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Canonical Forms of Pseudo-Orthogonal Matrices CANONICAL FORMS OF PSEUDO-ORTHOGONAL MATRICES BY G. ABRAHAM, B. M. ARTHUR AND P. SWAMIDHAS (Madras Christian College, Madras) Received May 15, 1971 (Communicated by Prof. P. L. Bhatnagar, F.n.sc.) ABSTRACT Canonical forms are found for all real four-dimensional matrices of the pseudo-orthogonal group which differs from the Lorentz group only in that its metric has two plus signs and two minus signs. INTRODUCTION REAL orthogonal matrices in n-dimensions have a very simple canonical form. We can construct a basis in which the matrix appears as the direct sum of a diagonal matrix, with + 1 and — I in the diagonal, and two- dimensional rotations.' In the same way, canonical forms for Lorentz matrices in n dimensions have been derived, 2 ' 3 the largest matrix in the direct sums being of order 3. A common feature of the canonical forms for orthogonal and Lorentz matrices is that each component matrix in the direct sum is of the form exp B, where B is antisymmetric (for orthogonal matrices) or pseudo-anti- symmetric (for Lorentz matrices), the latter being defined by Bt G'=—G'B where Bt is the transpose of B and G' is a diagonal matrix with one ele- ment — I and all others + 1. The canonical forms have been used by Wigner 3 for finding the represen- tations of the Lorentz group, which are fundamental in the theory of ele- mentary particles. Canonical forms for other pseudo-orthogonal groups have been derived' for similar applications. 1. In this paper we derive canonical-forms for real pseudo-orthoLor.c:l matrices L of order 4 defined by L tGL=G (1) I59 160 G. ABRAHAM AND OTHERS where G = [g;i] is a diagonal matrix with two elements + I and two ele- ments — 1. Matrices defined by (2) are matrices of isometric transforma- tions in a pseudo-Euclidean space in which the inner product of two vectors x = (ax, a2 , a3 , a4) and y = (b1 , b2 , b3 , b4) is defined by (x, y) = 2 Si aibj (2) 4,f A set of 4 vectors ei , e2 , e3 , e 4 is said to be a pseudo-orthonormal.basis if (ei , ej) = gij. The first step in the reduction of a isometric transformation to a canoni- cal form is to make _ list of the possible Jordan normal forms. This is considerably simplified by the following theorems: If an isometric transformation of a non-degenerate bilinear-metric space contains the elementary divisor (A — a)m ivitlr multiplicity k, then it also con- tains the elementary divisor (A — a, 1)m with multiplicity k. To derive a canonical form, we must obtain a pseudo-orthonormal set of basis vectors from linear combinations of the basis vectors x of the Jordan form. This is a gentr iisation of the familiar orthogonalising procedure for unitary spaces. We use Lagrange's algorithm for quadratic forms to transform the Gram matrix [(x1, x;)] to the diagonal form [g,j]. Only Jordan forms for which this is possible will be retained in our list. This method will be illustrated in the two following sections: 1. Let us denote by Ll the pseudo-orthogonal matrices L which have a complex eigenvalue ß with I ß = 1. From the theorem quoted in the previous section and from the fact that L is real, it immediately follows that L, has 4 different eigenvalues, ß, ß, ß -1 and ß'. It is simpler to derive at first a canonical form fur the corresponding pseudo-antisymmetric matrices B which are defined by Bt G = — GB, and have the property 6 that exp B is a pseudo-orthogonal matrix. Let us denote by B, the matrices B whose eigenvalues are of the form a, '., — a and — á, where a = a + ib, and neither a or b is zero. Obviously the Jordan forms for the L, and B, matrices are diagonal. Let the eigenvaiue equations of B, be Bi xi = aixi (3) Canonical Forms of Pseudo-orthogonal Matrices 161 where a 1 =a,a2 =6,a3 =—a,a 4 =—á and X1 =u1 +iu2,x2=hl,x3 — U3 + iu 4 , x., = .ti 3 . Then, using the antisymmetry property of B 1 , we can show that the Gram matrix [(ui, uj)] has the form 0 0 p q^ [(ui, uj)] = 00 q — p 1(4) p q 0 0 Lq —p 0 0 whose determinant is (p 2 + g 2)/16. Since x1 , x2, x3, x4 , form a basis, the matrix (4) is non-singular and so both p and q cannot be zero. Using the method of Lagrange (for diagonalising quadratic forms) we can . find 4 vectors 4 4 ei =27 akiuk = E tkixk (5) k=1 k=1 such that they form a pseudo-orthonormal basis. The fact that such a basis can be formed shows that the set of eigenvalues we are considering is an allowed one. If p > 0, we can define the vectors e= by the equations e, = p 1 (1 + µ 2)-1 (uß + 11 3 + /L112 y µx.ß) e3 p — -1 (1 + µ 2) - ' (ui — u3 + 1kí2 — e2 ` p-1 (u2 + u4), e4 = p-t (u2 — u4) (6) where µ = q/p. If p < 0, we must replace p} by (—p)r in (6). From (3) we derive Blut = au, — bu2 i Blu3 = — aua + b1f4 B,u2 = bul + au2, B,u4 = — bu3 — au4(7) From (6) and (7) we can find the matrix B 1 ,,, = [bti] defined by B,ej = E bijet (8) B1 , 0 is found to be Al 1B 02 (9) — LA1 Oz 162 Cl. ABRAHAM AND OTHERS where _ a —`, gb (1 + µ 2)t b Al — ^— (1 + 2)bb a — µb j (14) when p = 0, we can define the basis vectors by el = q- } (u1 + u3, e2 = q -1 (u2 — u3) e3 - ( 1 = q } u _` u4), e4 = q-} (u2 + u3) if q > 0, and by the same equations with (—q) 3 instead of q } if q < 0. Then we obtain a canonical form which is the same as (9) and (10) with µ=0. We now go back to the L l matrices. Put ß = exp a, a = a + ib, with a 0, b T 0. The eigenvalue equations can be written in the form Llxi' = ßixi' (11) where o __ Q p - P1 N" N2 = 03 N3 = N -13 N4 __ 1 and xi = U1 + (U2, xQ = xl ' , x3 ' = u3' + 1U41, x4 = x3 ' . Then, using the pseudo-orthogonal property of L l we can show that the Gram matrix [(ui', uï')] has the same form as that of [(ui, u1 given in (4) where now p = 2 ku,', u 41), q = 2 (u2', u3'). This enables us to define a basis es by putting ei' for ei and ui' for ui in (6). Since the vectors e;' are a linear combination of the vectors xi' we can put ei'. = E tki X)', Xi = E Skiek'. (12) k k From (11) and (12) we have Lej' = E (kj L (xk') = E tkjßkxk ' t = v rkiPkkei ' _ ligei where Sikßktkj (13) Canonical Forms of Pseudo-orthogonal Matrices 163 Similarly from (3), 5) and (8), we have bij = E Siikaktki (14) t If T = [tij], L = [lij], and D, D' are diagonal matrices with aj , aí respec- tively as diagonal elements, then from (13) and (14) L,, 0 = T-1 exp D'T = exp (T-' D'T) = exp B'1,0(15) where B',, 0 is defined as in (9) and (10) with a', b' instead of a and b. L,, 0 is therefore a canonical form for pseudo-orthogonal matrices L which have a complex eigenvalue ß whose modulus is not equal to 1. 3. We consider next the class of pseudo-orthogonal matrices L 2 which have the Jordan form 11 ta] (16) J2 = Voi eie, CO e- where 0 is real and non-zero. As in the previous section, we can construct a set of pseudo-orthogonal basis vectors from linear combinations of the basis vectors of the Jordan form (16). A canonical form for this class of matrices is X(8) —XC2+o) 0 A L2,0 = = exp 2 2 X (2 + o) X(0 — A2 02 where cos 0 sin 0 — µ sing ( ) X (e) — (17) sin 8 cos 9 — µ sin 8, _ µ 1 A2— C (18) The parameter µ arises in the same way as in the canonical form L, 0. We find only one more allowed Jordan form with all complex eigen- values, namely the diagonal form with four different eigenvalues eter, e-io,, eiB^, e- id=. Unlike the two previous cases, we can find a canonical form 164 G. ABRAHAM AND OTHERS which is the direct sum of two 2 x 2 matrices, namely the sum of two proper rotations in 2-dimensional Euclidean space, that is L3,o=R(01)4- R(02) (19) where os 8 — sin 8 R (B) _ — [sin B cos B1' All the pseudo-orthogonal matrices considered so far -have positive determinant and can be put in the exponential form. When the eigen- values are e'°, e-te,1, — 1, the matrix has negative determinant, and the canonical form is 1-4, 0 = R (0) -1-- 1 4-(—l). (20) 4. The remaining allowed Jordan forms have only real eigenvalues. For the Jordan form JS = [p e ± e^ ] + i. 0 :1:e-0 ] (21) where ;= 0, the corresponding canonical form is I-s,o f [P ( O) H(() ^)j exp r (22) A5@) Ash (0)•j where H (¢) = [sinh 0cosh ^^ ' P (q) = — e Ii (0) = [0 01 ' A5 = 1l — 11 (23) For the Jordan form Js = ^^ 0 1 J -f- L^ 0 + i J(24) the canonical form is 02 L5, 0 A61 = Ael 25) = rA' 6 E j [A6 Oa J Canonical Forms of Pseudo-orthogonal Matrices 165 where [01 (26) E — OJ , A6 = [— 1 1] .
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