QUATERNIONIC MATRICES INVERSION AND

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 inverse We also introduce the functional D M which

extends to quaternionic matrices the nonnegative number jdet M j

Key words Matrices

AMS sub ject classications A A

Introduction Much of the sp ectral theory of complex matrices do es not ex

tend to 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 Denote by R the ring

n  n

of ndimensional matrices over R and by I the 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

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

1

1

N D I D I MI D

n 1 n 2 2n

1

M I

2

and

I N

1

1

M D I D I NI D

n 2 n 2 2n

1

M I

2

and the result follows immediately

Pro of of Prop erty III We b egin by considering M invertible and constructing

the matrix

M I

2n  2n

H

I

We nd

M I

1

D M D IM I D I D I D

n n n n 2n

I

and noting that D I D I the rst part of prop erty III is demonstrated

n n

To complete the pro of we show that

D M M invertible

n

n  n

We move by induction on n The case n is trivial so assume M H n

If M we know that M is not invertible and D M by denition Otherwise

n

we may assume without loss of generality that

a B

M a H a

C D

Now

1

D M jaj D D C a B jaj D a

n n1 n S

hence using the induction hip othesis

D M D a a invertible M invertible

n n S S

Pro of of the Multiplicative Prop erty Let us prove eq If D M then M

n

is not invertible MN is not invertible hence D MN Otherwise consider

n

1

M I

2n  2n

H

N

1

Schur complements with resp ect to M gives

1

M I

1

D M D NM D NM D M D

n n n n 2n

N

Now Schur complements with resp ect to I gives

1

M I

D I D N D N D

n n n 2 n

N

In conclusion

D MN D NM D M D N

n n n n

Remarks We discuss additional prop erties of the functional D By using

the results of the previous section it is immediate to show that

Q

n

jM j IV For triangular matrices D M

ii n

i

Next we calculate D in terms of singular values The singular values decomp osi

tion SVD for complex matrices extends to quaternionic matrices in a straightforward

n  n

+ +

U Every way A matrix U H is called unitary if U U I where U

ij

ij

n n quaternionic matrix M has the SVD M U V where U and V are unitary

where are the singular values of

1 1 1 k 1 2 k

M In these terms the following holds

V D U if U is unitary

n

Q

n

VI D M

i n

i1

+

VII D M D M

n n

It can b e seen that the uniqueness claim in theorem follows from prop erty VI

n  n

Indeed every nontrivial multiplicative functional L H R should satisfy

+

Q

n

n

L and L U for U unitary Using the SVD it then follows

i i

i1

i1

that L D

A weak version of the Jordan canonical form extends to quaternionic matrices

n  n

Namley every matrix M H is similar over the quaternions to a complex Jor

dan matrix J dening a set of n complex eigenvalues However the eigenvalues C

i

are determined only up to complex conjugation For further discussion see The

n  1

Jordan canonical form is asso ciated with right eigenvalues M q H

q H which are determined only up to quaternionic similarity This is further dis

cussed in Thus D has the additional prop erty

Q

n

VIII D M j j if C are the right eigenvalues asso ciated with M

n i i

i1

Let us denote by Z M the complexication of the quaternionic matrix

M ie

M M

1

n  n

2

Z M M M j M M C

1 2 12

M M

2

1

It has b een shown in that if J is the complex Jordan form of M then J J is the

Jordan form of Z M Consequently the sp ectrum of Z M is f g

1 n

1 n

+

A quaternionic matrix H is called hermitian if H H Obviously for hermitian

quaternionic matrices the eigenvalues are uniquely determined and real This follows

from the fact that Z M is also hermitian In this case we can dene the following

real determinant

n

Y

jH j

i

r

i

n  1

+

It is easy to show that H is p ositive denite ie x H x for all non zero x H

if and only if all the eigenvalues are p ositive if and only if all the real determinants

of the principal minors are p ositive

We conclude this section by comparing the functionals D M and jH j introduced

r

in this pap er with the qdeterminant dened by

jM j det fZ M g

q

From previous considerations we have

n

Y

2 + 2 +

j j D M D M D M jM M j jM j

i

r q

i1

The case of matrices In this last section we discuss inversion adjoint

and determinant for quaternionic matrices

Inversion Let

a b

M

c d

b e an invertible matrix with quaternionic entries When a b c d are all non

zero four parallel applications of the Schur complement formula lead to a concrete

description of the inverse

a b

1

M

c d

where

1 1 1 1

a a bd c b c db a

1 1 1 1

c b ac d d d ca b

What happ ens if some of the entries is zero Assume for example that a The

1

invertibility of M implies that b c Consequently the element d ca b has

innite mo dulus In this case we can dene

1 1

d lim d ca b

a

A simple calculation

jaj

jd j lim lim lim

1 1 1 1

a a a

jd ca bj jcj jc d a bj jcj jac d bj

jaj

lim

a

jcj jbj

shows that d Thus

1

b

a b

1 1 1 1

a c db b c c b

c d

c

We conclude that Eqs remain valid under appropriate conventions when some

entries in M are zero We do not have a clear generalization of this phenomenon for

n

Adjoint Eqs are valid in every asso ciative ring R In case R is also

commutative Eqs reduce to the well known formula

AdjM

1

M

detM

In calculating the inverse of real and complex matrices is of great theoretical

imp ortance So far we have failed to generalize this formula to quaternion matrices

At rst sight it might make sense to conjecture a noncommuting generalization of

the general form

1

M P Adj M Q

with

P diagp p and Q diagq q

1 2 1 2

quaternionic diagonal matrices Nevertheless the constraints

1 1

p aq d p cq c

1 1 2 1

1 1

p bq b p dq a

1 2 2 1

which for commuative elds are satised if

I

P and Q I

detM

are not always satised in the quaternionic world It is easily checked that the uni

tary matrix cannot b e written in the form Whether a further weakening

b eyond of formula is valid for quaternionic matrices remains an op en

n  n

problem Even the denition of Adj M M H n is not clear

Determinant For n an application of theorem yields

a b

1 1 1 1

jaj jd ca bj jbj jc db aj jcj jb ac dj jdj ja bd cj D

2

c d

From this denition the functional D for the matrix is

2

i i j

p p

D D

2 2

j k j i

as required by unitarity For hermitian quaternionic matrices the real determinant

is given by

q

2

jq j R q H

1 2

q

r

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