Determinants of Blo ck Matrices

John R. Silvester

1 Intro duction

! !  

e f a b

. Their sum and N = Let us rst consider the 2  2 matrices M =

g h c d

and pro duct are given by

 !  !

a + e b + f ae + bg af + bh

M + N = and MN = :

c + g d + h ce + dg cf + dh

Here the entries a; b; c; d; e; f ; g ; h can come from a eld, such as the real numb ers, or more

n n

generally from a ring, commutative or not. Indeed, if F is a eld, then the set R = F

of all n  n matrices over F forms a ring non-commutative if n  2, b ecause its

elements can b e added, subtracted and multiplied, and all the ring axioms asso ciativity,

distributivity, etc. hold. If a;b;:::;h are taken from this ring R , then M; N can be

2 2 2n 2n

thought of either as memb ers of R 2  2 matrices over R  or as memb ers of F . It

is well-known fact, which we leave the reader to investigate, that whether we compute

with these matrices as 2n  2n matrices, or as 2  2 \blo ck" matrices where the blo cks

a;b;::: are n  n matrices, i.e., elements of R  makes no di erence as far as addition,

subtraction and multiplication of matrices is concerned. See for example [2], p. 4, or

2 2 2n 2n

[6], pp. 100{106. In symb ols, the rings R and F can be treated as b eing identical:

2 2 2n 2n 2 n n 2 2n 2n

R = F , or  F  = F . More generally, we can partition any mn  mn matrix

m n n m mn mn

as an m  m matrix of n  n blo cks:  F  = F .

The main p oint of this article is to lo ok at determinants of partitioned or blo ck

matrices. If a; b; c; d lie in a ring R , then provided that R is commutative there is a

determinant for M, which we shall write as det , thus: det M = ad bc, which of

R R

course lies in R . If R is not commutative, then the elements ad bc, ad cb, da bc,

da cb may not b e the same, and we do not then know which of them if any mightbe

n n

a suitable candidate for det M. This is exactly the situation if R = F , where F is a

R

eld or a commutative ring and n  2; so to avoid the diculty we take R to be, not

n n n n

the whole of the matrix ring F , but some commutative subring R  F . The usual

2 2 m m

theory of determinants then works quite happily in R , or more generally in R , and

m m n n

for M 2 R we can work out det M, which will b e an elementofR . But R  F , so

R

det M is actually a matrix over F , and we can work out det det M, which will b e an

R F R

n n m m m n n m mn mn

elementofF . On the other hand, since R  F ,wehave M 2 R   F  = F ,

so we can work out det M, which will also b e an element of F . Our main conclusion is

F

that these two calculations give the same result:

n n

Theorem 1. Let R be a commutative subring of F , where F is a eld or a commu-

m m

tative ring, and let M 2 R . Then

det M = det det M: 1

F F R 1

 !

A B

For example, if M = where A, B, C, D are n  n matrices over F which all

C D

commute with each other, then Theorem 1says

det M = det AD BC: 2

F F

Theorem 1 will be proved later. First, in section 2 we shall restrict attention to the

case m = 2 and give some preliminary and familiar results ab out determinants of blo ck

diagonal and blo ck triangular matrices which, as a by-pro duct, yield a pro of by blo ck

matrix techniques of the multiplicative prop erty of determinants. In section 3 we shall

prove something a little more general than Theorem 1 in the case m =2; and Theorem

1 itself, for general m, will b e proved in section 4.

2 The multiplicative prop erty

 !

A B

n n 2n 2n

Let M = , where A, B, C, D 2 F , so that M 2 F . As a rst case,

C D

! 

A O

, a blo ck-diagonal supp ose B = C = O, the n  n zero matrix, so that M =

O D

matrix. It isawell-known fact that

! 

A O

det = det A det D: 3

F F F

O D

The keen-eyed reader will notice immediately that, since

det A det D = det AD; 4

F F F

equation 3 is just 2 in the sp ecial case where B = C = O. However, we p ostp one

this step, b ecause with care we can obtain a proof of the multiplicative prop erty 4 as

aby-pro duct of the main argument.

One way of proving 3 is to use the Laplace expansion of det M by the rst n rows,

F

which gives the result immediately. A more elementary pro of runs thus: generalize to

the case where A is r  r but D is still n  n. The result is now obvious if r = 1, by

expanding by the rst row. So use induction on r , expanding by the rst row to p erform

the inductive step. Details are left to the reader. This result still holds if we know only

that B = O, and the pro of is exactly the same: we obtain

 !

A O

det = det A det D: 5

F F F

C D

By taking transp oses, or by rep eating the pro of using columns instead of rows, we also

obtain the result when C = O, namely,

! 

A B

= det A det D: 6 det

F F F

O D 2

In order to prove 4 we need to assume something ab out determinants, and we shall

assume that adding a multiple of one row resp ectively, column to another row resp ec-

tively, column of a matrix do es not alter its determinant. Since multiplying a matrix

on the left resp ectively, right by a unitriangular matrix corresp onds to p erforming a

number of such op erations on the rows resp ectively, columns, it do es not alter the de-

terminant. A uni triangular matrix is a triangular matrix with all diagonal entries equal

to 1. We shall also assume that det I =1, where I is the n  n identity matrix. So

F n n

now observe that

! ! !  !  !

I I A B C D I I I O

n n n n n

= ; 7

C D A B I I O I O I

n n n n

 !  !

C D A B

, since the rst three matrices on the left = det whence det

F F

A B C D

of 7 are unitriangular. From 5 and 6 it follows from this that

 !  !

O B A B

: 8 = det C det B = det det

F F F F

C D C O

But also

!  !  !

A AD A O I D

n

= :

I O I D O I

n n n

Here the second matrix on the left is unitriangular, so taking determinants and using 5

and the rst part of 8, we have

det A det D = det I det AD;

F F F n F

n n

and since det I =1, the multiplicativelaw 4 for determinants in F follows.

F n

3 Determinants of 2  2 blo ck matrices

Since we now know that det A det D = det AD, then also det C det B =

F F F F F

det B det C = det BC = det BC. From 5, 6 and 8, we obtain:

F F F F

 !

A B

Lemma 2. If M = , then

C D

det M = det AD BC 9

F F

whenever at least one of the blo cks A, B, C, D is equal to O.

Compare this with 2.

We shall now try to generalize somewhat. Supp ose the blo cks C and D commute,

that is, CD = DC. Then

! !  !  ! 

AD BC B AD BC B D O A B

: 10 = =

O D CD DC D C I C D

n 3

n n 2n 2n

We proved 4 in F for all n, so we can apply it in F to get, via 5 and 6,

det M det D = det AD BC det D;

F F F F

and so

det M det AD BC det D =0: 11

F F F

Now if det D is not zero or, in the case where F is a ring rather than a eld, if det D is

F F

not a divisor of zero, then 9 follows immediately from 11; but we do not actually need

this extra assumption, as we shall now show. Adjoin an indeterminate x to F , and work

in the p olynomial ring F [x]. This is another commutative ring, and a typical elementis

r r 1

a p olynomial a x + a x + :::+ a x + a , where a 2 F , all i, and addition, subtraction

0 1 r 1 r i

and multiplication of p olynomials is done in the obvious way. Now let us add to D the

scalar matrix xI , and for brevity write D = xI + D. Since a scalar matrix necessarily

n x n

commutes with C b ecause it commutes with every matrix, and also D commutes with

! 

A B

, then working C, it follows that D commutes with C. Thus, if we put M =

x x

C D

x

over F [x] in place of F yields equation 11 for M , that is,

x

det M det AD BC det D =0: 12

F x F x F x

But here we have the pro duct of two p olynomials equal to the zero p olynomial. The

second p olynomial det D = det xI + D is certainly not the zero p olynomial, but is

F x F n

n

monic of degree n, that is, it is of the form x + terms of lower degree. It is in fact the

characteristic p olynomial of D. This means that the left-hand p olynomial in 12, the

r

bracketed expression, must b e the zero p olynomial. For if not, it is of the form a x + terms

0

of lower degree, where a 2 F and a 6= 0. Multiplying the two p olynomials, we get

0 0

r +n

a x + terms of lower degree, which cannot be the zero p olynomial, a contradiction.

0

This argumentworks even when F is not a eld, but merely a commutative ring p ossibly

with divisors of zero. So now we have proved that det M det AD BC = 0;

F x F x

but D = D and M = M, so we just have to put x = 0 and we obtain the following

0 0

stronger version of Theorem 1:

 !

A B

n n

Theorem 3. If M = , where A, B, C, D 2 F and CD = DC, then

C D

det M = det AD BC: 13

F F

Other versions of Theorem 3 hold if di erent blo cks of M commute, so it is an exercise

for the reader to show that

if AC = CA then det M = det AD CB; 14

F F

if BD = DB then det M = det DA BC; 15

F F

if AB = BA then det M = det DA CB: 16

F F

Of course, if all four blo cks of M commute pairwise, then all of 13, 14, 15, 16

hold, the expressions on the right b eing equal. As a particular case, if R is a commutative

n n

subring of F and A, B, C, D 2 R , then any of 13{16 gives an expression for det M,

F

and this gives a pro of of Theorem 1inthe case m =2. 4

4 Determinants of m  m blo ck matrices

In this section we shall prove Theorem 1, but rst we do some preliminary calculations.

m m

Let R be any commutative ring, and let M 2 R , where m  2. We partition M as

! 

A b

m1 m1 m1 m1

, where A 2 R , b 2 R , c 2 R , and d 2 R . So here follows: M =

c d

b is a column vector, and c is a row vector. Supp ose c =c ;c ;:::;c , and consider

1 2 m1

the e ect of multiplying this on the rightby the scalar matrix dI . We have

m1

1 0

d 0 ::: 0

C B

0 d ::: 0

C B

C B

cdI  = c ;c ;:::;c 

. . .

m1 1 2 m1

.

C B

. . . .

.

A @

. . .

0 0 ::: d

= c d; c d;:::;c d

1 2 m1

= dc ;dc ;:::;dc  b ecause R is commutative

1 2 m1

= dc ;c ;:::;c =dc:

1 2 m1

In a similar way, AdI =dA. So now we have

m1

 ! !  !

A b dI 0 A b

m1 0

= 17

c d c 1 0 d

m1 m1

where A = dA bc 2 R . Notice here that bc, b eing the pro duct of an m 1  1

0

matrix and a 1  m 1 matrix, is an m 1  m 1 matrix. Applying det to

R

each side of 17, we get

m1

det Md = det A d: 18

R R 0

n n n n

If we now take R to be a commutative subring of F , then 18 is an equation in F ,

so we can apply det to each side and get

F

m1

det det Mdet d = det det A det d: 19

F R F F R 0 F

mn mn

Also, 17 is now an equation in F ,so we can apply det to each side and obtain

F

m1

det Mdet d = det A det d: 20

F F F 0 F

We now have everything we need ready for a pro of of Theorem 1.

Proof of Theorem 1. The result is trivial if m = 1, for then det M = M, and there

R

is nothing to prove. So assume m  2, partition M as ab ove, and use induction on m.

m1 m1

By the inductive hyp othesis, det det A  = det A , since A 2 R . From 19

F R 0 F 0 0

and 20 we deduce

m1

det M det det M det d =0: 21

F F R F

Now if det d is not zero or, in the case where F is a commutative ring rather than a

F

eld, if det d is not a divisor of zero, then our result follows immediately. To deal with

F

other cases, we rep eat our trick with p olynomials in the pro of of Lemma 2, in section 3. 5

Adjoin an indeterminate x to F , and work in the p olynomial ring F [x]. Add the scalar

matrix xI to d to de ne d = xI + d whichmay lo ok strange, but d, b eing in R , really

n x n

is an n  n matrix over F , and make the corresp onding adjustment to M, by putting

! 

A b

. Then exactly as for M, we have M =

x

c d

x

m1

det M det det M  det d  =0: 22

F x F R x F x

But det d is a monic p olynomial of degree n, and we conclude that the other term in

F x

22, det M det det M , is the zero p olynomial. Putting x =0 and noting that

F x F R x

d = d and M = M, we obtain det M = det det M, and the pro of of Theorem 1is

0 0 F F R

complete.

As an application, we can obtain the formula for the determinant of a tensor pro duct

 !

p p

11 12

2 2 n n

of matrices rather easily. For example, if P = 2 F and Q 2 F , the

p p

21 22

 !

p Q p Q

11 12

tensor product P Q is de ned to be the 2n  2n matrix . Here the

p Q p Q

21 22

blo cks, all b eing scalar multiples of Q, commute pairwise; one could in fact take R to b e

n n

the subring of F generated by Q, that is, the set F [Q] of all p olynomial expressions

r r 1

a Q + a Q + ::: + a Q + a I , where r  0 and a 2 F , all i. This is certainly a

0 1 r 1 r n i

n n

commutative subring of F and contains all four blo cks p Q. Applying Theorem 1, or

ij

Theorem 3, we have

det P Q = det p Qp Q p Qp Q

F F 11 22 12 21

2 2

p p Q p p Q = det

11 22 12 21 F

2

= det p p p p Q

F 11 22 12 21

2

= det det PQ

F F

n 2

= det P det Q :

F F

More generally:

m m n n

Corollary. Let P 2 F and Q 2 F . Then

n m

det P Q = det P det Q :

F F F

Proof. Take R = F [Q], as ab ove. If P =p , then by de nition

ij

1 0

p Q p Q ::: p Q

11 12 1m

C B

p Q p Q ::: p Q

C B

21 22 2m

m m

C B

2 R : P Q =

. . .

.

C B

. . . .

.

A @

. . .

p Q p Q ::: p Q

m1 m2 mm

Then det P Q is a sum of m! terms each of the form

R

m

Q :::p p Q=p Q :::p Qp p

mi 2i 1i mi 2i 1i

m m

2 1 2 1 6

for some p ermutation i ;i ;:::;i of 1; 2;:::;m. Thus

1 2 m

m

det P Q = det PQ :

R F

Then, by Theorem 1,

det P Q = det det P Q

F F R

m

= det det PQ 

F F

n m

= det P det Q 

F F

n m

= det P det Q :

F F

5 Acknowledgements

This pap er b egan when I set my linear algebra class the following exercise: if A, B, C,

n n

D 2 F and D is invertible, show that

 !

A B

1

det = det AD BD CD:

C D

The intended pro of was via the equation

 ! !  !

1

A B I O A BD C B

n

= ;

1

C D D C I O D

n

and the exercise to ok its inspiration from the theory of Dieudonn e determinants for

matrices over skew elds. See, for example, [1], chapter IV. It was a small step to

1

notice that, if also CD = DC, then 13 holds; but 13 do es not mention D , and so

the natural question to ask was whether 13 would hold even if D were not invertible

but still CD = DC. I eventually found a pro of of this, when F is a eld, by using

the principle of the irrelevance of algebraic inequalities. See [3], chapter 6. I passed

this around friends and colleagues, and Dr W. Stephenson showed me how to use monic

p olynomials to shorten the argument, and extend it from elds to rings. At the same

time, Dr A. D. Barnard suggested it might b e p ossible to extend the result to the m  m

case by assuming al l the pairs of blo cks commute. I am grateful to b oth of them for their

help, and their interest. I also wish to thank the referee for directing my attention to

some of the early literature on this sub ject, mentioned b elow.

Theorem 1 is not, I think, a new result, and I have seen what app ears to b e an abstract

version of it without pro of  at a much more advanced level. I have not b een able to

nd an elementary statement and pro of, as given ab ove, in the literature. The blo ck

matrix pro of of the multiplicative prop erty of determinants is essentially that given in

[2], chapter 4. The formula for the determinant of a tensor pro duct rst app ears in the

case m =4, n =2 in [11], and indeed is referred to in [7] as Zehfuss' theorem. The rst

pro of of the general case is probably that in [8], p. 117, though in [5], p. 82, the pro of is

attributed to [4], [9] and [10]. See also the references to these articles in [7], volume 2,

pp. 102{104, and volume 4, pp. 42, 62 and 216. 7

References

[1] E. Artin, Geometric Algebra,Interscience 1957.

[2] Frank Ayres, Jr., Matrices, Shaum's Outline Series, McGraw-Hill 1962.

[3] P.M. Cohn, Algebra, Volume 1, Wiley 1982 2nd edition.



[4] K. Hensel, Ub er die Darstellung der Determinante eines Systems welches aus zwei

anderen comp onirt ist, Acta Math. 14 1889, 317{319.

[5] C. C. MacDu ee, The theory of matrices, Chelsea 1956.

[6] L. Mirsky, An intro duction to linear algebra, Clarendon Press 1955.

[7] T. Muir, The theory of determinants in the historical order of development, Dover

reprint 1960.

[8] T. Muir, A treatise on determinants, Macmillan 1881.

[9] E. Netto, Zwei Determinantensatze, Acta Math. 17 1893, 199{204.

[10] R. D. von Sterneck, Beweis eines Satzes ub er Determinanten, Monatshefte f. Math.

u. Phys. 6 1895, 205{207.



[11] G. Zehfuss, Ub er eine gewisse Determinante, Zeitschrift f. Math. u. Phys. 3 1858,

298{301.

Department of Mathematics 7 September 1999

King's Col lege

Strand

London WC2R 2LS

[email protected] 8