17 Groups of Matrices

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1 17 Groups o Matrices In this paragraph, we examine some groups whose elements are matrices. The reader proaly knows matrices (whose entries are real or complex numers), ut this is not a prerequisite or understanding this paragraph. We give an elementary account o the theory o matrices as ar as needed here. Matrix theory will e taken systematically in Chapter 4, 43. We allow the entries to e elements o any ield. Fields will e ormally introduced in Chapter 3, 9 (Deinition 9.13). Until then, we shall e content with the ollowing deinition Temporary Deinition: A ield is one o the sets,, and p, where p is a prime numer. Ater having learned aout ields in Chapter 3, the reader may check that the theory in this paragraph carries over to the more general situation where the term "ield" is used in the sense o Deinition We note that K is a commutative group under addition,. whose identity element we shall denote y 0. (so that 0 is the numer 0 in case K is one o,,, and it is the residue class 0 = 0 + p in case K is p or some prime numer p), and that K\{0} is a group under multiplication.. This will e used many times in this paragraph Deinition: Let K e a ield. A matrix over K is an array ( a c o our elements a,,c,d o K, arranged in two rows and two columns, and enclosed within parentheses. (The plural o "matrix" is "matrices".) 170

2 Thus ( ) is matrix over (and also over an, ( 5 7 ) is a matrix over (and also over ). In addition, ( ) is a matrix over 7, when ars mean residue classes modulo 7. The set o all matrices over a ield K will e denotded y Mat (K). The suscript signiies that there are rows and columns in a matrix (in the sense o Deinition 17.). I K is a ield and A,B are matrices rom Mat (K), we say A is equal to B provided the corrsponding entries in A and B are equal. More exactly, A: ( a c d ) is equal to B: (a c d ) i and only i a = a, =, c = c, d = d.. In this case, we write A = B. A single matrix equation is equivalent to our. equations etween the elements o the underlying ield.. It is clear that matrix equality is an equivalence relation on Mat (K). In particular, it is legitimate to say that A and B are equal when A is equal to B.. In this deinition o matrix equality, the location o the entries are taken into account. Thus ( ) and ( ) are dierent matrices, although they are made up o the same numers. We introduce two inary operations on Mat (K), addition and multiplication. Addition is deined in the most ovious way Deinition: Let K e a ield. For any A = ( a c Mat (K), we deine the sum o A and B as the matrix, B = (e g h ) in ( a+e + c+g d+h ). The sum o A and B will e denoted y A + B. Taking sums in Mat (K) will e called addition (o matrices). Addition o matrices is essentially the addition in the underlying ield, carried out our times. Not surprisingly, many properties o addition in the ield are relected in matrix addition. For example, just like a ield is a group under addition, matrices over a ield orm a group under addition, too. 171

3 17.4 Theorem: Let K e a ield. Then Mat (K) is a commutative group under addition. Proo: We check the group axioms (i) For any matrices A = ( a c, B = (e g h ) in Mat (K), we have a + e, +, c + g, d + h K since a,,c,d,e,,g,h K and K is closed under addition. Hence A + B = ( a+e + c+g d+h ) K and Mat (K) is closed under (matrix) addition. (ii) Associativity o addition in Mat (K) ollows rom associativity o addition in K. Indeed, or any A = ( a c Mat (K), we have (A + B) + C = [( a c + (e g h )] + (k n = ( (a+e)+k (+)+m (c+g)+n (d+h)+p ) = ( a+(e+k) +(+m) c+(g+n) d+(h+p) ) = ( a c + (e+k +m g+n h+p ) = A + (B + C)., B = (e g m p ) = (a+e c+g h ), C = (k n + d+h ) + (k m n p ) m p ) in (iii) What can e the identity element? Well, proaly the matrix ( ), where 0 denotes the zero element o the ield K (or instance, when K is or some prime numer p, 0 is the residue class 0 = p ). Indeed, we have, or any A = ( a c A + ( ) = (a c and ( 0 0 p + ( ) = (a+0 c+0 Mat (K), +0 d+0 ) = (a c 0 0 ) is a right identity o Mat (K). The matrix (0 0 = A 0 0 ) will e called the zero matrix (over K) and will e designated y the symol 0. This should not e conused with the zero element o the underlying ield K. (iv) Any matrix A = ( a c (opposite) A in Mat (K), namely ( a c Mat (K) has a right inverse d ) (since a,, c, d K): 17

4 ( a c + ( a c a) d ) = (a+( c+( c) +( ) d+( d) ) = ( ) = 0. Thus Mat (K) is a group under addition. We inally check commutativity. (v) Commutativity o addition in Mat (K) ollows rom commutativity o addition in K. Indeed, or any A = ( a c Mat (K), we have A + B = ( a c + (e g h ) = (a+e + + c+g d+h )= (e+a g+c h+ = (e g So Mat (K) is a commutative group under addition., B = (e g h ) + (a c h ) in = B + A. The additive group Mat (K) is somewhat dull. It is just our copies o the additive group K.. More interesting matrix groups arise when the operation is multiplication. We introduce this operation now Deinition: Let K e a ield. For any A = ( a c, B = (e g Mat (K), we deine the product o A and B as the matrix h ) in ( ae+g a+h ce+dg c+dh ). The product o A and B will e denoted y A. B or simply y AB.. Taking products in Mat (K) will e called multiplication (o matrices). This deinition looks izarre. One would expect the product o A and B, with the notation o Deinition 17.5, to e ( ae cg dh ). Some motivation or Deinition 17.5 can e gained as ollows. With each matrix ( a c (over, say), there is associated a coordinate transormation x = ax + y y = cx + dy o the Euclidean plane. Carrying out the transormations associated with ( a c, (e g h ) successively, we otain 173

5 x = ax + y y = cx + dy x = ex + y y = gx + hy, which gives x = a(ex + y ) + (gx + hy ) = (ae+g)x + (a+h)y y = c(ex + y ) + d(gx + hy ) = (ce+dg)x + (c+dh)y, so the product o the matrices is the one which is associated with the successive application o the transormation. I matrix multiplication is new to you, you are urged to write down matrices over perorming this operation. and multiply them in order to acquire dexterity in We collect some asic properties o matrix multiplication. in the next theorem. Let us recall that K\{0} is a group under multiplication.. The identity element o this group will e denoted y 1.. Thus 1 is the numer 1 when K is one o,,, and the residue class 1 = 1 + p when K = p or some prime numer p Theorem: Let K e a ield, whose zero element is 0 and whose identity element is 1. (1) Mat (K) is closed under matrix multiplication. () (AB)C = A(BC) or all A,B,C Mat (K). (3) Let I = ( ). Then AI = IA = A or all A Mat (K). (4) A(B + C) = AB + AC and (B + C)A = (BA + CA) or all A,B,C Mat (K). Proo: Let A = ( a c Mat (K)., B = (e g h ), C = (k n m p ) e aritrary elements o (1) Since a ield is closed under addition and multiplication,. ae + g, a + h, ce + dg, c + dh K whenever a,,c,d,e,,g,h K. So AB Mat (K) or all A,B Mat (K) and Mat (K) is closed under multiplication. () This is routine calculation. We evaluate (AB)C and A(BC): 174

6 (AB)C = [( a c (e g h )](k m n p ) = (ae+g ce+dg a+h c+dh )(k m n p ) (ae+g)k + (a+h)n (ae+g)m + (a+h)p = ((ce+dg)k + (c+dh)n (ce+dg)m + (c+dh)p ) = ( aek+gk+an+hn aem+gm+ap+hp cek+dgk+cn+dhn cem+dgm+cp+dhp ), (i) A(BC) = ( a c [(e g h )(k m n p )] = (a c (ek+n em+p gk+hn gm+hp ) = ( a(ek+n)+(gk+hn) a(em+p)+(gm+hp) c(ek+n)+d(gk+hn) c(em+p)+d(gm+hp) ) = ( aek+an+gk+hn aem+ap+gm+hp cek+cn+dgk+dhn cem+cp+dgm+dhp ). (ii) Since addition is commutative in K, the matrices (i). and (ii) are equal. Hence (AB)C = A(BC) or all A,B,C Mat (K). (3) We compute AI = ( a c as claimed. IA = ( 1 0 ( ) = (a1+0 c1+d0 0 1 )(a c = (1a+0c 0a+1c a0+1 c0+d1 ) = (a c = A, 1+0d 0+1 = (a c = A, (4) We have A(B + C) = ( a c = ( a c [(e g (e+k g+n h ) + (k n +m h+p ) = ( a(e+k)+(g+n) c(e+k)+d(g+n) = ( ae+ak+g+n ce+ck+dg+dn = ( ae+g+ak+n ce+dg+ck+dn = ( ae+g a+h = ( a c ce+dg (e g = AB + AC. m p )] a(+m)+(h+p) c(+m)+d(h+p) ) a+am+h+p c+cm+dh+dp ) a+h+am+p c+dh+cm+dp ) c+dh ) + (ak+n ck+dn h ) + (a c (k m n p ) am+p cm+dp ) The proo o (B + C)A = (BA + CA) ollows similar lines and is let to the reader. 175

7 Theorem 17.6 seems promising. Three o the group axioms are satisied, with I as the identity. It remains to investigate whether every matrix over a ield has a right inverse. Suppose K is a ield and A = ( a c Mat (K). Then A has a right inverse X = ( x y z u ) in Mat (K) i and only i AX = I, which is equivalent to (1) ax + z = 1, () ay + u = 0, (3) cx + dz = 0, (4) cy + du = 1. We multiply the equation (1) y d, (3) y and add them side y side. Using associativity o addition in K, distriutivity o multiplication over addition, and commutativity o multiplication in K, we get (ad c)x = d. We multiply () y d, (4) y and add them. We multiply (1) y c, (3) y a and add them. We multiply () y c, (4) y a and add them. We get (ad c)y =, (ad c)z = c, (ad c)u = a. We emphasize again that commutativity o multiplication in K is used crucially to derive these equations. The element ad its importance, we give it a name. c appears in each one o these equations. In view o 17.7 Deinition: Let K e a ield and A = ( a c element ad as det A. Mat (K). Then the c in K is called the determinant o A, written as det(a) or We have shown: i K is a ield and A = ( a c in Mat (K) is a right inverse o A, then Mat (K), and i X = (x z y u ) (det A)x = d (det A)y = (det A)z = c (det A)u = a. (D) 176

8 These equations impose certain conditions on a matrix having a right inverse. We cannot expect that every matrix has a right inverse. Those having a right inverse are charecterized very simply as the matrices with a nonzero determinant Theorem: Let K e a ield and A = ( a c right inverse i and only i det A unique right inverse o A, namely the matrix ( (det A) 1 d (det A) 1 c where (det A) 1 is the inverse o det A group K\{0}. Mat (K). Then A has a 0. I this is the case, then there is a (det A) 1 (det A) 1 a ) K\{0} in the multiplicative Proo: First we assume det A = 0 and show that A has no right inverse. Indeed, i det A = 0 and A had a right inverse, then the equations (D) would ecome d = 0 = 0 c = 0 a = 0, and A = ( a c would e the zero matrix (0 0 inverse X = ( x y z u ) would yield ( ) = I = AX = ( )(x y z u ) = (0x+0z 0x+0z 0 0 ). The existence o a right 0y+0u 0y+0u ) = ( ), hence 1 = 0 in K, a contradiction. Thus A has no right inverse i det A = 0. Now let us assume det A Since det A 0 and show that A has a unique right inverse. K\{0} and K\{0} is a group under multiplication, det A has an inverse in K\{0}, which we denote y (det A) 1. This is the nonzero element o the ield K such that (det A) 1 (det A) = (det A)(det A) 1 = 1 = the identity element o K\{0}. So we can solve or x,y,z,u in (D) y multiplying the equations in (D) y (det A) 1. We get x = (det A) 1 d, y = (det A) 1, z = (det A) 1 c, u = (det A) 1 a. Thus, i A has a right inverse at all, this right inverse must e the matrix written in the enunciation o the theorem (in particular, A has a unique 177

9 right inverse). It is easy to check that this matrix is indeed a right inverse o A: ( a c d)( (det A) 1 d (det A) 1 c (det A) = ( 1 (ad c) (det A) 1 (cd dc) = ( ) = I. (det A) 1 (det A) 1 a ) (det A) 1 ( a+a) (det A) 1 ( c+da)) Hence A does have a unique right inverse and it is the matrix given in this theorem. We will prove presently that the matrices with right inverses orm a group under multiplication. From Lemma 7.3, it will then ollow that the unique right inverse o a matrix with a nonzero determinant is also the unique let inverse o the same matrix. We shall reer to is as its inverse. The rule or inding the inverse o A = ( a c d ) is simple: interchange a and d, then put a minus sign in ront o and c, and multiply each entry y (det A) 1 [i.e., divide each entry y det A]. For example, the inverse o 1 1 ) Mat ( ) is ( 5 = ( 1/4 1/ /8 5/8 ( ) Mat ( 7 ) is ( ) = ( ) since the determinant is equal to 18 = 4 and 4 1 =. ) and that o 17.9 Theorem: Let K e a ield.. (1) det (AB) = (det A)(det B) or all A,B Mat (K). () det I = 1 ( K).. (3) I AX = I, then det X = (det A) 1.. Proo: (1) We use the notation o Deinition We get det (AB) = (ae + g)(c + dh) (a + h)(ce + dg) = aec + aedh + gc + gdh ace adg hce hdg = aedh adg + gc hce = ad(eh g) c(eh g) 178

10 = (ad c)(eh g) = (det A)(det B). () det I = = 1 0 = 1. (3) This ollows rom (1) and (): i AX = I, then 1 = det I = det(ax) = (det A)(det X), so det X = (det A) 1. The ormula det AB = (det A)(det B). is known as the multiplication rule o determinants.. Loosely speaking, the determinant o a product is the product o the determinants.. By induction on n, it is extended to n actors: det (A 1 A... A n ) = (det A 1 )(det A )... (det A n ). We inally have a group o matrices under multiplication Theorem: Let K e a ield. Then {A Mat (K): det A 0} is a group under matrix multiplication. Proo: We check the group axioms. Let us call our set G or revity. (i) For A,B G, we have det A 0 det B. In the ield K, product o nonzero elements is nonzero (K\{0} is a group, and closed under multiplication). So det AB = (det A)(det B) 0 y Theorem 17.9(1) and consequently AB G. Thus G is closed under multiplication. 17.6(). (ii) Associativity o multiplication in G ollows rom Theorem (iii) I is a right identity element o G, or det I = 1 0 y Theorem 17.9(), so I G; and AI = A or all A G y Theorem 17.6(3). (iv) Any A G has a right inverse in G. Indeed, i A G, then det A 0, so A has a right inverse X in Mat (K). As det X = (det A) 1 0 (Theorem 17.9(3)), we see X G. Thus A has a right inverse in G. Thereore, G is a group. 179

11 17.11 Deinition: Let K e a ield. The group o Theorem is called the general linear group (o degree ) over K, and is written as GL(,K). Since GL(,K) is a group, the unique right inverse o any matrix A in GL(,K) is also the unique let inverse o that matrix (Lemma 7.3). It will e called the inverse o A, and will e written as A 1, in conormity with the usual terminology and notation. The matrix I will e called the identity matrix. Elements o GL(,K) are called invertile matrices or regular matrices. Matrices whose determinants are zero are called singular. The next theorem urnishes another matrix group Theorem: Let K e a ield. Then {A Mat (K): det A = 1} is a group under matrix multiplication. Proo: Let us call this set S or revity. As 1 0 in K, we get S GL(,K). We use the sugroup criterion (Lemma 9.) to check that S is a sugroup o GL(,K). (i) For A,B S, we have det A = 1 = det B, thereore det AB = (det A)(det B) = 1.1 = 1 y Theorem 17.9(1) and consequently AB S. Thus S is closed under multiplication. (ii) For any A S, we have det A = 1, so det (A 1 ) = (det A) 1 = 1 1 = 1 y Theorem 17.9(3) and A 1 S. Thus S is closed under the orming o inverses. Thereore, S is a sugroup o GL(,K) Deinition: Let K e a ield. The group o Theorem 17.1 is called the special linear group (o degree ) over K, and is written as SL(,K). We close this paragraph with a group that plays an important role in numer theory and in complex analysis. 180

12 17.14 Theorem: The set {( a c Mat ( ): a,,c,d, ad c = 1} is a group under matrix multiplication. Proo: Let us call this set H or revity. Clearly H SL(, ). We check that H is a sugroup o SL(, ). AB = ( ae+g ce+dg and B = (e g h ) are elements o H. Then (i) Suppose A = ( a c a+h c+dh ). Here the entries o AB, namely ae+g, ce+dg, a+h, c+dh are integers, ecause a,,c,d,e,,g,h are integers. Also, det A = 1 = det B, thereore det AB = (det A)(det B) = 1.1 = 1 y Theorem 17.9(1) and AB H. Thus H is closed under multiplication. (ii) Let A = ( a c H. Then det A = 1 and so A 1 = ( d c a ) y Theorem The entries d,, c, a o A 1 are integers, ecause a,, c, d are integers. Also, we have det A = 1, so det (A 1 ) = (det A) 1 = 1 1 = 1 y Theorem 17.9(3) (or det (A 1 ) = da ( )( c) = ad c = 1). So A 1 H and H is closed under the orming o inverses. Thereore, H is a sugroup o SL(, ) Deinition: The group o Theorem is called the special linear group (o degree ) over, or the modular group, and is written as SL(, ) or as. Exercises 1. Let K e a ield. Show that GL(,K) is not an aelian group.. Find all elements o GL(, ). What is the order o GL(, )? 3. Write down the multiplication tale o GL(, ). Compare it (eventually ater reordering the rows and columns). with the multiplication tale o S

13 4. Find all elements o SL(, 3 ). What is the order o SL(, 3 )? 5. Write down the multiplication tale o SL(, 3 ) 6. Let K e a ield and let A = ( a c Mat (K). When a = 0 =, we have det A = 0. In case (a,) (0,0),. prove that det A = 0 i and only i there is an element k in K such that c = ka, d = k.. Use this result and show that GL(, p ) = (p 1)(p p). 7. Determine how many elements in GL(, ) have the same determi- p nant. Find the order o SL(, p ). 8. Show that {( 1 a 0 ) Mat (K): 0} is a sugroup o GL(,K). 9. Prove that {( a 0 Mat (K): ad Its elements are called triangular matrices. 0} is a group under multiplication. 10. Let K e a ield. For any A = ( a c Mat (K), we deine the trace o A to e the element a + d o K. (sum o the entries in the upper-let lower-right diagonal).. Show that the trace o AB is equal to the trace o BA or all A,B Mat (K). 11. Let K e a ield. For any A = ( a c transpose o A to e the matrix ( a Show that det A t = det A and (AB) t = B t A t or all A,B Mat (K), we deine the c Mat (K), which is written At. Mat (K). 1. Let m and put Mat ( m ) = {( a c d ): a,,c,d }. Show that the m theory in the text, until Theorem 17.8,. remains valid or the elements o Mat ( m ), which are called matrices over m. In place o Theorem 17.8, prove that A inverse i and only i det A m. Mat ( m ) has a unique right Put GL(, m ) = {A Mat ( m ): det A m }. Show that GL(, m ) is a group under multiplication. Prove that Theorem 17.1 remains true i "K" is replaced y " m ". 18

14 13. Develope a theory o matrices over y modiying the theory o matrices over. How do you deine GL(, )? 14. Let H = {( a conjugate o x a ): a, } Mat ( ), where x is the complex. Prove that H is closed under addition and multiplication. Show that H\{0} is a group under multiplication. 15. I K is a ield and A = ( a c Mat (K), we write A = ( a c d ). Let 1 = ( ), i = ( i 0 0 i ), j = ( ), k = (0 i i 0 ) Mat ( ). Thus 1 is the identity matrix over.. Show that ij = k, jk = i, ki = j. Prove that {1, 1,i, i,j, j,k, k} is a group under multiplication,. called a quaternion group o order 8 and is denoted as Q 8. Show that Q 8 has exactly one element o order. Find all sugroups o Q

A matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and

Section 5.5. Matrices and Vectors A matrix is a rectangular array of objects arranged in rows and columns. The objects are called the entries. A matrix with m rows and n columns is called an m n matrix.