tries Definition of tri mtri is regulr rry of numers enlosed inside rkets SCHOOL OF ENGINEERING & UIL ENVIRONEN Emple he following re ll mtries: ), ) 9, themtis ), d) tries Definition of tri Size of tri Rows nd Columns of tri tri ddition Slr ultiplition of tri tri ultiplition rnspose tri he Zero tri 9 he Identity tri he Determinnt nd Inverse of tri he Determinnt of tri he Inverse of tri n lterntive ethod for Clulting the Determinnt of tri Eigenvlues nd Eigenvetors of tri lgerilly we denote mtri with pitl letter, eg numer ourring in mtri is lled n element Eh element in mtri n e identified y its row nd olumn numers For emple the element in position, ) is the element in row nd olumn of the mtri Emple ) For the mtri the element in position, ) is 9 ) he element t lotion, ) is Size of tri We n identify the size dimension) of mtri using numer pir in the form r, where r is the numer of rows in the mtri nd the numer of olumns mtri with the sme numer of rows s olumns is lled squre mtri for ovious resons) Dr Clum donld Emple he mtries in Emple hve the following sizes, ), ), ) nd d)
Rows nd Columns of tri nother useful pproh to the struture of mtri is to look t the rows nd olumns of the mtri hese re simple nd ovious onepts; ut we need to know tht the rows re numered strting from the top ie row ) nd the olumns re numered strting from the left hnd side of the mtri ie olumn ) Slr ultiplition of tri mtri n e multiplied y numer nd this proedure is referred to s slr multiplition We perform slr multiplition y multiplying eh element in the mtri y the numer Slr multiplition should not e onfused with mtri multiplition whih will e defined lter nd Row rd Column We n define ddition nd multiplition etween pirs of mtries s long s they re of pproprite dimensions Emple ), ) tri ddition ddition is strightforwrd wo mtries n e dded if nd only if they re of the sme size; the result is hieved y dding orresponding elements of the mtries he result of the ddition is mtri of the sme size Emple ) 9 ) ) 9 w y z w w z y w z 9 9, It is importnt to reognise tht we n not dd mtries with different dimensions For emple, we n not dd mtri to mtri Emple ) Simplify, tri ultiplition tri multiplition n only e rried out etween mtries whih re onformle for mtri multiplition wo mtries nd, with sizes m n nd p q respetively, re onformle for multiplition if nd only if n p ; ie the numer of olumns of is the sme s the numer of rows of he result is written Note tht mtri multiplition my e defined for ut not neessrily for Hene, mtri multiplition is not in generl ommuttive he result of multiplying n m n mtri nd n n q mtri is n m q mtri note the outside size symols give the size of the result) Emple Whih of the following mtries re onformle for mtri multiplition,, C?, C, C nd re vlid multiplitions; wheres we nnot lulte, C, or CC
o multiply two mtries, onformle for mtri multiplition, involves n etension of the dot produt proedure desried erlier in the setion on vetors o form the result of multiplying on the left) y on the right) ie to form the produt ) we view s mtri omposed of rows nd s mtri mde up of olumns he entries in the produt mtri re determined y forming dot produts - following the method desried previously o form the i, )-th element of we form the dot produt of row i of mtri with olumn of mtri Emple Let nd he produt mtri n e lulted s hs size nd hs size, ie the numer of olumns of is the sme s the numer of rows of Hene, will hve size For emple, to otin the element in position, ) of the produt mtri we tke the dot produt of row of with olumn of, ie o otin the element in position, ) of the produt mtri we tke the dot produt of row of with olumn of, ie in vetor form we hve [ ] his proess n e ontinued to generte the omponents of the produt mtri ) ) ) ) ) ) ) ) Emple 9 Let nd If possile lulte nd In this se we n lulte oth nd Why?) he result of oth the multiplitions will e mtri Why?) nd 9 Note: this is n emple of very importnt result in mtri rithmeti, ie in generl Emple Evlute the following mtri produts: ), ), ) y ), ) ) y y Emple Let nd 9 Clulte the mtri produts nd ), In this se mtri multiplition is ommuttive,, nd this prtiulr produt mtri is lled the identity mtri see Setions 9 nd for further disussions
rnspose tri he mtri otined from y interhnging the rows nd the olumns of is lled trnspose nd is denoted Emple Consider the mtri : he mtri Find is otined y interhnging rows nd olumns of the mtri Hene, he Determinnt nd Inverse of tri Consider the following rithmeti evlutions Note tht row of is olumn of nd row of is olumn of One wy of interpreting this lst sttement is tht ny non-zero) numer hs ssoited with it multiplitive inverse, nd tht numer times its inverse equls the identity) We n mke n nlogous sttement for squre mtries tht will prove useful lter on Provided the determinnt defined elow) of squre mtri is non zero, there eists nother squre mtri sme size s ) lled the inverse of nd denoted nd it hs the property tht he Zero tri he zero mtri is mtri for whih every element is zero Stritly there re mny zero mtries one for eh possile size of mtri 9 he Identity tri his refers to the identity with respet to mtri multiplition he identity mtri is only defined for squre mtries: it hs on the min digonl the digonl strting t top left nd going to ottom right) nd zeros everywhere else he mtri is usully represented y I Note tht some tet ooks inlude susript n, nd write I n, to indite the size of the squre mtri Emple ) he identity mtri is, I, ) We met the identity mtri is, ) he identity mtri is, I in Emple, I I where I is similrly sized identity mtri In this se mtri multiplition is ommuttive We now look t how to determine inverse mtries for the se For generl mtri the inverse is simply written down using the formul d d det ) where det ) d is lled the determinnt of the mtri Note tht det ) is often written However, re must e tken not to onfuse this nottion with the modulus We nnot over-emphsise the importne of the requirement tht d for the eistene of n inverse of the generl mtri he determinnt of mtri n e used to d determine the eistene, or otherwise, of mtri inverse: y heking tht it is non-zero mtri with n inverse is lled invertile mtri with no inverse is sid to e non-invertile or singulr
Emple Clulte the determinnt of eh of the following mtries; hene identify whih mtries re invertile nd for the invertile mtries lulte the inverse ), ), ), he Determinnt of tri We now demonstrte how to lulte the determinnt of mtri Define the mtri o lulte the determinnt we proeed s follows: ) d), e) Determinnt is zero No inverse Epnding y Row : det) det det det Note the sign on the entrl term We do not need to simplify this epression ny further s we lredy know how to lulte the determinnt of mtri lterntively we n epnd on the seond or third rows s follows: ) Determinnt is tri hs n inverse, Epnding y Row : det) det det det ) Determinnt is zero No inverse d) Determinnt is - tri hs n inverse, e) Determinnt is tri hs n inverse, For those mtries for whih n inverse eists you should hek tht I Note: Geometrilly, the solute vlue of the determinnt of mtri is the d re of prllelogrm whose edges re the vetors, ) nd, d) y Epnding y Row : det) det det det In the three ses desried ove note the rry of signs tht prefi the oeffiients i, ie In similr mnner we n epnd on ny of the olumns of using the sign rry ove Epnding y Column : det) det det det Epnding y Column : det) det det det Epnding y Column : det) det det det Note: he vlue of the determinnt will lwys e the sme regrdless of whih row or olumn we perform the epnsion on 9
Emple Clulte the determinnt of the mtri 9 : Epnding on the first row gives: Emple det) det det 9 det 9 9 ) 9 ) ) 9 Clulte the determinnt of the mtri : Epnding on the first row gives: det ) det det det ) ) ) s we sw erlier the vlue of the determinnt of squre mtri n e used to determine if mtri is invertile: if the determinnt is non-zero the mtri is invertile; otherwise the mtri is NO invertile Hene, the mtri in Emple is not invertile while the mtri in Emple is invertile provided tht we hve Note: When lulting the determinnt of mtri we usully epnd long the row or olumn ontining most zeros in order to minimize the rithmeti So, in Emple we ould hoose to epnd long row or epnd down olumn he Inverse of tri y the Coftor ethod We now etend the ide of n inverse mtri to the se In generl, the inverse of mtri is given y the formul d ) det ) det ) where the mtri d ) is known s the doint of In order to lulte the inverse of mtri, if it eists, we must therefore otin the determinnt of nd the doint of he derivtion of the doint mtri requires us to lulte quntities known s the minors nd oftors of the mtri he following prgrphs illustrte the methodology y wy of n emple reduing the proedure to five distint steps Emple Determine the inverse of the mtri if it eists SEP : Clulte the determinnt of Epnding long the third row the determinnt of is Sine det ), det ) det )det det eists SEP : Clulte the mtri of minors he minor of entry i, denoted y i, is otined s follows: remove the i th row th remove the olumn the minor i is the determinnt of the remining mtri he minor of entry )
he minor of entry he minor of entry he minor of entry he minor of entry ) SEP : Clulte the oftor mtri i he oftor of entry, denoted y C, is defined s C i ) i i o otin the oftor mtri of ) we simply hnge signs of the elements of the mtri of minors lulted in Step using the sign mtri: Hene, we otin the oftor mtri of ) i he minor of entry he minor of entry ) SEP : Clulte the doint mtri he doint of is determined s the trnspose of the oftor mtri, ie d ) he minor of entry he minor of entry Hene, the mtri of minors 9 ) SEP : Clulte the inverse mtri he inverse of the mtri is lulted s d ) det ) so tht whih simplifies to give It is strightforwrd to hek tht I
Emple Determine the inverse of the mtri 9 if it eists SEP : Clulte the determinnt of Epnding long the first row the determinnt of is Sine det ), det ) ) eists ) he minor of entry he minor of entry he minor of entry 9 9 9 ) SEP : Clulte the mtri of minors he minor of entry i, denoted y i, is otined s follows: remove the i th row th remove the olumn the minor i is the determinnt of the remining mtri he minor of entry he minor of entry 9 9 9 he minor of entry he minor of entry Hene, the mtri of minors 9 9 9 9 9) he minor of entry he minor of entry 9 9 9 SEP : Clulte the oftor mtri i he oftor of entry, denoted y C, is defined s C i ) i i o otin the oftor mtri of ) we simply hnge signs of the elements of the mtri of minors in Step using the sign mtri: i
Hene, we otin the oftor mtri of ) n lterntive ethod for Clulting the Determinnt of tri Rule of Srrus n lterntive pproh for lulting the determinnt of mtri SEP : Clulte the doint mtri he doint of is determined s the trnspose of the oftor mtri, ie SEP : Clulte the inverse mtri he inverse of the mtri is lulted s so tht d ) d ) det ) We will hek tht our inverse is orret s follows: 9 I s required ws developed y the Frenh mthemtiin Pierre Srrus 9-) he Rule of Srrus involves the following steps: rewrite the first two olumns of the mtri to the right of it using the left to right digonls tke the produts,, using the right to left digonls tke the produts,, omine the ove nd lulte det ) Note: Geometrilly, the solute vlue of the determinnt of mtri is the volume of prllelepiped whose edges re the vetors u,, ), v,, ) nd w,, ) v w u
Emple 9 Clulte the determinnt of the mtri nd find if it eists 9 Emple If nd, then We rewrite s nd lulte 9 det ) 9 9 9 Here we hve tht nd so we sy tht is n eigenvetor of orresponding to the eigenvlue λ he geometri effet in this emple is tht the vetor hs een strethed y ftor of ut its diretion remins unhnged s λ > Note tht ny vetor k, where k is onstnt, is n eigenvetor orresponding to the eigenvlue s det ) the mtri is not invertile Geometrilly this result mens tht the vetors,, ),,, ), nd,, 9) re oplnr, ie they lie in the sme plne, so the volume of prllelepiped sed on them is equl to For emple,, eigenvlue λ 9 /,, et re ll eigenvetors orresponding to the Eigenvlues nd Eigenvetors of tri Eigenvlues nd eigenvetors hve mny importnt pplitions in siene nd engineering inluding solving systems of differentil equtions, stility nlysis, virtion nlysis nd modelling popultion dynmis Let e n n) mtri n eigenvlue of is slr rel or omple) suh tht I) for some non-zero vetor In this se, we ll the vetor n eigenvetor of orresponding to the eigenvlue Geometrilly Eq I) mens tht the vetors nd re prllel he vlue of determines wht hppens to when it is multiplied y, ie whether it is shrunk or strethed or if its diretion is unhnged or reversed Clultion of Eigenvlues If is ) or ) mtri it is usully reltively strightforwrd to lulte its eigenvlues nd eigenvetors y hnd So, how do we lulte them? We hve tht I, where I is the identity mtri, so we n rewrite Eq I) s I - I λ I) We note tht the eqution λ I) n only hold for non-zero vetor if the mtri λ I) is singulr does not hve n inverse) Hene, the eigenvlues of re the numers for whih the mtri λ I) does not hve n inverse In other words the numers stisfy the eqution det λ I) II) nd, s noted ove, they n e rel or omple 9
Emple Find the eigenvlues of the following mtries i) ii) iii) C s λ i) λ I λ λ λ Hene, det λ I) λ) λ) ) ) λ λ λ We ll λ λ the hrteristi polynomil of the mtri nd the eigenvlues of stisfy the hrteristi eqution det λ I), ie λ λ λ ) λ ) λ nd λ Hene, λ nd λ re the eigenvlues of the mtri Note: We n esily hek our nswer s follows: Let tr) denote the tre of mtri, ie the sum of the elements on the min digonl hen the sum of the eigenvlues equls the tre of the mtri Here we hve tht tr) nd the sum of the eigenvlues is s required λ iii) C λ I λ λ λ Hene, det C λ I) λ) λ) )) λ λ Now solve det C λ I) to find the eigenvlues of C, ie λ λ ± Hene, the eigenvlues of the mtri C re λ nd λ he following emple demonstrtes short-ut pproh tht n e dopted when lulting the eigenvlues of speifi types of mtries Emple Find the eigenvlues of the following mtries i) ii) iii) C s We first lulte the eigenvlues using the method desried ove efore identifying short-ut pproh for these speil types of mtries λ i) Solving det λ I) gives λ λ ) λ) λ, λ λ ii) λ I λ λ λ Hene, det λ I) λ) λ) ) ) λ λ λ Now solve det λ I) to find the eigenvlues of, ie λ λ λ ) λ ) λ nd λ λ ii) Solving det λ I) gives λ λ iii) Solving det C λ I) gives λ λ ) λ) λ, λ λ ) λ) λ, λ Hene, λ nd λ re the eigenvlues of the mtri
In this emple: tri is lower-tringulr mtri nd hs the property tht ll of its entries ove the min digonl re tri is n upper-tringulr mtri nd hs the property tht ll of its entries elow the min digonl re tri C is digonl mtri nd hs the property tht ll of its entries not on the min digonl re Note here tht it is possile to hve some of the entries on the min digonl equl to zero Short-ut: In ll three ses - lower tringulr, upper-tringulr nd digonl - the eigenvlues re simply the entries on the min digonl nd so we n ust red them off without the need for ny lultions Notes For n n) mtri the hrteristi eqution is polynomil of degree n In the emples ove we sw tht the hrteristi eqution in the se is qudrti he sum of the eigenvlues is equl to the tre of the mtri he produt of the eigenvlues is equl to the determinnt of the mtri If is n eigenvlue of mtri then is not invertile If is n eigenvlue of n invertile mtri then is n eigenvlue of λ he mtri nd its trnspose,, hve the sme eigenvlues Repeted Eigenvlues In the emples presented up to now the eigenvlues hve een distint ut it is possile for mtri to hve repeted eigenvlues Emple Find the eigenvlues of the mtri o find the eigenvlues we solve 9 9 λ 9 det λ I) 9 λ λ )9 λ) 9 λ λ λ ) λ ) λ repeted) Comple Eigenvlues It is possile for rel-vlued mtri to hve omple eigenvlues nd eigenvetors) s illustrted y the following emple Emple Find the eigenvlues of the mtri o find the eigenvlues we solve λ det λ I) λ λ λ ±
Clultion of Eigenvetors One we hve lulted the eigenvlues we n find the eigenvetors y solving the mtri eqution for eh eigenvlue in turn Note here tht eigenvetors must e non-zero he eqution given ove is often written s Emple λ I) Find the eigenvlues nd eigenvetors of the mtri First we find the eigenvlues y solving: λ det λ I) λ λ ) λ) λ λ λ ) λ ) λ, λ We now lulte the eigenvetors orresponding to the eigenvlues y solving the eigenvetor eqution, Cse : o find n eigenvetor orresponding to eigenvlue λ we solve λ hese re simultneous equtions nd we note tht one eqution must e multiple of the other If not then you hve mde mistke! Here Eq ) is times Eq ) oth the equtions give nd if we let α, sy, for some non-zero numer α, then α nd we find the first eigenvetor to e of the form α α α Note tht there re infinitely mny non-zero eigenvetors depending on the vlue hosen for α Setting α gives n eigenvetor orresponding to the eigenvlue λ s We n hek our nswer y showing tht λ We hve Hene, λ s required nd λ Cse : o find n eigenvetor orresponding to eigenvlue λ we solve λ oth these equtions give Let α, α ) then α nd so ) ) α α α
Setting α gives Hene, λ s required It is reltively strightforwrd to hek tht, ie nd λ In summry, we therefore hve the eigenvlue/eigenvetor pirs, Emple λ, ; λ, Find the eigenvlues nd eigenvetors of the following mtri : In Emple prt ii) we found the eigenvlues of to e λ nd λ We now lulte the eigenvetors orresponding to these eigenvlues y solving the eigenvetor eqution, Cse : o find n eigenvetor orresponding to eigenvlue λ we solve λ oth these equtions give Note tht for system we do not tully need to introdue the prmeter s we did in the previous emple We n simply hoose onvenient numeril vlue for either of the omponents or of the eigenvetor So here we n let, sy, giving hen n eigenvetor orresponding to the eigenvlue λ is Cse : o find n eigenvetor orresponding to eigenvlue λ we solve λ oth these equtions give Let, sy, so tht hen n eigenvetor orresponding to the eigenvlue λ is In summry, we therefore hve the eigenvlue/eigenvetor pirs, λ, ; λ,
Systems With Zero s n Eigenvlue We hve previously noted tht n eigenvetor nnot e the zero vetor,, ut it is possile to hve n eigenvlue λ Cse : o find n eigenvetor orresponding to eigenvlue λ we solve λ Emple Find the eigenvlues nd eigenvetors of the mtri o find the eigenvlues we need to solve λ det λ I) λ λ ) λ) λ λ λ λ ) λ, λ We now find the eigenvetors orresponding to these eigenvlues: Cse : o find n eigenvetor orresponding to eigenvlue λ we solve λ oth these equtions give Let, sy, so tht n eigenvetor orresponding to the eigenvlue λ will then e oth these equtions give Let, sy, then n eigenvetor orresponding to the eigenvlue λ will then e o summrise we hve: λ, ; λ, Emple Find the eigenvetors of the mtri in Emple We previously found tht hd omple eigenvlues, Cse : o find n eigenvetor orresponding to eigenvlue λ λ nd λ λ we solve 9
If, for emple, we multiply the first eqution y oth equtions give Let, sy, then n eigenvetor orresponding to the eigenvlue λ will then e Cse : o find n eigenvetor orresponding to eigenvlue λ we solve λ If, for emple, we multiply the seond eqution y oth equtions give Let, sy, then n eigenvetor orresponding to the eigenvlue λ will then e o summrise we hve: λ, ; λ, utoril Eerises ) Simplify the following i) ii) iii) iv) ) Simplify the following mtri produts i) ii) iii) [ ] iv) [ ] ) Simplify the following nd omment on your nswers i) ii) ) i) Whih of the following mtries n e squred?, ii) In generl, whih mtries n e squred?
) i) Given tht, find the inverse mtri nd lulte the mtri produts nd Comment on your results ) Given tht nd, find the inverse mtri nd lulte the mtri produts ii) Given tht lulte the mtri produt ) Determine when the following mtri is invertile nd lulte its inverse k ) For the mtri show tht ) ) Given tht D, find the inverse mtri D ) Consider the mtries D nd P mtri P nd lulte the mtri produt P D P Determine the inverse ) Let nd Evlute ) nd Comment on your nswer nd ) Find the eigenvlues of eh of the following mtries: i) ii) iii) iv) 9) Let nd Evlute ) nd Comment on your nswer ) Evlute the determinnt of eh of the following mtries ) Find the eigenvlues nd eigenvetors of eh of the following mtries: i) ii) iii) iv) i) ii) iii) ) Find the eigenvlues nd eigenvetors of eh of the following mtries: ) Whih of the mtries in Question re invertile? Justify your nswer i) ii) iii) iv) ) For eh invertile mtri in Question determine the inverse
nswers ) i) ; ii), iii) ; iv) ) i) ; ii) ; iii) 9 iv) ) i) ii) he onlusion from this emple is tht mtri multiplition is not ommuttive, so tht the order in whih mtries re multiplied is importnt ) i) Only the first mtri n e squred sine it is onformle for multiplition 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 onformle for mtri multiplition if p m ) i) so det) whih is non-zero nd so the mtri is invertile hen We hve tht I oth mtri produts give the identity mtri ii) so nd 9 ) k so k ) det his is non-zero provided k nd in this se the inverse is k k ) ) s required ) so ) In generl for mtries nd we hve tht ) 9) nd so ) lso, nd so ) In generl for n n squre mtries nd we hve tht ) ) i) det det det D ii) det det det D
iii) D det det det ) i) λ, λ ii) λ, λ iii) λ, λ iv) λ, λ ) hey re ll invertile eept iii), sine only iii) hs zero determinnt ) i) ii) ) i) λ, ii) λ, λ, ; λ, ) iii) λ, ; λ, iv) λ, ; λ, ) D ) i) λ, ; λ, Note: If D is n n n n digonl mtri then its inverse is given y ii) λ, iii) λ, ; λ, ; λ, D n n provided tht none of the digonl elements re zero iv) λ, ; λ, ) P nd P D P