pd matrix and do not  
want t o  r e f l e c t  coordinates o r  change the  handedness of the  co- 
ord ina te  sys tem.  
The f a c t o r i z a t i o n  of a pd matrix of dimension chree i s  thus 
given by 
(10) T P = S A S ,  
where 
> 2 
A 
0 
i A =  i o  
and 
5 
- S O  
c(pl = cos cpl , scpl = sin (pl 
Thus, it is clear that by using this representation under the 
restrict ions 
A > o  i i = 1, 2, ..., n 
- 7 7 -  < ' p i < - T ,  i = 1, 2, ..., n - 1 - 
(n - 1) (n - 2) 
2 
7T -T 
9 - -  < O i < 2 ,  i = 1, 2, ..., 2 ,  - 
the candidate Q matrices are guaranteed to be positive-definite. 
The constraints, Eq. ( 1 4 ) ,  can be removed by defining: 
6 
I 
I 
1 
“‘i A = e  i 
1 I 
i i 
8 = ?(- I- + 0 mod 2 ~ )  ,
q,. real 9 
I 
1 
1 
0 .  real 
! s only appear as arguments of ‘i and recognizing that since the 
trigometric functions, they can be arbitrary real numbers. 
7 
COMPUTER PROGRAM 
A computer program which implements t he  preceding numerical 
a lgori thm has  been developed and i s  being used i n  conjunction with 
t h e  problem of f ind ing  an " o p t i m a l "  quadra t ic  Liapunov funct ion 
f o r  a gth order quas i - l inear  d i f f e r e n t i a l  equation. The dimen- 
s ion  of the problem, however, i s  read  a s  data and i s  a r b i t r a r y .  
To use t h e  program, one must spec i fy  the  parameters: ql; 
1 1 1 
q2;  
t i o n s  of Eq. (15) hold.  The names given t o  these  a r r a y s  are:  
... 'ln; ol ;  e2; 0 . .  Qk; Cpl; (P2; ... C p n - l '  where t h e  r e s t r i c -  
q a r ray  E XLAM (I) 
1 
8 a r ray  E THETA (I) 
cp a r ray  =- PHI (I) . 
I f  it i s  des i red  t o  p r i n t  out t he  orthogonal matrix, 
i s  used i n  Eq. (ll), t h e  cont ro l  va r i ab le  PP i s  set equal t o  
1.0, otherwise it i s  set  equal t o  zero.  This matrix [SI i s  
c a l l e d  B ( 1 ,  J) i n  the  program. The f i n a l  requi red  output of 
t h e  program i s  Q ( I ,  J ) ,  a n  n x n pos i t i ve -de f in i t e  symmetric 
matrix. Enclosed i n  the appendix i s  t h e  complete computer pro-  
gram l i s t i n g .  
S, which 
REFERENCES 
1. Geiss, G .  R .  and Abbate, J. V . ,  Study On Determining S t a b i l i t y  
Domains f o r  Nonlinear Dynamical Systems,  Grumman Research D e -  
partment Report RE-282, February 1967. 
2.  Kel ley ,  H. J .  and Myers, G .  E . ,  Conjugate Direction Methods f o r  
Parameter Optimization, presented a t  18th Congress of t h e  I n t e r -  
na t iona l  Astronaut ical  Federation, Belgrade, Yugoslavia, 1967. 
3 .  Kopp, R. E . ,  M c G i l l ,  R.,  Moyer, H. G . ,  and Pinkham, G . ,  "Several 
Tra jec tory  Optimization Techniques," i n  Computing Methods i n  
Optimization Problems, A.  V. Balakrishnan and L. W.  Neustadt, 
Edi tors ,  Academic Press, New York, N . Y . ,  1964. 
4 .  Gantmacher, F.  R . ,  The Theory of Matrices, V o l .  I, Chelsea Pub- 
l i s h i n g  Co., New York, 1959. 
5. Murnaghan, F .  D . ,  Lectures on Applied Mathematics, Vol. 111: 
The Unitary and Rotation Groups, Spartan Books, Washington, 
D . C . ,  1962. 
APPENDIX 
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Parameterization of the set of positive- definite matrices and an algorithm for its generation

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