Chapter 3. Linear Equations and Matrices

Transcription

1 Vector Spaces Physcs 8/6/05 hapter Lear Equatos ad Matrces wde varety of physcal problems volve solvg systems of smultaeous lear equatos These systems of lear equatos ca be ecoomcally descrbed ad effcetly solved through the use of matrces Lear depedece of vectors Let us cosder the three vectors E, E ad E gve below E E,, E E,, (-) E E,, These three vectors are sad to be learly depedet f there est costats c, c, ad c, ot all zero, such that j j 0 c E c E c E E c (-) The secod form the equato above gves the -th compoet of the vector sum as a mpled sum over j, vokg the Este summato coveto Substtutg the values for the compoets E from equato (), ths vector equato s equvalet to the three lear equatos: j c c c c c c 0 0 c c c 0 These relatos are a specal case of a more geeral mathematcal epresso, (-) c d (-) where c ad d are vectors represeted as colum matrces, ad s a sort of operator whch we wll represet by a square matr The goal of the et three chapters s to solve equato (-) by the audacous leap of fath, -

2 Vector Spaces Physcs 8/6/05 What s c d (-5)? Dvdg by a matr!?! We wll come to ths later B Defto of a matr matr s a rectagular array of umbers Below s a eample of a matr of dmeso ( rows ad colums) j,,, j, (-6) We wll follow the "R coveto" for umberg elemets of a matr, where j s the elemet of matr ts -th row ad j-th colum s a eample, the matr above, the elemets whch are equal to - are,, ad The traspose of a matr T The traspose of a matr s obtaed by drawg a le dow the dagoal of the matr ad movg each compoet of the matr across the dagoal to the posto where ts mage would be f there were a mrror alog the dagoal of the matr: Ths correspods to the terchage of the dces o all the compoets of : mrror le Fgure - The traspose operato moves elemets of a matr from oe sde of the dagoal to the other T j Traspose (-7) Eample: alculate the traspose of the square matr gve below: (-8) Soluto: j traspose (-9) -

3 Vector Spaces Physcs 8/6/05 D The trace of a matr smple property of a square matr s ts trace, defed as follows: Tr( ) Trace (-0) Ths s just the sum of the dagoal compoets of the matr Eample: Fd the trace of the square matr gve (-8) Soluto: Tr 6 (-) E ddto of Matrces ad Multplcato of a Matr by a Scalar These two operatos are smple ad obvous you add correspodg elemets, or multply each elemet by the scalar To add two matrces, they must have the same dmesos B j Bj ddto (-) j cj c Multplcato by a Scalar (-) j F Matr multplcato Oe of the mportat operatos carred out wth matrces s multplcato of oe matr by aother For ay two gve matrces ad B the product matr B ca be defed, provded that the umber of colums of the frst matr equals the umber of rows of the secod matr Suppose that ths s true, so that s of dmeso p, ad B s of dmeso q The product matr s the of dmeso p q The geeral rule for obtag the elemets of the product matr s as follows: B B,, p, j, q Matr Multplcato (-) j k kj k B j k kj (Este's verso) -

4 Vector Spaces Physcs 8/6/05 Ths llustrated below, for a matr ad B a matr B y 8 0 z (-5) Eample: alculate the three mssg elemets y, ad z the result matr above Soluto: k * * ( ) *( ) * 0; y ( ) * ( ) *( ) * *( ) 6 B k z * * *( ) ( ) * 5 There s a tactle way of rememberg how to do ths multplcato, provded that the two matrces to be multpled are wrtte dow et to each other as equato (-5) Place a fger of your left had o, ad a fger of your rght had o Bj Multply together the two values uder your two fgers The step across the matr from left to rght wth the fger of your left had, smultaeously steppg dow the matr B wth the fger of your rght had s you move to each ew par of umbers, multply them ad add to the prevous sum Whe you fsh, you have the value of j For stace, calculatg the eample of equato (-5) ths way gves = 6 Este Summato oveto For the case of matrces operatg o - compoet colum vectors, we ca use the Este summato coveto to wrte matr operatos: matr multplyg vector: y y (-6) matr multplyg matr: j j -

5 Vector Spaces Physcs 8/6/05 j B k B kj (-7) The rules for matr multplcato may seem complcated ad arbtrary You mght ask, "Where dd that come from?" Here s part of the aswer Look at the three smultaeous lear equatos gve (-) above They are precsely gve by multplyg a matr of umercal coeffcets to a colum vector c of varables, to wt: c 0; c 0 (-8) c 0 c 0 The square matr above s formed of the compoets of the three vectors E, E ad E, placed as ts colums: E E E (-8a) Ths matr represetato of a system of lear equatos s very useful Eercse: Use the rules of matr multplcato above to verfy that (-8) s equvalet to (-) G Propertes of matr multplcato Matr multplcato s ot commutatve Ulke multplcato of scalars, matr multplcato depeds o the order of the matrces: Matr multplcato s thus sad to be o-commutatve B B (-9) Eample: To vestgate the o-commutatvty of matr multplcato, cosder the two matrces ad B : B, (-0) alculate the two products B ad B ad compare - 5

6 Vector Spaces Physcs 8/6/05 Soluto: 5 B (-) 0 but B (-) 7 The results are completely dfferet Other propertes The followg propertes of matr multplcato are easy to verfy B B ssocatve Property (-) B B H The ut matr useful matr s the ut matr, or the detty matr, Dstrbutve Property (-) 0 0 I 0 0 j 0 0 (-5) Ths matr has the property that, for ay square matr, I I (-6) I has the same property for matrces that the umber has for scalar multplcato Ths s why t s called the ut matr I Square matrces as members of a group The rules for matr multplcato gve above apply to matrces of arbtrary dmeso However, square matrces (umber of rows equals the umber of colums) ad vectors (matrces cosstg of oe colum) have a specal terest physcs, ad we wll emphasze ths specal case from ow o The reaso s as follows: Whe a square matr multples a colum matr, the result s aother colum matr We thk of ths as the matr "operatg" o a vector to produce aother vector Sets of operators lke ths, whch trasform oe vector a space to aother, ca form groups (See the dscusso of groups hapter ) The key characterstc of a group s that multplcato of oe member by aother must be defed, such a way that the group s closed uder multplcato; ths s the case for square matrces ( addtoal requremet s the estece of a verse for each member of the group; we wll dscuss verses soo) - 6

7 Vector Spaces Physcs 8/6/05 Notce that rotatos of the coordate system form a group of operatos: a rotato of a vector produces aother vector, ad we wll see that rotatos of -compoet vectors ca be represeted by square matrces Ths s a very mportat group physcs J The determat of a square matr For square matrces a useful scalar quatty called the determat ca be calculated The defto of the determat s rather messy For a matr, t ca be defed as follows: a b a b det ad bc c d (-7) c d That s, the determat of a matr s the product of the two dagoal elemets mus the product of the other two Ths ca be eteded to a matr as follows: a d g b e h c f e a h f d b g f d c g e h (-8) a( e hf ) b( d gf ) c( dh ge) Eample: alculate the determat of the square matr of eq (-8) above Result: ( 6) ( ) ( ) (-9) - 7

8 Vector Spaces Physcs 8/6/05 m={{,,},{,,},{,,}} MatrForm[m] cm a b Iverse[m] MatrPower[m,] Det[m] Tr[m] Traspose[m] Egevalues[m] Egevectors[m] Egevalues[N[m]],Egevectors[N[m]] m=table[radom[],{},{}] defg a matr dsplay as a rectagular array multply by a scalar matr product matr verse th power of a matr determat trace traspose egevalues egevectors umercal egevalues ad egevectors matr of radom umbers Table - Some mathematcal operatos o matrces whch Mathematca ca carry out The form (-8) for the determat ca be put a more geeral form f we make a few deftos If s a square matr, ts (,j)-th mor Mj s defed as the determat of the (-)(-) matr formed by removg the -th row ad j-th colum of the orgal matr I (-8) we see that a s multpled by ts mor, -b s multpled by the mor of b, ad c s multpled by ts mor We ca the wrte, for a matr, j j j M (-0) [Notce that j occurs three tmes ths epresso, ad we have bee oblged to back away from the Este summato coveto ad wrte out the sum eplctly] Ths epresso could be geeralzed the obvous way to a matr of a arbtrary umber of dmesos, merely by summg from j = to j The determat epressed wth the Lev-vta tesor Lovg the Este summato coveto as we do, we are pqued by havg to gve t up the precedg defto of the determat For matrces we ca offer the followg more elegat defto of the determat If we wrte out the determat of a matr terms of ts compoets, we get (-) Each term s of the form jk, ad t s ot too hard to see that the terms where (jk) are a eve permutato of () have a postve sg, ad the odd permutatos have a egatve sg That s, - 8

9 Vector Spaces Physcs 8/6/05 (Yes, wth the Este summato coveto force) (-) The Meag of the determat The determat of a matr s at frst sght a rather bzarre combato of the elemets of the matr It may help to kow (more about ths later) that the determat, wrtte as a absolute value,, s fact a lttle lke the "sze" of the matr We wll see that f the determat of a matr s zero, ts operato destroys some vectors - multplyg them by the matr gves zero Ths s ot a good property for a matr, sort of lke a character fault, ad t ca be detfed by calculatg ts determat K The determat epressed as a trple scalar product You mght have oted that (-) looks a whole lot lke a scalar trple product of three vectors I fact, f we defe three vectors as follows: j k jk the we ca wrte,,, (-) ad the matr ca be thought of as beg composed of three row vectors: j k jk (-) (-5) Thus takg the determat s always equvalet to formg the trple product of the three vectors composg the rows (or the colums) of the matr L Other propertes of determats Here are some propertes of determats, wthout proof Product law The determat of the product of two square matrces s the product of ther determats: det( B) det( )det( B) (product law) (-6) Traspose Law Takg the traspose does ot chage the determat of a matr: - 9

10 Vector Spaces Physcs 8/6/05 det( T ) det( ) (traspose law) (-7) Iterchagg colums or rows Iterchagg ay two colums or rows of a matr chages the sg of the determat Equal colums or rows If ay two rows of a matr are the same, or f ay two colums are the same, the determat of the matr s equal to zero M ramer's rule for smultaeous lear equatos osder two smultaeous lear equatos for ukows ad : or (-8) (-9) or The last two equatos ca be readly solved algebracally for ad, gvg, (-0) (-) by specto, the last two equatos are ratos of determats:, (-) Ths patter ca be geeralzed, ad s kow as ramer's rule: - 0

11 Vector Spaces Physcs 8/6/05 -,,,,,, (-) s a eample of usg ramer's rule, let us cosder the three smultaeous lear equatos for,, ad whch ca be wrtte ; (-) Frst we calculate the determat : (0) ) ( ) ( (-5) The values for the are the gve by

12 Vector Spaces Physcs 8/6/05 ( ) ( 7) 9 ( 7) ( ) (9) () (0),,, (-6) So, the soluto to the three smultaeous lear equatos s (-7) Is ths correct? The check s to substtute ths vector to equato (-), carry out the multplcato of by, ad see f you get back N odto for lear depedece Fally we retur to the questo of whether or ot the three vectors E, E ad E gve secto above are learly depedet The lear depedece relato (-) ca be re-wrtte as E E E 0, (-a) a specal case of equato (-8) above, (-8) where the costats are all zero From ramer's rule, equato (-), we ca calculate, ad they wll all be zero! Ths seems to say that the three vectors are learly depedet o matter what But the ecepto s whe = 0; that case, ramer's rule gves zero over zero, ad s determate Ths s the stuato (ths s ot qute a proof!) where the (a) above are ot zero, ad the vectors are learly depedet We ca summarze ths codto as follows: -

13 Vector Spaces Physcs 8/6/05 E 0 E, E, E E E are learly depedet (-8) Eample: s a llustrato, let us take the vectors E ad E from eq (-), ad costruct a ew vector E by takg ther sum: 5 E E E (-9) E, E ad E are clearly learly depedet, sce E + E - E = 0 We form the matr by usg E, E ad E for ts colums: E E E Now the acd test for lear depedece: calculate the determat of : 5 5 (-50) () ( ) 5( ) 0 (-5) Ths cofrms that E, E ad E are learly depedet There s a fal observato to be made about the codto (-8) for lear depedece The determat of a matr ca be terpreted as the cross product of the vectors formg ts rows or colums So, f the determat s zero, t meas that E E E 0, ad ths has the followg geometrcal terpretato: If E s perpedcular to E E, t must le the plae formed by E ad E ; f ths s so, t s ot depedet of E ad E O Egevectors ad egevalues For most matrces there est specal vectors V whch are ot chaged drecto uder multplcato by : V V (-5) I ths case V s sad to be a egevector of ad s the correspodg egevalue I may physcal stuatos there are specal, terestg states of a system whch are varat uder the acto of some operator (that s, varat asde from beg multpled by a costat, the egevalue) Some very mportat operators represet the tme -

14 Vector Spaces Physcs 8/6/05 evoluto of a system For stace, quatum mechacs, the Hamltoa operator moves a state forward tme, ad ts egestates represet "statoary states," or states of defte eergy We wll soo see eamples mechacs of coupled masses where the egestates descrbe the ormal modes of moto of the system Egevectors are geeral charsmatc ad useful So, what are the egestates ad egevalues of a square matr? The egevalue equato, (-5) above, ca be wrtte I V (-5) 0 or, for the case of a matr, V 0 V 0 (-5) V 0 s dscussed the last secto, the codto for the estece of solutos for the varables V, V, V s that the determat (-55) vash Ths determat wll gve a cubc polyomal the varable, wth geeral ( ) () () (),, For each value, the equato three solutos, I V ca be solved for the compoets of the -th egevector Eample: Rotato about the z as 0 (-56) () V I hapter we foud the compoets of a vector rotated by a agle about the z as Icludg the fact that the z compoet does ot chage, ths rotato ca be represeted as a matr operato, R (-57) z where cos s 0 R z s cos 0 (-58) 0 0 Now, based o the geometrcal propertes of rotatg about the z as, what vector wll ot be chaged? vector the z drecto! So, a egevector of R s V 0 0 z (-59) -

15 Vector Spaces Physcs 8/6/05 Try t out: Rz cos s 00 0 V s cos V s a egevector of Rz, wth egevalue (-60) PROBLEMS Problem - osder the square matr B (a) alculate the traspose of B T (b) Verfy by drect calculato that B detb det Problem - osder the two matrces ad B If s the product matr, B, verfy by drect calculato that det det B det det B Problem a There s a useful property of the determat, stated Problem - above, whch s rather hard to prove algebracally: det det B det det B, where, B ad are square matrces of the same dmeso Let's cosder a umercal "proof" of ths theorem, usg Mathematca (See Table - for a collecto of useful Mathematca operatos) (a) Fll two matrces - call them ad B - wth radom umbers (b) alculate ther determats (c) alculate the product matr B, ad the determat of (d) See f the theorem above s satsfed ths case - 5

16 Vector Spaces Physcs 8/6/05 (e) other theorem cocerg square matrces s stated Problem - above Test t, for both of your radom matrces Problem - Usg tesor otato ad the Este summato coveto, prove the followg theorem about the traspose of the product of two square matrces: If s the product matr, B, the T T T B [s a startg pot, uder the Este Summato oveto, B ] j l lj Problem - alculate det I ad Tr I, where I s the detty matr Problem -5 Startg wth the defto (-0) of the determat, but geeralzed to dmesos by carryg out the sum over j from to, use the fact that terchagg two rows of a matr chages the sg of ts determat to prove the followg epresso for the determat of a matr: j j j M, where ca take o ay value from to Ths theorem says that, whle the determat s usually calculated as a sum of terms wth the frst factor comg from the top row of the matr, that the frst factor ca fact be take from ay chose row of the matr, f the correct sg s factored j Problem -6 Verfy by multplyg out soluto of equato (-) that the s of equato (-7) are a Problem -7 osder the system of lear equatos below for,, ad osder ths to be a matr equato of the form c 0 6 Frst, wrte dow the matr The, usg ramer's rule, solve for,, ad Fally, as a check, multply out ad compare t to c Problem -8 osder the system of lear equatos below for the compoets (,,, ) of a four-dmesoal vector, - 6

17 Vector Spaces Physcs 8/6/05 These equatos ca be represeted as a matr equato of the form c, wth a matr, ad ad c -elemet vectors) (a) Wrte dow the matr ad the vector c (b) Use Mathematca to solve the system of equatos; that s, fd the ukow vector (Ht: frst calculate the verse of the matr ) (c) heck the soluto by multplyg by ad comparg to c (Do ths check by had, ot wth Mathematca) Problem -9 (a) Show that the three vectors gve equato (-) of the tet are learly depedet (b) Make a ew E by chagg oly oe of ts three compoets, such that E, E, ad E are ow learly depedet Problem -0 osder the rotato matr dscussed the chapter, cos s 0 R z s cos (a) osder the vector V 0 R alculate z V, ad thus show that V s a egevector of Rz [Here the secod compoet of V s, the comple umber whose square s equal to -] (b) What s the egevalue correspodg to the egevector V of part (a)? Problem - I ths chapter we defed addto of matrces ad multplcato of a matr by a scalar Do square matrces form a vector space V? Refer to Table - for the formal propertes of a vector space (a) Demostrate whether or ot ths vector space V s closed uder addto ad multplcato by a scalar (b) Do the same for the estece of a zero ad the estece of a egatve (c) Show that property (-6) s satsfed [Ht: Ths s most ecoomcally doe usg de otato] (d) vector space wthout a metrc s ot very terestg ould the matr product as we have defed t be used as a er product? Dscuss - 7

18 Vector Spaces Physcs 8/6/05-8