Exponential representations and canonical forms of matrix groups

Int. J. Engng Sci. Vol. 6, pp. 59-63.

Pergamon Press 1968. Printed in Great Britain

RESEARCH

NOTE

Exponential representations and canonical forms of matrix groups* I. INTRODUCTION NUMEROUSproblems in applied mathematics, physics and engineering science can be cast in terms of matrices C whole elements satisfy a system of conditions such as C’GC = G, where G is a given nonsingular matrix. If G happens to be the identity matrix, then the c’s are orthogonal matrices whose properties are well known. If, on the other hand, G is an arbitrary nonsingular matrix, then the C-matrices can have many and diverse properties. This note provides a direct method ofobtaining the properties of such C-matrices by giving explicit representations in exponential form. These exponential representations are obtained from the solution of a linear system of matrix relations, as opposed to the quadratic system CLGC = G and depend explicitly on G, G1 and G-l. In actual calculation, only G-l has to be obtained, in contrast to the usual methods which require the right and left eigenvectors of G, and hence a significant economy of effort results. We deal exclusively with n-by-n matrices over the complex number field, which we denote by capital Latin letters. Inversion and transposition are denoted by a superior - I and a superior t, respectively: complex conjugation is denoted by a superior asterisk. The identity is denoted by E. The diagonal matrix, whose first s-diagonal entries are - I and whose remaining n-s diagonal entries are + I, is denoted by D(s, n). All groups under consideration are matrix groups. Definition 1. Let G be a given nonsingular matrix and let Q denote the groupt nlhose elements C satisfy C’GC = G,

(1)

then G is said to be the generating kernel of V. 2. EXPONENTIAL

REPRESENTATIONS

Lemma I. Let G be nonsingular, then G exp (A) = exp (GAG?)G, (2) exp(A)G

= G exp(G-‘AC).

ProoJ: The result follows from the series representation for all positive integersj.

of exp (A) and the identity G(A)’ = (GAG-‘)jG

Lemma 2. Let G and U be nonsingular, then C=exp(UQ)

(3)

sa tisJies C’GC = G

(4)

for all Q and all R such that Qt=-GuQG-lrl_1-l+RU’-‘,

GUQG-‘R

(5)

= RGUQG-’

(6)

exp(R) = E.

(7)

*This note is based, in part, on research performed for the United States Air Force under Project RAND at The RAND Corporation, Santa Monica, California and in part on results obtained under NASA grant NGR 15-005-021. tThe collection of matrices Q that satisfy (I) obviously forms a group since EE imply C,C, E Q and C1 E Q implies C,-* E Q. 59

Q,

Q and

c,

E Q

60

Research Notes Proof. Substituting (3) and (4) and using (2). we have G=exp(Q’U’)Gexp(UQ) = exp(Q’U’) exp(GUQG-‘)G

and hence exp(Q’U’) exp(GUQG-l)

= E

since G is nonsingular. Thus exp(QtUL) = exp(-GUQG-I). If we set B = -GUQCT-~, then exp(B) = exp = (B)E = exp (B) exp (R) for any R satisfying (7). If (6) is satisfied, then B and R commute, in which case exp(B) = exp(B + R). It thus follows that necessary and sufficient conditions for (3) to satisfy (4) are (6), (7) and exp (Q’U’) = exp (-G UQG-‘+ R) . It thus suffices to require equality of the arguments of exp (.), from which we obtain (5). Remark. The set of R’s that satisfy (6) and (7) for all G, U and Q is not vacuous, for R = 2vikE for all integers k satisfies these requirements. Theorem

1.

Let G czzzd U be notzsinyulur, therz the component d of tlze group g with generrzting kernel G thtrt is continuously connected to the identity element is girerz by exp ( UQ), where Q ranges through ull matrices thczt satisfy

Q( = -GUQG-lU’-1,

(8)

Proof. By the definition of V, V forms a subgroup of the general linear group G L (2n) and hence the component of % that is continuously connected to the identity element is characterised by its germ. If we write C=E+(6C)+0(6’), so that (6C) defines the corresponding

element of the germ of %, (I) requires (6C)‘G =-G(K).

(9)

If we take R to be the zero matrix, then conditions (5) and (6) of Lemma 2 are identically satisfied. Hence Lemma 2 shows that exp( UQ) satisfies (3) for all Q such that (8) holds. Since Q = 0 satisfies (8). we consider (SQ) in a sufficiently small neighborhood of the zero matrix. We then have exp(UGQ) = E+CMQ+0(6’), with (USQ)‘G

= -G

( U6Q)

(10)

from (8). Thus, if SQ satisfies (IO), then CSCj = USQ satisfies (9) and conversely. The component of M that is continuously connected to the identity elements and its Lie algebra are then easily shown to agree with exp( UQ) and its Lie algebra on any sufficiently small neighborhood of the identity under the identification (6C) = U&Q. Hence, since exp (UQ) is analytic in UQ and U is nonsingular they agree over the entire component. Remurk. Since U is an arbitrary nonsingular matrix, we obtain a different representation of g for each choice of U. We consequently have an (n* - 1)-fold infinity of such representations. Theorenz 2. Let G be the generating kernel of??, tlzen the component g of V that is continuously idenfity element is given by exp (G-IQ). where Q ranges through ull matrices that satisfy

Q’ = -QG-‘G“,

connected

to the

(11)

Hence, if G is symmetric, then Q rarzges through ull skewsymmetric matrices and if G is skewsymmetric, then Q ranges through all symmetric matrices. Proof. Since the U appearing in Theorem 1 was an arbitrary nonsingular matrix, we may set U = G-l.

When this is substituted into (8), we obtain ( 1 I). The remaining conclusions are obvious. Remark. For the component @, the above theorem has reduced the quadratic problem C’GC = G to the linear problem Q’ = - QG-‘G’ and this reduction only requires us to compute G-’ from the given G. The resulting economy of calculation afforded by this reduction is the underlying reason for this note. 3. APPLICATIONS Under Theorem 2, the proof is almost immediate that any real proper orthogonal matrix can be represented as exp (A), where A is real and skewsymmetric. If C is any real proper orthogonal matrix, then CLEC = E and hence the generating matrix is G = E = G- ‘. By Theorem 2, we then have C = exp(Q) represents the proper (complex) orthogonal group for all skewsymmetric Q. Now, the real proper orthogonal

61

Research Note

group is the proper subgroup of the orthogonal group that is obtained under the restriction C* = C. Since C* = exp(Q*) forC ’ = exp( Q) it suffices to take Q* = Qin order to obtain the required result. For the nrouer Lorentz group, we have C’D (1,4) C = D (1.4). Since D(s, n)= D(s, n)-I. we have C = exp(D(i, d)Q), by Theorem 2, for all skewsymmetric 4-by-4 matrices Q and the condition C* = C can be satisfied if we take Q* = Q. Consequently, any element of the proper Lorentz group can be represented by

exp

for real-valued a, b, c, d, e and J It is thus surprising that one would want to go through all of the work involved in attempting to represent the proper Lorentz group by means of skewsymmetric matrices in mimicry of the orthogonal group [ I] when this is not the natural course at all. For the proper hyper-Lorentz group we have l
C’D(s, n)C = D(s, n) and hence C=

AB

exp

Bt

M

[

.

1

where A is an s-by-s matrix such that A’=-A, B is an s-by-(/r-s) matrix and M is an (n-s)-by-(n--s) matrix such that M’ = -M. Hence any such C-group contains the two orthogonal subgroups

L1 A0

exp

0

E

EO

and

exp o M [

1

The sympletic group Y of order 2n-by-2n is defined by the generating kernel

Theorem 2 then gives A’=A,

B’=

B

for any Y E 9. If G denotes either of the Caughy-Green deformation tensors of a material body, then CLGC = G defines the proper invariance group Q of the deformation. Since G is symmetric and nonsingular, Theorem 2 gives Q*=Q. C = exp(G-IQ), Q’ = -Q, In general, such an invariance group can not be realised as a group of Jacobian matrices of coordinate transformations. 4. CANONICAL FORMS FOR SYMMETRIC GENERATING KERNELS Throughout the remainder of this note we confine our attention to groups with symmetric generating kernels. Lemma3. Let G be symmetric

and nonsingular,

then there exists a complex-valued G = StD(0,

Let G be symmetric, that

nonsingular

and real-valued,

nonsingular

matrix

S such that

n)S = StS.

(12)

then there exists a real-valued

G = StD(s,

nonsingular

n)S,

matrix

S such

(13)

where s is the index of G. Proof: This isjust the

well-known theorem on transformations of bases on a vector space over the complex number field and over the real number field, respectively; see, for instance [2, p. 341. Theorem

3.

Any group V with symmetric generating complex numberfield, under the mapping

kernel

c = s-‘WS.

is isomorphic

W E P(n,

to 0 (n, c), the orthogonal

c),

group over the

(14)

Research Note

62 where G = 9s.

Hence,

all such groups have only two components,

S’exp

and they are given by

(Q)S = exp (S-*QS)

and

S-lexp(Q)SD(l,n), where Q ranges through all skewsymmetric matrices. Proof. SinceG is symmetric and nonsingular, Lemma 3 shows that there exists a nonsingular S such that

( 12) holds. When this is substituted into (l), we have S’-‘CLsLSCS-’ = (SCS-‘)L(SCS1)

= E.

Hence, under the substitution (14), (1) is equivalent to WLW= E. Since S is nonsingular, (14) is a group isomorphism and WV+’= E implies W E b(n, c), we establish the isomotphism of Q and D(n, c). 1f.X is the component of fin, c) that is continuously connected to the identity, then Theorem 2 gives and hence ( 14) and G = S’S give S’exp

(Q)S = exp(S’QS)

for all Q such that Q’ = -Q as the corresponding component of V. The other component of Q is obtained by noting that a representative element of 0 (n, c) /Z is D ( 1, n) . This theorem shows thatS_’ exp(Q)S{E, D( 1, n)}, Q’ = -Q may be taken as the canonical form of any group %’over the complex number field with symmetric generating kernel, and hence 0 (n, c) is the canonical group over the complex number field with symmetric generating kernel. We now restrict our considerations to matrices ouer the real numberfield. Theorem

4.

Any group V over the real number held with symmetric Lorentz group 2 (s, n) (W’D(s, over the real numberfield

n)W = D,

generating W E Y(s,

kernel G is isomorphic

to the hyper-

n))

under the mapping c = S_‘WS,

(15)

where s is the index of G and S satisfies (13). Hence the component of +Zthat is continuously connected to the identity element is given by S-’ exp (D(s,n)Q)S = exp (G-‘SQS), where Q ranges through all real-valued skewsymmetric matrices. Proof Under the hypotheses, Lemma 3 shows that there exists a nonsingular matrix S such that ( 13) holds,

where s is the index of G. When this is substituted into (l), we have (SCS-‘) D(s, n)(SCS-*)

= D(s, n).

Under the substitution ( 1S), which is a group isomorphism since S is nonsingular, ( 1) is equivalent to WID(s,

n) W = D(s, n)

and hence V and Y (s, n) are isomorphic. The remaining results follow from Theorem 2 and (13). It is evident that the groups {Z (s, n), 0 < s s n} from a canonicnl collection for the set of all groups +? over the real number field with symmetric generating kernels. This collection can be reduced in number, however, Theorem 5. The groups _Y(s, n) and _F(n - s, n) are isomorphic under the map ’ W = PWP-‘, permutation matrix such that -D(s, n) = P’D(n-s. n)P. Proof

If W E B(s.

where P is a symmetric

n), W’D(s, n) W = D(s, n).

Multiplying (16) by (-I),

(16)

we have WL(-D(s,

n)) W = -D(s,

n).

it follows from the definition of D(s, n) and the elementary properties of permutation there exists a symmetric permutation matrix P such that Now,

-D(s,

n) = P’D(n-s,n)P.

(17)

matrices that (18)

63

Research Note When this is substituted into (17), we have, since P’ = P = P, P-‘W’PD(n-s,

n)PWP’

= (PWP-‘)‘D(n-s,

n) (PWP-1) = D(n-s,

s)

(19)

and the result follows. Corollary. Every group Q over the real numberfield with symmetric generating kernel with index s is isomorphictoPEP(p,n),withp=min(s,n-s). Theorem 6. Let Q be a group over the real number field with symmetric generating kernel G of index s and let N (s, n) denote the number of components of Q. We have (for n > 0) N(s, n) =

2 for 4 for

s=O,s= O
n,

Proof. By Theorem 4, N(s, n) is equal to the number of components of .Lp(s, n) and by Theorem 5, N(s, n) = N(n- s, n). For s = 0, _Y(O, n) is the orthogonal group over the reals, and hence N(0, n) = 2. For 0 < s < n, we have the hyper-Lorentz group, from which a trivial calculation based on Givens’ results[3] yields N (s, n) = 4 for 0 < s < n. This theorem shows that the structure of the components of _Y (s, n), 0 c s < n is identical with the structure of the components of OEP ( 1,4), the Lorentz group. In fact, with / (s, n) as the component of _Y’(s, n) that is continuously connected to the identity, representative elements of 5? (s, n)// (s, n) are E,D(l,n),diag(D(O,s),D(l,n-s)),diag(D(l,s),D(l,n-s)). We can thus obtain all elements of all components ofoEP(s, n) and hence of Q. Let R be any diagonal matrix such that exp(R) = E, let G be any diagonal matrix whose diagonal entries are + l’s and-l’s, and set U = G-l. It is then easily shown that Q = $RG satisfies (S), (6) and (7), and hence Lemma 2 shows that C = exp(G-IQ) = exp()R) satisfies C’GC = G. It then follows that for

R=

1

2sriD(O,s) 0 0 0’ [ we have C = exp(+R) = D(s, n) and hence the four representative elements of PEP(s, n)//(s, n) can be generated by appropriate choices of Q, R and U which satisfy the conditions of Lemma 2. With the above results and those for the orthogonal group in mind, we make the following conjecture. Let Q be a group with generating kernel G and set II = G-‘, then {C = exp (G-‘Q)}, where Q ranges through the set of all solutions of(S), (6) and (7) covers every component of Vat least once. Division of Mathematical Sciences Purdue University Lafayette, Indiana

D. G. B. EDELEN

REFERENCES 111 S. L. BAZANSKI, Decomposition of the Lorentz transformation matrix into skew-symmetric tensors. J. Math.Phys.6, 1201-1203 (1965). [2] J. A. SCHOUTEN, Ricci-Calculus, 2nd edition. Springer (1954). [3] W. GIVENS, Factorization and signature of Lorentz metrics. Bull. Am. math. Sot. 46,s l-85 (1940). (Received 19 April 1967)