Introduction to Matrix Algebra

Transcription

1 Itrodutio to Mtri Alger George H Olso, Ph D Dotorl Progrm i Edutiol Ledership Applhi Stte Uiversit Septemer Wht is mtri? Dimesios d order of mtri A p q dimesioed mtri is p (rows) q (olums) rr of umers, or smols, d will itself e represeted upper-se, old, itli smol; eg, A, β For iste, mtri, A, e represeted, s smoli rr, A, or s tul rr of umers 6 A I either se, the order of A is sid to e O the other hd, the mtri, d d d d d d D, d d d d d d is mtri Tht is, its order is A mtri hvig ol oe olum or oe row is lled vetor d is represeted lower-se, old, itli smol; eg,, β Hee the d mtries,, d

2 β, re oth vetors A is row vetor; β is olum vetor The umers (or smols) iside mtri re lled elemets Thus, i the first mtri, A, give ove, d re elemets of the mtri A Similrl, i the seod mtri, A, give erlier, the umers d 6 re elemets Elemets i mtri re ideed (or referred to) susripts tht give their row d olum lotios For, iste, i the first mtri, A, ove, is the elemet i the seod row of the third olum Similrl, i the mtri, D, ove, elemet d is foud i the fourth row, first olum I geerl, ij is the i,j th elemet of A Whe referrig to elemets of vetors, however, the first row (or olum) is ssumed Hee, i the vetor,, ove, is its seod elemet; i β, is the third elemet I geerl, elemets of vetors, re idited the j th elemet of row (ie, q) vetors d the i th elemet of olum (ie, p ) vetors A mtri i whih the umer of rows (p) equls the umer of olums (q) is lled squre mtri; otherwise, providig it is ot vetor, it is lled retgulr mtri The turl order of retgulr mtri hs more rows th olums (ie, p > q) Hee, A, mtri, is preseted i its turl order The trspose of mtri is otied iterhgig its rows d olums Thus, the trspose of A, represeted s A, is A, mtri As orete emple, osider

3 7 X, whih, whe trsposed, eomes X The turl order of vetor is p olum vetor Here, the olum vetor,, is preseted i its turl order The trspose of,, is row vetor As metioed erlier mtri where p = q is squre mtri At lest two prtiulr tpes of squre mtries re prtiulrl importt, smmetri mtries d digol mtries A smmetri mtri is squre mtri i whih the elemets ove the mi digol re mirror imges of the elemets elow the mi digol (the mi digol is tht set of elemets ruig from the upper left-hd side of squre mtri to the lower right-hd I the smmetri mtri V, give elow, the elemets i old represet the mi digol Note tht the elemets ove the mi digol re mirror imges of the elemets elow the mi digol 6 8 V Digol mtries re squre mtries i whih ll elemets eept the mi digol re zero () For iste,

4 6 7 D is smmetri digol mtri A importt digol mtri is the idetit mtri The idetit mtri is digol mtri i whih ll the elemets log the mi digol re uit () For emple, I is idetit mtri Smmetri mtries hve importt properties For iste, smmetri mtri is equl to its trspose Tht is, for the mtri give little erlier, V = V

5 Mtri Opertios Additio The dditio of two mtries, A d B, is omplished ddig orrespodig elemets, eg, ij + ij ; hee, for the mtries, A d B C A B It is ovious (or should e ovious) tht the orders of the two mtries eig dded eed to e idetil Here, for iste, oth mtries re of order Whe the orders of the two mtries re ot idetil, dditio is ot impossile Additio of mtries is ommuttive Hee A + B = B + A As orete emple, let 7 A 8, d 8 B

6 The, C = A + B = B + A = = Multiplitio Multiplitio is somewht more omplited Whe multiplig mtries we eed to distiguish mog multiplitio slr, vetor multiplitio, premultiplitio, d post-multiplitio, ier-produts d outer-produts But first, let us defie slr A slr is simpl sigle umer, suh s, 76, or - Smolill, we tpill use the smol,, to represet slr Slr multiplitio Slr multiplitio is esil show emple Give the mtri X d the slr,, the produt, X, is give X Note tht ll the elemets i X re multiplied the slr,

7 As orete emple, let X 7 d = The 7 X 7 I the ove epressio, the produt, X, results from pre-multiplig X the slr ; wheres the produt X results from post-multiplig X the slr, Sie slr multiplitio is ommuttive, X will lws e equl to X This is ot the se with mtri multiplitio, however Ol i erti speil ses will the produts formed pre-multiplig d post-multiplig two mtries, X d Y e equl I geerl XY YX Mtri Multiplitio First, ote tht give two mtries, eg, A of order m, d B of order p q, multiplitio is ol possile whe the umer of olums i the pre-multiplier is equl to the umer of rows i the post-multiplier Hee the produt, AB, is ol possile whe m = p Similrl, the produt BA is ol possile whe = q Whe m = p the produt, C = AB will hve order q Similrl, whe q =, the produt, C = BA will hve order p m We illustrte this defiig the two mtries, X, d Y

8 Note tht X hs order, d Y hs order Here, ol the produts, XY (ot XY ) d Y X (ot YX) re possile Post-multiplig X Y, ields XY where the produt is of order As more orete emple, let The X d Y XY 6 6 6, mtri The ol other produt possile etwee these two mtries is Y X :

9 Y X , whih is mtri Whe A d B re oth squre mtries of the sme order, the ll the produts, AB, BA, A B, AB, B A, BA, A B, d B A re possile Furthermore, eept is erti speil ses, the mtries formed these vrious produts will e differet from eh other As eerise, tr omputig ll possile produts of the followig two squre mtries: A d B Vetor Multiplitio Hvig lered how to ompute mtri multiplitio, vetor multiplitio is es It follows the sme rules of mtri multiplitio eept tht here oe of the mtries is vetor For iste, give the olum vetor,, d the mtri, Y, the produts Y d Y re the ol possile produts Hee,

10 i i i i Y To put this i orete terms, let d Y The 6 Y Some speil ses of vetor d mtri multiplitio re prtiulrl importt For iste, let the vetor, e defied s vetor with ll elemets equl to : d Y {Note, d re olum vetors} The, 8 Y If we let

11 X d Y The, 6 Y X Note lso tht X X Ier-produts d outer-produts Give the two mtries, A d B, Verif tht the followig two produts re possile, AB d BA (of ourse, A B d B A lso re possile) The orders of the two produts re ot idetil AB hs order,, while BA hs order The produt AB is outer-produt while the produt BA is ier-produt I geerl, wheever oth ierprodut d outer-produt eist for two mtries, the order of the ierprodut will e less th the outer-produt Furthermore, i geerl, the ierprodut will ot e idetil to the outer-produt For emple, let A d B The

12 8 6 6 AB d 8 BA Ier- d outer-produts re prtiulrl importt i vetor multiplitio Let,, d X The i i i X, {Note the summtio is over rows} d j j j j 6 X {Here the summtio is over olums} Also, it is es to verif tht ij 8 X where the summtio is over rows d olums