QUATERNIONIC MATRICES INVERSION AND DETERMINANT y y NIR COHEN AND STEFANO DE LEO Abstract We discuss the Schur complement formula for quaternionic matrices M and give an ecient metho d to calculate the matrix inverse We also introduce the functional D M which extends to quaternionic matrices the nonnegative number jdet M j Key words Quaternions Matrices Determinants AMS sub ject classications A A Introduction Much of the sp ectral theory of complex matrices do es not ex tend to quaternion matrices without further mo dications In particular the determinant cannot b e dened and the eigenvalues have several p ossible extensions which do not necessarily resp ect the fundamental theorem of algebra Quaternionic mathematical structures have recently app eared in the physical litterature mainly in the context of quantum mechanics and group the ory The relevance of quatenionic calculus to several iussues in theoretical phycis has rekindled the interest in quaternionic linear algebra A usefull thecnical pro cedure in approaching quaternionic applications in physics has b een the replace ment of quaternionic matrices by complex matrices of double size and the consequent sp ectral analysis of the latter matrix This indirect pro cedure usually suces to solve the problem at hand but it sheds little light on the structure of quaternionic matrices We aim to discuss inversion and determinant for quaternionic matrices in view of new classications of quaternionic groups and p ossible applications in grand unica tion theory The main results of this pap er can b e summarized by 1 Inversion The formula M Adj M detM cannot b e generalized for quater nionic matrices Nevertheless the Schur complement formula provides an ecient metho d to calculate the matrix inverse Determinant The determinat for n n quaternionic matrices M cannot b e de ned in a consistent way Instead the nonnegative number jdetM j can b e extended to quaternionic matrices by the functional D M dened in terms of quaternionic M entries Notation Quaternions introduced by Hamilton can b e represented by four real quantities q a i b j c k d a b c d R and three imaginary units i j k satisfying i j k ij k Submitted to Electronic Journal of Linear Algebra on July y Department of Applied Mathematics IMECC University of Campinas CP Campinas SP Brazil nirimeunicampbr deleoimeunicampbr We will denote by Req a and Imq q a i b j c k d the real and imaginary parts of q The quaternion skeweld H is an asso ciative but noncommutative algebra of rank over R endowed with an involutory op eration called quaternionic conjugation q a i b j c k d Req Im q satisfying pq q p q p H The quaternion norm jq j is dened by jq j q q a b c d Among the prop erties of the norm to b e used in subsequent sections we mention here the following jpq j jq pj jq j jpj and j pq j j q pj Every nonzero quaternion q has a unique inverse 1 q q jq j Two quaternions p and q are similar if 1 q s p s s H By replacing s by u sjsj we may always assume s to b e unitary The usual complex conjugation in C may b e obtained by choosing s j or s k A necessary and sucient condition for the similarity of p and q is given by Req Rep and jIm q j jImpj An equivalent condition is R eq R ep and jq j jpj Every similarity class contains a complex number unique up to conjugation Namely every quaternion q is similar to Req i jImq j Since Req Req and Imq Imq the quaternions q and q are similar In fact it can b e seen that for s H to satisfy 1 q s q s given q H it is necessary and sucient that Req s Res 1 However there exists no single s H such that q s q s for all q H n n Matrix inversion Let R b e an asso ciative ring Denote by R the ring n n of ndimensional matrices over R and by I the identity matrix in R A matrix n n n n n M R is called invertible if MN NM I for some N R which is n necessarily unique It is p ossible to show that for quaternionic matrices MN I n implies NM I n The Schur complments pro cedure is a p owerful computational to ol in calcu lating inverses of matrices over rings Let us write a generic ndimensional matrix M in blo ck form A B M C D k k Assuming that A R is invertible one has 1 I A B A I k k M 1 I A CA I nk S nk with 1 A D CA B S The invertibility of A ensures that the matrix M is invertible if and only if D is S invertible and the inverse is given by 1 1 I A I A B k k 1 M 1 1 CA I A I nk S nk The inversion of an ndimensional matrix is thus reduced to inversion of two smaller matrices k k (n k ) (n k ) A R and A R S We shall call A the Schur complement of A in M S To invert ndimensional quaternionic matrices we may apply the following pro n n cedure Let us consider a quaternionic matrix M H a B M C D If H a we can apply directly Eq Otherwise we p erform a columnrow p ermutation by constructing a new matrix a B 1 1 P M Q H a 1 1 1 C D 1 1 By using Eq and applying recursively the columnrow p ermutations P and Q m m we nd 1 M L U where a a 1 n n1 1 Y a B m m I P L m1 m I nm m1 n1 Y Q I U m m1 1 C a I m nm m m1 Determinant The identity detMN detM detN and the fact that M is invertible if and only if detM are central in the study of complex matrices For twodimensional quaternionic matrices at rst glance one could introduce four dierent denitions ab cd ab dc ba cd ba dc However has little to do with invertibility For example for the invertible matrix i p j k two expressions in vanish and the other two are nonzero Another example is the invertible matrix i j p j i whose four determinants calculated by are all zero In fact as shown b elow the complex functional n n det C C cannot extend to quaternions More precisely there is no functional n n d H H n n which is multiplicative dMN dM dN for all M N H and such that Q n n d z z for all z C As a counterexample consider the matrices i i i i1 i1 i i M N S i i j Since SM NS we conclude that dS dM dN dS hence dM and dN should b e similar This is a contradiction b ecause obviously Re fdM g Re fdN g Nevertheless the real functional n n jdetM j C R + can b e extended to quaternionic matrices Theorem There exists a unique non trivial functional n n D H R n + which is multiplicative ie D MN D NM D M D N n n n n This functional has the fol lowing properties A B k k 1 D A D D CA B as long as A H and D A I D k nk k n C D II D I MN D I NM n n 1 III D M D M if N invertible D M otherwise n n n Proof For n we dene D q jq j q H 1 Obviously Eq and prop erties I II I I are trivially satised For n we use induction to dene D Assume that D has b een dened for all k n For all n k i j n set n n M fM H M g ij ij ij M R by We dene the functional D + ij n ij D M jM j D M ij n1 ijS n (n 1) (n 1) where M H is the Schur complement of M in M We claim that ijS ij 2 ij i j n are compatible namely all the n functionals D n k l ij M M M M M D D ij k l n n There are four cases to check Case i k j l Trivially true Case i k j l We shall only consider the case i k j and l All other cases follow by rowcolumn p ermutations So we have a b E A C D F M a b H ab G H J The two relevant Schur complements are D F C 1 b E M a 11S H J G and C F D 1 a E M b 12S G J H It is easy to see that 1 1 1 1 a b a E a b a E 1 ja bj D M M n1 11S 12S I I Therefore using eq for D which holds by induction we have n1 11 1 D M jaj D M jaj D M ja bj jbj D M n1 11S n1 12 S n1 12 S n 1 2 D M n as required Case i k j l This case is treated along similar lines Case i k j l We shall only consider the case i j and k l All other cases follow by rowcolumn p ermutations So we have a b E A c d F M a b c d H ad G H J 11 2 2 We want to prove that D M D M For b or c by using the results in n n and 1 1 1 2 2 2 11 21 22 D M D M D M or D M D M D M n n n n n n trivially holds Let us consider the only remaining case b c a E A M d F a d H ad G H J 11 The functional D is given by n d F 1 1 11 1 1 D M jaj D jaj jdj D J Ga E H d F n2 1 n n1 H J Ga E 2 2 Similar calculations for the functional D lead to n a E 2 2 1 1 1 1 D M jdj D jaj jdj D J H d F Ga E n2 1 n n1 G J H d F 1 1 22 Consequently D M D M the compatibility has b een estabilished and we n n may thus sp eak unambiguously of D M n Pro of of Prop erty I We shall use induction on k The case k is equivalent to and has just b een proved For k we consider A B k k M H C D and assume without loss of generality that a A Then 11 a B B A 1 A B A C D B D D A A 2 n n C D C C D 1 2 1 By Schur complements the invertibility of A implies that of A D C a B S A A A By the induction hypothesis we get 1 1 D C a B B C a B A B A A A 2 A 1 jaj D D n1 n 1 1 C C a B D C a B C D 2 1 A 1 1 1 jaj D D C a B D X k 1 A A A nk D A D X k nk where X is a Schur complement dened by 1 1 1 1 1 X D C a B C C a B D C a B B C a B 1 1 2 1 A A A A 2 A 1 1 1 a B a B A 1 C C D 1 2 1 1 I a C a I B A 2 S 1 D CA B With this Prop erty I is proved Pro of of Prop erty II Let us introduce the following matrix I N 1 2n 2n H M I 2 By using prop erty I and Schur complements with resp ect to I and I we have 1 2 I N