SCHOOL OF ENGINEERING & BUIL ENVIRONMEN MARICES FOR ENGINEERING Dr Clum Mcdonld
Contents Introduction Definitions Wht is mtri? Rows nd columns of mtri Order of mtri Element of mtri Equlity of mtrices Opertions with mtrices Mtri ddition Mtri sutrction Sclr multipliction of mtri Mtri multipliction Mtri multipliction nd the sclr (dot) product Specil mtrices rnspose mtri Properties of trnspose mtrices he identity mtri he zero mtri Digonl mtrices Upper nd lower tringulr mtrices Symmetric mtrices he determinnt nd inverse of mtri Properties of inverse mtrices he determinnt nd inverse of μ mtri he determinnt μ mtri Determinnts nd the Rule of Srrus he inverse of μ mtri y the cofctor method 7 Eigenvlues nd eigenvectors of mtri 7 Clcultion of eigenvlues 7 Rel distinct eigenvlues 7 Repeted eigenvlues 7 Zero eigenvlues 7 Comple eigenvlues of rel mtrices 7 Clcultion of eigenvectors 7 Some properties of eigenvlues nd eigenvectors Eercises & Answers
Mtrices Introduction his unit introduces the theory nd ppliction of mthemticl structures known s mtrices With the dvent of computers mtrices hve ecome widely used in the mthemticl modelling of prcticl rel-world prolems in computing, engineering nd usiness where, for emple, there is need to nlyse lrge dt sets Emples of the pplictions of mtrices occur: in ll res of science to solve (lrge) systems of equtions in computer grphics to project three dimensionl imges onto two dimensionl screens nd pply trnsformtions to rotte nd move these screen ojects in cryptogrphy to encode messges, computer files, PIN numers, etc in usiness to formulte nd solve liner progrmming prolems to optimise resources suject to set of constrints Definitions Before we undertke clcultions involving mtrices it is firstly necessry to present some definitions nd terminology Wht is mtri? A mtri is n ordered rectngulr rry of numers nd/or vriles rrnged in rows nd columns nd enclosed in rckets For our purposes these elements will tke the form of rel numers In generl, mtrices re denoted y upper cse letters Emple he following re ll mtrices: (i) A (ii) B 9 8 8 7 (iii) 8 K (iv) M 8
Rows nd columns of mtri A useful interprettion of the structure of mtri is to consider the rows nd columns of the mtri hese re simple nd ovious concepts; ut we need to know tht the rows re numered strting from the top (ie Row ) nd the columns re numered strting from the left hnd side of the mtri (ie Column ) For emple we hve, Row Column Order of mtri he size, lso clled the order or dimension, of mtri is identified y numer pir in the form m n, where m is the numer of rows in the mtri nd n the numer of columns A m m n n mn We sy tht the mtri A is m y n mtri A mtri with the sme numer of rows s columns, ie m n, is clled squre mtri Emple he mtrices in Emple hve the following sizes: (i) A is mtri, ie rows nd columns (ii) B is mtri, ie rows nd columns (iii) K is mtri, ie rows nd columns (iv) M is mtri, ie rows nd column
Element of mtri Ech element, or entry, in mtri is denoted y lower cse letter, with pproprite suscripts, indicting its row nd column position Hence, element i j is locted in the i-th row nd j-th column of the mtri A s shown elow: A m m n n mn Emple For the mtri 8 A, 7 9 (i) element, s it is locted t Row, Column (ii) element, s it is locted t Row, Column We cn esily drw prllels etween mtrices nd computer rrys s used in progrmming lnguges For emple, in C if A is n rry we would use the synt A[][ ] to inde the element in the second row nd third column of A, in Mple we would use the nottion A[, ] nd in MALAB we would write A(, ) Equlity of mtrices wo mtrices A nd B re equl if nd only if: they re of the sme size, ie oth re m n mtrices their corresponding elements re the sme, ie i j i j for i,, m nd j,, n
Emple Determine the vlues of w,, y nd z tht gurntee the mtrices A nd B re equl, y A, B z 7 w Solution We require w,, y, z 7 z Opertions with mtrices We now look t some sic rithmetic opertions with mtrices Mtri ddition wo mtrices cn e dded if nd only if they re of the sme size o dd two mtrices we dd corresponding elements he result of the ddition is mtri of the sme size Mtri ddition is commuttive, ie A B B A, see prts (ii) nd (iii) in the following emple Emple In ech of the following crry out the specified ddition (i) 9 8 8 7 8 9 8 7 8 9 9, 7 (ii) (iii)
Mtri sutrction wo mtrices A nd B cn e sutrcted if nd only if they re of the sme size o form B A sutrct ech element of B from the corresponding element of A he result of the sutrction is mtri of the sme size As for sutrction of rel numers, mtri sutrction is not commuttive, ie A B B A, see the net emple Emple In ech of the following crry out the specified sutrction (i) 7 8 (ii) 7 8 Sclr multipliction of mtri Any mtri cn e multiplied y numer (sclr) nd this procedure is referred to s sclr multipliction Sclr multipliction is performed y multiplying ech element in the mtri y the numer Sclr multipliction must not e confused with mtri multipliction which will e defined lter Emple 7 Simplify ech of the following y performing the sclr multipliction (i) 8, (ii)
7 Emple 8 Let, B A nd 9 C If possile simplify ech of the following: (i) B A, (ii) A B (iii) A C Solution (i) B A 8 7 (ii) A B 8 (iii) We re unle to clculte A C s the mtrices hve different sizes Here C is mtri while A is mtri
Mtri multipliction Mtri multipliction cn only e crried out etween mtrices which re conformle for mtri multipliction wo mtrices A nd B, with sizes m n nd p q respectively, re conformle for multipliction if nd only if n p ; ie the numer of columns of A is the sme s the numer of rows of B he result of multiplying m n mtri, A, nd n p q mtri, B, where n p, is m q mtri nd we write the product s AB Note tht mtri multipliction my e defined for A B ut not necessrily for B A Hence, mtri multipliction is not in generl commuttive Note: If the inner dimensions n nd p re equl we cn multiply the mtrices nd the resulting product mtri hs size given y the outer dimensions, ie m q A B Result AB ( m n) ( p q) ( m q) n p If required further resources on multipliction of mtrices topic cn e found t: https://wwwkhncdemyorg/mth/preclculus/preclc-mtrices/multiplying-mtrices-ymtrices/v/multiplying--mtri-y--mtri http://wwwmthcentrecuk/topics/mtrices/mtri-rithmetic/ 8
Emple 9 Determine which of the following mtrices re conformle for mtri multipliction A, B, C Solution he mtri A is he mtri B is he mtri C is (i) First consider the mtri product, A B A B ( ) ( ) Not equl he inner dimensions re not equl nd so we cnnot perform the mtri multipliction he numer of columns in A () does not equl the numer of rows in B ( ) (ii) Now consider the mtri product, A C A C ( ) ( ) Equl he inner dimensions re equl nd so we cn perform the mtri multipliction nd the product mtri will hve size given y the outer dimensions, ie Eercise Confirm the following: BA, AC, CB nd BB re vlid multiplictions; wheres we cnnot clculte the products AB, BC, AA or CC 9
Mtri multipliction nd the sclr (dot) product o multiply two mtrices, conformle for mtri multipliction, involves n etension of the dot (sclr) product procedure from vector lger o form the result of multiplying A (on the left) y B (on the right) ( ie to form the product AB ) we view A s mtri composed of rows nd B s mtri mde up of columns he entries in the product mtri re determined y forming dot products o determine the element in Row i / Column j, ie position ( i, j ), of AB we form the dot product of Row i of mtri A with Column j of mtri B Emple Let A nd B he product mtri AB cn e clculted s A hs size nd B hs size, ie the numer of columns of A is the sme s the numer of rows of B he product mtri, AB will hve size For emple, to otin the element in Row /Column, ie position (, ) of AB we tke the dot product of Row of A with Column of B, ie
o otin the element in Row /Column, ie position (, ) of AB, we tke the dot product of Row of A with Column of B In vector form we hve, ) ( ) ( ) ( his process cn e continued to generte the components of the product mtri, AB ) ( ) ( ) ( ) ( ) ( ) ( ) ( Emple Let A nd B If possile clculte AB nd BA Solution In this cse A nd B re oth (squre) mtrices nd so we cn clculte AB nd BA he result of oth multiplictions will e mtri We hve ) (
8 B A nd 9 A B Note: his is n emple of very importnt result in mtri rithmetic In generl, for squre mtrices A nd B, we hve tht BA AB Emple If possile evlute the following mtri products: (i), (ii), (iii) y (iv) ( ) (v) ( ) Solution (i) he product cn e formed s the first mtri hs size nd the second mtri hs size, ie the numer of columns in the first mtri () is the sme s the numer of rows () in the second mtri he product mtri will hve size Multipliction gives, ) ( ) ( ) ( ) ( ) (
(ii) ( Eercise: Check this nswer ) (iii) y y ( Eercise: Check this nswer ) (iv) ( ) he product cn e formed s the first mtri hs size nd the second mtri hs size, ie the numer of columns in the first mtri () is the sme s the numer of rows () in the second mtri he product mtri will hve size Multipliction gives, ) ( ) ( ) ( 8 (v) ( ) he product cnnot e formed s the first mtri hs size nd the second mtri hs size, ie the numer of columns in the first mtri () is not the sme s the numer of rows () in the second mtri
Emple Let 8 A nd 9 B Clculte the mtri products AB nd BA Solution (i) AB, BA Eercise: Verify these clcultions Unlike Emple we find tht in this cse mtri multipliction is commuttive nd BA AB his prticulr squre mtri, with s on the min digonl (top left to ottom right) nd s everywhere else, is known s the identity mtri see Section for further discussions Emple Determine the vlues of nd y tht stisfy the following mtri eqution y Solution Epnding the left-hnd-side (LHS) gives, y y y A We therefore hve three equtions he first two of these re oth equtions in one vrile which cn esily e solved for nd y he third eqution cn e used to check our nswers, y y y
Specil mtrices here re severl specil mtrices tht we should e wre of s they will e needed in future clcultions rnspose mtri he mtri otined from mtri A y interchnging the rows nd the columns of A is clled the trnspose of A nd is denoted A We refer to this mtri s, A trnspose Emple Let A 7 Write down the mtri A Solution he mtri A is otined y interchnging rows nd columns of the mtri A Hence, A 7 Note tht Row of A is Column of Alterntively, Column of A is Row of A nd Row of A is Column of A, etc A Properties of trnspose mtrices ( A ) A the trnspose of trnspose mtri equls the originl mtri ( AB ) B A the trnspose of mtri product equls the product of the trnspose mtrices, with the order of multipliction reversed ( A B) A B the trnspose of mtri sum equls the sum of the trnspose mtrices ( k A ) k A the trnspose of mtri multiplied y sclr equls the sclr multiplied y the trnspose of the mtri, k is sclr
he identity mtri We know tht when ny numer is multiplied y the numer the vlue of the originl numer is unchnged, eg 9 9,, etc In this contet we cll the identity element for multipliction We now define n identity element for mtri multipliction so tht when mtri is multiplied y the identity it remins unchnged his identity element is clled n identity mtri nd is only defined for squre mtrices Although there is only single multiplictive identity, ie, when working with numers there re mny different identity mtrices depending on the size of the mtri in question, eg,, etc he identity mtri hs s on the min digonl ( the digonl strting t top left nd going to ottom right ) nd zeros everywhere else see Emple he mtri is usully represented y the letter I Note tht some tetooks include suscript n, nd write I n, to indicte the size of the identity mtri Emple (i) he identity mtri is, I (ii) We met the identity mtri, I in Emple he zero mtri he zero mtri is mtri for which every element is zero Strictly speking there re mny zero mtrices, one for ech possile size of mtri Here re the nd zero mtrices he zero mtri is the identity mtri for mtri ddition
7 Emple 7 If 9 A then 9 9 he mtri A is unchnged y ddition of the zero mtri Digonl mtrices A squre mtri is clled digonl mtri if ll the entries tht do not lie on the min digonl re zero Note tht it is llowed for some entries on the min digonl to equl zero Emple 8 he following mtrices re ll emples of digonl mtrices: (i), (ii), (iii) he identity mtri is specil cse of digonl mtri where ll the digonl entries re equl to Upper nd lower tringulr mtrices An upper tringulr mtri is squre mtri in which ll the entries elow the min digonl re zeros Note tht some entries on the min digonl nd/or ove the min digonl cn equl zero Emple 9 he mtrices elow re ll emples of upper tringulr mtrices (i), (ii) 9, (iii)
8 A lower tringulr mtri is squre mtri in which ll the entries ove the min digonl re zeros Note tht some entries on the min digonl nd/or elow the min digonl cn equl zero Emple he mtrices elow re ll emples of lower tringulr mtrices (i), (ii) 9, (iii) Symmetric mtrices A squre mtri is clled symmetric mtri if it is equl to its own trnspose, ie A A Emple he following mtrices re ll symmetric (i) (ii) (iii) 9 You re now redy to ttempt the multiple choice eercise t the link elow http://wwwmthsusydeduu/u/ug/jm/mah/quizzes/quiz7html In the net section we look t how to clculte the determinnt nd inverse ( if it eists ) of mtri
he determinnt nd inverse of mtri Consider the following rithmetic evlutions for numers, One wy of interpreting the ove is tht ny (non-zero) numer hs ssocited with it multiplictive inverse Furthermore, ny numer multiplied y its inverse equls, the multiplictive identity for sclrs We cn mke n nlogous sttement for (some) squre mtrices tht will prove useful lter For generl mtri, A provided c d det ( A ) d c, there eists nother mtri, clled the inverse of A, denoted A, where A d det ( A) c he quntity det ( A ) is clled the determinnt of A nd is often written, A he determinnt of mtri cn therefore e used to determine the eistence (or otherwise) of mtri inverse y checking tht it is non-zero In mnner similr to tht oserved erlier for sclrs, A A A A I 9
where I is the identity mtri with the sme size s the squre mtri A Also, A I I A A Note tht A must not e interpreted s, A A mtri with n inverse is clled invertile or non-singulr A mtri with no inverse is sid to e non-invertile or singulr We now look t how to determine inverse mtrices, where they eist, for the cse Emple Clculte the determinnt of ech of the following mtrices Hence, identify which mtrices re invertile nd for ech invertile mtri clculte its inverse (i) 7 A (ii) B (iii) C (iv) M Solution (i) det ( A ) ( ) ( ) No inverse (ii) det ( B ) 7 7 Mtri hs n inverse B Eercise: Check tht B B B B I 7 7 (iii) det ( C ) No inverse
(iv) det ( M ) ( ) ( ) Mtri hs n inverse, M Eercise: Check tht M M M M I Properties of inverse mtrices If A nd B re invertile mtrices: ( A ) A the inverse of n inverse mtri equls the originl mtri ( AB ) B A the inverse of mtri product equls the product of the inverse mtrices, with the order of multipliction reversed ( k A ) A the inverse of mtri multiplied y sclr equls the k inverse of the sclr multiplied y the inverse of the mtri, k is sclr ( A ) ( A ) the inverse of trnspose mtri equls the trnspose of the inverse mtri Aside: Geometriclly, the solute vlue of the determinnt of mtri A is c d the re of prllelogrm whose edges re the vectors (, ) nd ( c, d ) y O
In the net section we look t how to clculte the determinnt nd inverse of mtri he determinnt nd inverse of mtri here re numer of techniques ville for clculting the inverse of mtri In our work we shll focus on the cofctor method Before looking t finding the inverse of mtri, when it eists, we need to know how to clculte the determinnt of mtri he determinnt mtri Define mtri A o clculte the determinnt of A we cn epnd y ny row, or y ny column of A, nd we will otin the sme vlue for det ( A ) Epnding y Row : det ( A ) Note the minus sign on the centrl term, which is eplined elow We cn now evlute the determinnt, det ( A ), s we lredy know how to clculte determinnts Epnding y Row : det ( A) Epnding y Row : det ( A ) In the three cses descried ove note the rry of signs tht prefi the coefficients i j, ie In similr mnner we cn epnd on ny of the columns of A using the sign rry ove
Epnding y Column : det ( A ) Epnding y Column : det ( A) Epnding y Column : det ( A ) Note: he vlue of the determinnt will lwys e the sme regrdless of which row or column we perform the epnsion on Emple Clculte the determinnt of the mtri, A 7 8 9 Solution Epnding on the first row gives: det ( A ) 8 9 7 9 7 8 ( 9 8 ) ( 9 7 ) ( 8 7 ) 9
Emple Clculte the determinnt of the mtri, M Solution: Epnding on the first row gives: det ( M ) ( ) ( ) ( ) As we sw erlier the vlue of the determinnt of squre mtri cn e used to determine if mtri is invertile If the determinnt is non-zero the mtri is invertile, otherwise the mtri is NO invertile Hence, the mtri in Emple is not invertile while the mtri in Emple is invertile provided tht we hve Note: When clculting the determinnt of mtri we usully epnd long the row or column contining most zeros in order to minimize the rithmetic So, in Emple we could choose to epnd long row or epnd down column Determinnts nd the Rule of Srrus An lterntive pproch for clculting the determinnt of mtri A c c c ws developed y the French mthemticin Pierre Srrus (798-8) he Rule of Srrus involves the following steps:
rewrite the first two columns of the mtri to the right of it c c c c c using the left to right digonls tke the products,, c c c : c c c c c using the right to left digonls tke the products,, c c c, c c c c c comine the ove nd clculte the determinnt s follows: ) det( c c c c c c A he Rule of Srrus is essentilly the sme s the method we descried previously Aside: Geometriclly, the solute vlue of the determinnt of mtri is the volume of prllelepiped whose edges re the vectors u ),, (, v ),, ( nd w ),, ( c c c u v w
he inverse of mtri y the cofctor method We now etend the ide of n inverse mtri to the cse In generl, the inverse of mtri A is given y the formul A det ( A) dj ( A) det ( A ) where the mtri dj (A) is known s the djoint mtri of A In order to clculte the inverse of mtri A, if it eists, we must therefore otin the determinnt of A nd the djoint of A he derivtion of the djoint mtri requires us to clculte the mtri of minors of A nd mtri of cofctors of A he following prgrphs illustrte the methodology y wy of n emple reducing the procedure to five distinct steps Emple Determine the inverse of the mtri, A if it eists Solution SEP : Clculte the determinnt of A Epnding long the third row the determinnt of A is det( A ) ( ) Since det( A ), A eists
SEP : Clculte the mtri of minors he minor of entry i j, denoted y remove the i th row th remove the j column M i j, is otined s follows: the minor M i j is the determinnt of the remining mtri he minor of entry is: M ( ) he minor of entry is: M he minor of entry is: M he minor of entry is: M he minor of entry is: M ( ) he minor of entry is: M he minor of entry is: M ( ) 7
he minor of entry is: M 9 ( ) he minor of entry is: M 8 Hence, the mtri of minors is: SEP : Clculte the cofctor mtri i j he cofctor of entry i j, denoted y C i j, is defined s C i j ( ) M i j o otin the cofctor mtri cof (A) we simply chnge signs of the elements of the mtri of minors clculted in Step using the sign mtri: Hence, we otin the cofctor mtri cof (A) 8
9 SEP : Clculte the djoint mtri he djoint of A is defined to e the trnspose of the cofctor mtri, ie (A) dj SEP : Clculte the inverse mtri he inverse of the mtri A is clculted s, ) ( ) det ( A dj A A so tht A which simplifies to give 8 8 8 A It is strightforwrd to check tht I A A A A We hve, I A A 8 8 8 s required
Emple Determine the inverse of the mtri A 9 if it eists 7 Solution SEP : Clculte the determinnt of A Epnding long the first row the determinnt of A is 9 9 det ( A ) ( ) ( ) 7 9 Since det( A ), A eists SEP : Clculte the mtri of minors he minor of entry is: M 9 8 7 7 he minor of entry is: 9 M 9 7 7 he minor of entry is: 9 M 9 7 he minor of entry is: M 9 7 7 7 he minor of entry is: M 9 7 7
he minor of entry is: M 9 7 he minor of entry is: M 9 ( ) 8 7 he minor of entry is: M 9 8 9 7 he minor of entry is: M 9 ( 9) 9 7 Hence, the mtri of minors is: 8 8 SEP : Clculte the cofctor mtri i j he cofctor of entry i j, denoted y C i j, is defined s C i j ( ) M i j o otin the cofctor mtri cof (A) we simply chnge signs of the elements of the mtri of minors in Step using the sign mtri: Hence, we otin the cofctor mtri cof (A) 8 8
SEP : Clculte the djoint mtri he djoint of A is determined s the trnspose of the cofctor mtri, ie 8 dj (A) 8 SEP : Clculte the inverse mtri he inverse of the mtri A is clculted s A dj( A) det ( A) so tht 8 A 8 Check the inverse is correct s follows: 8 A A 9 8 I 7 s required
Emple 7 Clculte the determinnt of the mtri, A nd find 7 8 9 A if it eists Solution As n lterntive we use the Rule of Srrus to clculte the determinnt Writing 7 8 9 7 8 det ( A ) 9 7 8 9 8 7 8 9 7 8 As det ( A ) the mtri A is not invertile Aside: Geometriclly the result tht det ( A ) mens tht the vectors (,, ), (,, ), nd (7, 8, 9) re coplnr, ie they lie in the sme plne, so the volume of prllelepiped sed on them is equl to
7 Eigenvlues nd eigenvectors of mtri Eigenvlues nd eigenvectors hve mny importnt pplictions in science nd engineering including solving systems of differentil equtions, stility nlysis, virtion nlysis nd modelling popultion dynmics Let A e n n mtri An eigenvlue of A is sclr λ (rel or comple) such tht A λ (I) for some non-zero vector In this cse, we cll the vector n eigenvector of A corresponding to λ Geometriclly Eq (I) mens tht the vectors A nd re prllel he vlue of λ determines wht hppens to when it is multiplied y A, ie whether it is shrunk or stretched or if its direction is unchnged or reversed Emple 8 If A nd, then A Here we hve tht A nd so we sy tht the eigenvlue λ is n eigenvector of A corresponding to he geometric effect in this emple is tht the vector hs een stretched y fctor of ut its direction remins unchnged s λ > Note tht ny sclr multiple of the vector eigenvlue λ is n eigenvector corresponding to the
7 Clcultion of eigenvlues If A is mtri it is reltively strightforwrd to clculte its eigenvlues nd eigenvectors y hnd So, how do we clculte them? We know tht I, where I is the identity mtri, so we cn rewrite Eq (I) s A λ I A λ I ( A λ I ) If the mtri ( A λ I) is invertile, ie det( A λ I), then the only solution to the ove eqution is the zero vector, ie We re not interested in this cse s n eigenvector must e non-zero he eqution ( A λ I) cn only hold for non-zero vector if the mtri ( A λ I) is singulr (does not hve n inverse) Hence, the eigenvlues of A re the numers λ for which the mtri ( A λ I) does not hve n inverse In other words the numers λ stisfy the eqution det ( A λ I ) (II) nd they cn e rel or comple
7 Rel distinct eigenvlues We firstly look t the cse where n n n mtri hs n distinct eigenvlues Emple 9 Find the eigenvlues of the following mtrices : (i) A (ii) B (iii) C 7 7 Solutions λ (i) A λ I λ 7 7 λ λ Hence, det ( A λ I ) ( λ)( λ) ( )(7) λ λ 7 λ We cll λ λ the chrcteristic polynomil of the mtri A he eigenvlues of A re the roots of the chrcteristic eqution det ( A λ I ), ie λ λ ( λ )( λ ) λ nd λ Hence, λ nd λ re the eigenvlues of the mtri A Note: We cn esily check our nswer s follows: Let tr(a) denote the trce of mtri A, ie the sum of the elements on the min digonl he sum of the eigenvlues of A must equl the trce of the mtri Here we hve tht tr ( A ) ( ) nd the sum of the eigenvlues is, s required
(ii) λ B λ I λ 7 7 λ λ Hence, det ( B λ I ) ( λ)(7 λ) ( )( ) λ λ 7 7 λ Now solve det ( B λ I ) to find the eigenvlues of B, ie λ λ 7 ( λ )( λ ) λ nd λ Hence, λ nd λ re the eigenvlues of the mtri B (iii) λ C λ I λ λ λ Hence, det ( C λ I ) ( λ)( λ) ()() λ λ he eigenvlues of C stisfy det ( C λ I ), ie λ λ ± Hence, λ nd λ re the eigenvlues of the mtri C he following emple demonstrtes short-cut pproch tht cn e dopted when clculting the eigenvlues of specific types of mtrices Emple Find the eigenvlues of the following mtrices: (i) 7 A (ii) B (iii) C 8 7
In this emple we note tht: mtri A is digonl mtri (see Section ) nd hs the property tht ll of its entries not on the min digonl re mtri B is n upper-tringulr mtri (see Section ) nd hs the property tht ll of its entries elow the min digonl re mtri C is lower-tringulr mtri (see Section ) nd hs the property tht ll of its entries ove the min digonl re Note tht, in ech cse, some of the entries on the min digonl cn e zero Solution In ll three cses digonl, upper-tringulr nd lower tringulr - the eigenvlues re simply the entries on the min digonl nd so we cn just red them off without the need for ny clcultions Hence, he eigenvlues of mtri A re: λ, λ 8 he eigenvlues of mtri B re: λ, λ he eigenvlues of mtri C re: λ, λ We verify our nswers using the method descried erlier λ (i) Solving det( A λ I) gives 8 λ ( λ )(8 λ) λ, λ 8 λ 7 (ii) Solving det( B λ I) gives λ ( λ )( λ) λ, λ λ (ii) Solving det( C λ I) gives λ ( λ)( λ) λ, λ 8
7 Repeted eigenvlues In the emples presented up to now the eigenvlues hve een distinct ut it is possile for mtri to hve repeted eigenvlues Emple Find the eigenvlues of the mtri, 9 A 9 Solution o find the eigenvlues we solve λ 9 det ( A λ I ) 9 λ ( λ )(9 λ) 9 λ λ ( λ ) ( λ ) λ (repeted) he eigenvlue λ is sid to hve lgeric multiplicity, ie the numer of times it is root of the chrcteristic eqution 9
7 Zero eigenvlues We hve previously noted tht n eigenvector cnnot e the zero vector,, ut it is possile to hve n eigenvlue λ Emple Find the eigenvlues nd eigenvectors of the mtri, A Solution o find the eigenvlues we need to solve λ det ( A λ I ) λ ( λ )( λ) λ 7λ λ ( λ 7) λ, λ 7 his emple shows tht it is possile for to e n eigenvlue of mtri Note tht if is n eigenvlue of mtri then the mtri is not invertile Hence, the mtri A in this emple cnnot e inverted
7 Comple eigenvlues of rel mtrices It is possile for rel-vlued mtri to hve comple eigenvlues (nd eigenvectors) s illustrted y the following emple Emple Find the eigenvlues of the mtrices: (i) A (ii) B Solution λ (i) det ( A λ I ) λ λ λ j, λ j λ (ii) det ( B λ I ) λ λ λ Solve using the qudrtic formul, or y completing the squre, to otin, λ j, λ j Note: For mtri with rel entries its comple eigenvlues lwys occur in comple conjugte pirs
7 Clcultion of eigenvectors Once we hve clculted the eigenvlues we cn find the eigenvectors y solving the mtri eqution A λ (III) or equivlently, s we sw ove, ( A λ I ) for ech eigenvlue in turn Emple Find the eigenvlues nd eigenvectors of the mtri, 7 A Solution First we find the eigenvlues y solving: λ 7 det ( A λ I ) λ ( λ )( λ) 7 λ λ ( λ )( λ ) λ, λ We now clculte the eigenvectors corresponding to the eigenvlues y solving Eq (III)
Cse : o find n eigenvector corresponding to eigenvlue λ we solve, A λ 7 7 7 7 () () hese re simultneous equtions nd we note here tht one eqution will lwys e multiple of the other - if not then you hve mde mistke! Here Eq () is 7 times Eq () Both equtions give If we let α, sy, for some non-zero rel numer α, then α nd we find the first eigenvector to e of the form α α α Note tht there re infinitely mny non-zero eigenvectors depending on the vlue chosen for α Setting α gives n eigenvector corresponding to the eigenvlue λ s We cn check our nswer y showing tht A A 7 nd λ Hence, A λ s required
Cse : o find n eigenvector corresponding to eigenvlue λ we solve, A λ 7 7 7 7 Both these equtions give 7 Let α, sy, for some non-zero rel numer α, then 7α nd so 7α α 7 α Setting α gives 7 It is strightforwrd to check tht A In summry, we therefore hve the eigenvlue/eigenvector pirs, λ, ; λ, 7
Emple Find the eigenvlues nd eigenvectors of the mtri, B 7 Solution In Emple 9 prt (ii) we found the eigenvlues of B to e λ nd λ We now clculte the eigenvectors corresponding to these eigenvlues y solving the eigenvector eqution, A λ Cse : o find n eigenvector corresponding to eigenvlue λ we solve, A λ 7 7 8 Both these equtions give Note tht for μ system we do not ctully need to introduce the prmeter s we did in the previous emple We cn simply choose convenient numericl vlue for either of the components or of the eigenvector So here we cn let, sy, giving hus n eigenvector corresponding to the eigenvlue λ is
Cse : o find n eigenvector corresponding to eigenvlue λ we solve A λ 7 7 8 Both these equtions give Let, sy, giving hen n eigenvector corresponding to the eigenvlue λ is In summry, we therefore hve the eigenvlue/eigenvector pirs, λ, ; λ, Emple Find the eigenvlues nd eigenvectors of the mtri, A
Solution o find the eigenvlues we need to solve λ det ( A λ I) λ ( λ )( λ) λ 7λ λ ( λ 7) λ, λ 7 We now find the eigenvectors: Cse : o find n eigenvector corresponding to eigenvlue λ we solve, A Both these equtions give Let, sy, giving Hence, n eigenvector corresponding to the eigenvlue λ is Cse : o find n eigenvector corresponding to eigenvlue λ 7, solve A 7 7 7
7 7 Both these equtions give Let, sy, giving Hence, n eigenvector corresponding to the eigenvlue λ 7 is o summrise we hve: λ, ; λ 7, Emple 7 Find the eigenvectors of the mtri, A Solution In Emple we found tht A hd comple eigenvlues, λ j nd λ j If mtri A with rel entries hs comple eigenvlue λ then we know tht its comple conjugte λ is lso n eigenvlue of A Furthermore, it cn e shown tht if is n eigenvector corresponding to λ then its comple conjugte, formed y tking the comple conjugtes of the entries of, is n eigenvector corresponding to λ We now use this result to find the eigenvectors of the mtri A 8
Cse : o find n eigenvector corresponding to eigenvlue λ j we solve A λ j j j If, for emple, we multiply the first eqution y j oth equtions give j Let, sy, then j An eigenvector corresponding to the eigenvlue λ j will then e j Cse : o find n eigenvector corresponding to eigenvlue conjugtes of the entries of giving, j λ j simply tke the comple o summrise we hve: λ j, j ; λ j, j 9
7 Some properties of eigenvlues nd eigenvectors Let A e rel n n mtri A will hve ectly n eigenvlues which my e repeted nd will e rel or occur in comple conjugte pirs An eigenvlue cn e zero ut n eigenvector cnnot e the zero vector, he sum of the eigenvlues of A equls the sum of the min digonl entries of A, ie the trce of A he product of the eigenvlues of A equls the determinnt of A If is n eigenvlue of A then A is not invertile If λ is n eigenvlue of n invertile mtri A, with s corresponding eigenvector, then is n eigenvlue of A, gin with s corresponding eigenvector λ k If λ is n eigenvlue of A, with s corresponding eigenvector, then λ is n eigenvlue of k A, gin with s corresponding eigenvector, for ny positive integer k he mtri A nd its trnspose, A, hve the sme eigenvlues ut there is no simple reltionship etween their eigenvectors Summry o clculte the eigenvlues nd eigenvectors of n n mtri A we proceed s follows: Clculte the determinnt of the mtri A λ I, it will e polynomil in λ of degree n Find the roots of the polynomil y solving det ( A λ I ) he n roots of the polynomil re the eigenvlues of the mtri A For ech eigenvlue, λ, solve ( A λ I ) to find n eigenvector
utoril Eercises Q Simplify the following: (i) (ii) (iii) (iv) Q Simplify the following mtri products: (i) (ii) (iii) ( ) (iv) ( ) 7 Q Simplify the following nd comment on your nswers: (i) (ii)
Q (i) Which of the following mtrices cn e squred?, M A (ii) In generl, which mtrices cn e squred? Q (i) Given tht A, find the inverse mtri A nd clculte the mtri products AA nd A A Comment on your results (ii) Given tht A clculte the mtri product A A Q Determine when the following mtri is invertile nd clculte its inverse: k A Q7 For the mtri A show tht A A ) ( Q8 Let A nd 8 B Evlute B A ) ( nd B A Q9 Let A nd B Evlute ) ( B A nd A B
Q Evlute the determinnt of ech of the following mtrices: (i) (ii) (iii) Q Which of the mtrices in Question re invertile? Justify your nswer Q For ech invertile mtri in Question determine the inverse Q Given tht A, find the inverse mtri A nd clculte the mtri products AA nd A A Q Given tht D, find the inverse mtri D Q Consider the mtrices D nd 8 P Determine the inverse mtri P nd clculte the mtri product P P D A
Q Find the eigenvlues of ech of the following mtrices: (i) (ii) (iii) Q7 Find the eigenvlues nd eigenvectors of ech of the following mtrices: (i) (ii) (iii) (iv) (v) (vi) (vii) 7 (viii) 7 (i) ()
Answers S (i) ; (ii), (iii) ; (iv) S (i) ; (ii) 8 9 ; (iii) 9 (iv) S (i) (ii) 8 he conclusion from this emple is tht mtri multipliction is not commuttive, so tht the order in which mtrices re multiplied is importnt S (i) Only the first mtri, A, cn e squred since it is conformle for multipliction with itself (ii) In generl, to squre mtri of size p m requires multiplying n p m mtri y n p m mtri hese re only conformle for mtri multipliction if p m S (i) A so det(a) which is non-zero nd so the mtri is invertile hen A We hve tht I A A AA Both mtri products give the identity mtri
(ii) A so A nd 9 AA M S k A so k A ) det( he mtri A is invertile provided ) det( A, ie provided k In this cse the inverse is, k k A S7 A hen A A ) ( s required S8 8 8 B A 8 B A so B A B A 8 ) ( In generl for mtrices A nd B we hve tht B A B A ) ( S9 7 8 B A nd so 8 7 ) ( B A Also, A nd B so ) ( 8 7 B A A B In generl for n n squre mtrices A nd B we hve tht, ) ( A B B A
7 S We shll epnd on Row throughout ut note tht we could epnd on ny row or column nd the determinnt of the mtri would e the sme in ech cse (i) D (ii) D (iii) D S hey re ll invertile ecept (iii), since only (iii) hs zero determinnt S (i) (ii) S A ; AA A A S D
Note: If D n n is n n digonl mtri then its inverse is given y D n n provided tht none of the digonl elements re zero S P nd A P D P Note: We shll see lter tht the mtri D hs the eigenvlues of A on its min digonl while the columns of P contin the corresponding eigenvectors written in the sme order s the eigenvlues pper in D he process of clculting the mtrices P nd D is clled digonlistion S (i) λ, λ (ii) λ, λ (iii) λ j, λ j 8
S7 (i) λ, λ, (ii) λ, ; λ, (iii) λ, ; λ, (iv) λ, ; λ, (v) λ, ; λ 8, (vi) λ, ; λ, (vii) λ, ; λ, 7 (viii) λ, ; λ 7, (i) λ, ; λ, () λ, ; λ, 9