Introduction to matrix algebra

R STUMNTS TEXT wm@ INTRODUCTIO;; TO MATRIX ALGEBRA SCHOOL MATHE#ATlCS STUDY YALE UNIWRSCTY ?RESS GROUP School Mathematics Study Group Introduction to Matrix Algebra Ii~troductionto Matrix Algebra Strrdetlt's Text Prcparcd u n h r th ~est~pen.kjon of rhc Panel on Sample T e x r h k s of rhc School Mathcmarm Srudy Group: Frank B Allen Edwirl C Douglas Donakl E Richmond Cllarlm E,Rickart Hcnry Swain Robert J Walkcr Lyonr Township High School Tafi S C h I Williamr Collcgc Yale Univcrriry New Trier Township High S c h d Cornell University Ncw Havcn a d h d o n , Y a1c University Press Coyyrig!lt O I*, r p r by Yak Ut~jversiry Printed in rhc Ur~itcdStarm af America A]] r i ~ h t srescwcd Ths h k may not ltrc repcduccd, in whdc or in part, in any form, wirho~~t written permission from the publishen FinanciaZ slspporr for the S c b l Mathemaria Study Group has km providcd by rhe Narional Scicncu: Foundation The i n c r e a s i n g contribution of mathematics to the culture of the d c r n w r l d , as well as i t s importance as a v i t a l part of s c i e n t i f i c and humanistic tdueation, has mndc i t essential that the m a t h e ~ u t i e n our echool@ be bath wZ1 selected and e l l taught t j i t h t h i s i n mind, the various mathc~atical organLtaeions i n the United Seatss cooperated i n the forumtion of the School F ~ t k m a e i c sStudy Group (SPSC) SMG includes college and unirersiley mathmaticlans, eencbera of m a t h w t i e s a t a l l lavels, experts i n education, and reprcscntarives of science and tcchrmlogy Pha general objective o f SMG i a the i m p r o v m n t a f thc? teaching of =themtics in the sehmla of t h i s country The l a t i o n u l Science Foundation has provided subs tantiaL funds for the support of this endeavor One of the pterequlsitea for the improvement of t h e teaching of math&matira in our ~ c h m t si e on improved c u r r i e u ~ ~ n which s takea account of thc incrcasing use of oathematice Ln science and teehnoLogy end i n other areas o f knowledge and a t the a m timc one v f i i ~ href l o e t a recent advancta i n mathem~tica i t s c l f One of thc f l r n t projects undartaken by SrSC was t o enlist a group of outstanding mathematiclans and nrarhematica teachern t o prepare a nariea of textbooks uhich would illustrmre such A n improved curriculum The professional arathcmatLcLanlr in SMSG belleve that the mathematics p r P eented In this t t x r is vaLuahLc fat a l l w2 l-ducatcd citlzene i n our society t o know 6nd t h a t it ia tqmrtant f o r the prceollage atudent: to learn i n preparation far advanced vork i n thc f i e l d A t thc time t i n e , teachers in SWC believe t h a t it is presented in rrueh a farm that i t can r ~ a d F l ygrasped by students ! ! In most inntanees the mnterlal will have a inmiltar n o t e , he the prcsentation and the point of vicv vill be diftcrsnt Samc matcrial w i l l be e n t i r e l y neu t o ehc traditional curriculum, This i s ar i t should be, for ~ A a t h c m & t i ~i & a a ltving and an cvcwrawinff subject, and Rat a dead and frozen product of ant i q u i t y This heakthy fusion o f the old and Ehc ncu should lead students to a better undcsstsnding of the basic concepts and structure of mthemtics and provide a f i m r foundation for understanding and use of ~ t h c m a t i c sin a gcitntlfic sociery It la n o t intendad that this book be regarded an the mly d c f i n i t l v a way of presenting good mathcmatica t o student# a t t h i s Level Instead, I t should be thought of as e ample of the kind of i q r w e d currieulua that we need and as a source of nuggestiana for the authora of c-rclal textbka It is sincerely hoped that these t e x t s wLl1 lead the way coward inspiring a more meaningful reaching o f Hathemnckea, t h e Wcca and Servant of tho Sctencea 220 and then we have Since is a complex number, the unit matrix is s t i l l If z a x 4- iy, then we write and call this dumber the complex conjugate of A quaternion i~ m element q [G q] , We denote by (a) Q ( q ) = x2 Hence conclude that, since q = or simply the conjugate of of the particular form r t C and w E C the s e t o f all quaternions Ohow that and only i f of K z, + y + uZ + v2 x, y , u, and v if r - x + iy are real numbers, and w = u 6(q) = if Show that if q E Q, then q has an inveree if and only if q and exhibit the form of q-' if it exists (b) Four elements o f names : + iv Q + 0- , are of particular importance and we g i v e them special z (c) where Show t h a t if z x f iy and w ='u + iv, then q =X + yIU + u v * v w (d) Prove the following i d e n t i t i e s involving I, U, V and W: v'PV2iJi-f and UV=W=-W, and VW=U=-WV, (el Use the preceding two exercises t o show then q + r , q - r, and q r are all elements of W=V uw that if q E Q Q The conjugate of the element q = and the norm and [ 1, where r = x + iy, v - u + lv, trace are given respectively by and (f) Show that i f q f Q, and i f q is invertible, then and r E Q, 222 9, and if q-l From t h i s c ~ n lcu d e that if q -Z (g) Show that each q (h) Show that i f q -1 r s a t i s f i e s the quadratic equation Q q E Q, cnis t a , then then Note t h a t t h i s may be proved by using the result t h a t i f then and then ueing the results given in ( d ) ( i ) Show t h a t i f q E Q and r e Q, then tqrl = Iql I r l and The geometry o f quaterniona constitutes a very i n t e r e s t i n g subject requires the representation of a quaternion as a p o i n t w i t h coordinates (a, b, e, d) in four4imenaional spaces It The subset of elements, QL {q: E Q and Iql = I), is a group and is represented geometrically as the hyperephere w i t h equation Nonassociative Algebras The algebra o f matrices (we r e s t r i c t our attention in t h i s exercise to the set M of cation x matrices) has an associative but not a comutative multipli- "Algebras" with nonassociative multiplication have become increasingly example, in mathema tical gene t i c s important in recent years-for Genetics is a eubdiacipline o f biology and is concerned w i t h transmission o f hereditary traits Nonassociative "algebras" are important also in the study of quantum mechanics, a subdiscipline o f p h y s i c s We give first: a simple algebra (named a f t e r the Norwegian geometer Sophus LLe) If A E M and read this (a) and "A op B," "op" (i) AoB = - BOA, (ii) AoA = Q, (iv) being an abbreviation f o r operation o: Ao(BoC) + Bo(CoA) + Co(AoB) AoI = = IoA G i v e an example ro show that and hence that o i s not example of a L i e we write Prove the following properties of liii) (b) E bf, = Ao(BoC) 2, and (AoB)oC are d i f f e r e n t an associative operation, D e s p i t e these strange properties, o behaves nicely relative to ordinary metrix addition (c) Show that o dietributes over addition: (d) Show that o behaves nicely r e l a t i v e to multiplication by a number and 224 It will be recalled that and is defined as the element A' is termed the multiplicative inverse of B A satisfying the r e l a t i o n s h i p s AB = - I BA But i t m u s t elso be recalled thar t h i s d e f i n i t i o n was motivated by the fact that A1 = A = that is, by the fact t h a t (e) IA, I is a multiplicative unit Show rhar there is no o unit thar We know, from the foregoing work, o i s neither c o m u t a t i v e n o r Here is another kind of operation, called Jordan multiplication: associative If A f M and wedefine B f M, AjB = (AB + BA) - We see at once that AjB = B j A , so that Jordan mulriplicarion is a comutative operation; b u t i t i s n s associative (f) Determine all of the properties o f the operation For example, does j j that you can d i s t r i b u t e over a d d i t i o n ? The Algebra of Subsets We have seen that there are interestfng algebraically defined subsets of x matrices One such subset, f o r example, is the set M, the set of all Z, which is isomorphic wf th the set of complex numbers Much of higher mathematics i e concerned w i t h the "global structure" of "algebras," and generally this involves the consideration of eubsets of the "algebras" being s t u d i e d t h i s exercise, we shall generally underscore l e t t e r s t o denote subsets of If and g are subsets of M, then In M is the s e t of a l l elements of the form where A + B , A E and B E -B In set-builder notation this may be written A + B = - {A+B * By an a d d i t i v e subset of : A s and B E B] is meant a subset M A CM such that Determine which of the f o l l o w i n g are additive subsets o f (a) (i) M: (213 (ii> (11, (iii) M, (iv) 2, (v) %, (vi) the s e t of a l l (i) if B= (ii) (Lii) A+ 0, - and if with x (d) E with are subsets of A - 1, M, then of M, then A - + C-C B-+ C - C -B & and B are addttive subsets M denote the s e t o f a l l column vectors R and Show t h a t i f M whose entries are nonnegative V y c &(A) ( A + B ) +C, = is also an addittve subset of Let and M !+A, (B + C-) Prove t h a t if (c) in t h e set of all elements of Prove t h a t if A, (b) A R v is a fixed element of V, then then A: A E M M is an a d d i t i v e subset of set = Notice also that i f If A - and B are subsets is the Av and M, of AV = then (-A)" = then of all AB, A and E B E g Using set-builder notation, we can w r i t e this in the form AB = A subset A of M IAB: A A - and B E 01 is multiplicative if (e) Which of the s u b s e t s in p a r t (a) are multiplicative? (f) Show t h a t i f (ii) and i f A -, B, - ACE, and are subsets of C - then M, then A C C BC - A B of -, M such t h a t (g) Give an example of two s u b s e t s (h) Determine which o f the following subsets are multiplicative: and (iii) the s e t of a l l elements o f M w i t h negative entries, (iv) the s e t o f all elements of M f o r which t h e upper left-hand M o f t h e form e n t r y is l e s s than 1, (v) the set of all elements of with Osx, Osy, x + y