1 4 6 is symmetric 3 SPECIAL MATRICES 3.1 SYMMETRIC MATRICES. Defn: A matrix A is symmetric if and only if A = A, i.e., a ij =a ji i, j. Example 3.1.

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1 SPECIAL MATRICES SYMMETRIC MATRICES Def: A mtr A s symmetr f d oly f A A, e,, Emple A s symmetr Def: A mtr A s skew symmetr f d oly f A A, e,, Emple A s skew symmetr Remrks: If A s symmetr or skew symmetr, the A s squre mtr If A s skew symmetr, the the elemets o the dgol of A re ll zero d eh off dgol elemet s mus ts symmetr prter Produts of Symmetr Mtres Produts of symmetr mtres re ot geerlly symmetr If A d B re symmetr mtres of the sme order the the trspose of the produt AB s ( AB ) B A BA Se BA s geerlly ot the sme s AB, ths mes AB s geerlly ot symmetr Emple Let A d B 7 7 The AB Ad ( AB ) Remrks: If A s symmetr, the A s symmetr If A s r r mtr, the A S + K (deomposto s uque) where S s symmetr d K s skew symmetr Also, A + A s symmetr b A A s skew symmetr

2 Let A d B be symmetr mtres A + B s symmetr b AB s symmetr f d oly f AB BA Propertes of AA d A A ) Produts of mtr d ts trspose lwys est d re symmetr If A s r mtr, the ) AA s symmetr, se ( AA ) ( A ) A AA ) A A s symmetr, se ( A A ) ( A ) ( A ) A A Note: AA d A A re ot eessrly equl Remrk: Mtr multplto esures tht elemets of AA re er produts of rows of A wth themselves d wth eh other: Suppose A r r r r, A r r r r the AA rr r r r Q: How bout A A? Remrk: AA d A A hve dgol elemets tht re oegtve se the sum of squres s lwys oegtve ) A A O mples A O ( AA O mples A O ) Proof: ) tr( A A ) O mples A O ( tr( AA ) O mples A O ) Proof:

3 Remrk: Results ) d ) re seldom useful for the ske of some prtulr mtr A, but they re ofte helpful developg other results mtr lgebr whe A s futo of other mtres Emple For mtres P, Q r Proof: r Produts of Vetors, X s, P X X Q X X mples P X Q X ) The er produt of vetors d y s slr, thus t s lwys symmetr y y k, k slr ) The outer produt of vetors d y s ot eessrly symmetr y s ot geerlly equl to y Emple Let d y d the outer produt s Sums of Outer Produts Cosder A r b b,, d B s The the er produt of d y s [ ] where b hs s dmeso b where hs r dmeso,,

4 The AB rs [ ] b b b b sum of outer produts of olums of A wth orrespodg rows B Emple Let A d B b b The AB Spel se: B A AA Elemetry Vetors Def: A vetor wth uty for ts th elemet d zeros elsewhere s lled elemetry vetor Notto: e ) ( where dtes the posto of d deotes ts order Emple 7 e ) (, e ) (, e ) (, e ) (

5 Remrks: ( ) E e Emple 8 Let e ( ) ) e ) ( ull mtr eept for the (,)th elemet beg uty d e ( ) ) I e e ( ) ( ) E The E e ) ( e ( ) Emple 9 I e ) ( e ( ) + e ) ( e ( ) + e ( ) e ( ) + e ( ) e ( ) E + E + E + E ) Let A be r mtr, the ( e r ) A th row of A ( b A e ) th olum of A Emple Let A 7 9, e ( ), d e ( ) The e ( ) A d A e ( )

6 MATRICES WITH EQUAL ELEMENTS Def: Vetors whose every elemet s re lled summg vetors They be used to epress sum of umbers mtr otto s er produt Notto: Emple ) [ ], [ ] ) A A A Remrk: The er produt of summg vetor wth tself s slr, the vetors order, e, Def: Let J deote the outer produt of summg vetors, r d s The, J s mtr wth ll elemets equl to oe Notto: J rs r s, J Emple Remrks: ) λ J rs ) J rs J st mtr wth ll elemets equl to λ s J rt

7 ) r J rs ) J rs r s s s r ) J d J J ) J J d J J Def: Let C I J I J The C s lled the eterg mtr Emple : C Note: ) C C d C C ) C O ) C J J C O Remrk: The me d sum of squres bout the me for the dt,,, be epressed terms of vetors d J mtres Let [ ] Eerse: Show tht The ) Smple me mtr form ) Eh observto s devto from mtr form [ ] C s the smple vre 7

8 IDEMPOTENT MATRICES Def: A mtr A s dempotet f d oly f A A Emple Idetty mtres, squre ull mtres, J Def: A mtr A stsfyg A O s lled lpotet, d tht for whh A I ould be lled upotet Emple ) A ) B s lpotet s upotet Remrks: Idempotet mtres re eessrly squre, ow, A does ot est Whe A s dempotet, A r A, r,, Theorem: Let A d B be dempotet mtres, the ) A + B s dempotet f AB BA O b) AB s dempotet f AB BA ) ( I A ) s dempotet, but ot ( A I ) ORTHOGONAL MATRICES Def: The orm of rel vetor [ ] Emple + Let [ ] s defed s 8

9 Def: A vetor s sd to be ut vetor (orml vetor) whe ts orm s uty, e, Emple Let [ 8] Note: Gve o ull vetor, let u se Emple u u Let [ ] the u s the ormlzed form of u [ ] Def: The o ull vetors d y re sd to be orthogol whe ) Emple ) Let [ ] d y [ ] ) Let [ ] d y [ ] The y The y y (or y Def: The vetors d y re defed s orthoorml whe they re orml d orthogol, e, y y d y y Emple Let [ ] d y [ 9 ] Def: A group, or olleto, of vetors ll of the sme order s lled set of vetors Def: A set of vetors for,,, s sd to be orthoorml set of vetors whe every vetor the set s orml, for ll, d whe every pr of dfferet vetors the set s orthogol, for,,, 9

10 Remrks: The vetors of orthoorml set re ll orml, d prwse orthogol A mtr A whose rows osttute orthoorml set of vetors s sd to hve r orthoorml rows, whereupo AA I r But the detty mtr I Coversely, whe A r but AA my ot be detty mtr A A s ot eessrly hs orthoorml olums A A I Emple Let A The AA d A A Def: Let A be squre mtr, the A s sd to be orthogol mtr f AA I A A A orthogol A hs orthoorml rows d orthoorml olums Remrk: se squre mtres wth orthoorml rows hve orthoorml olums Emple 7 ) Show tht A ) I s orthogol mtr s orthogol mtr Remrk: Let A d B be orthogol mtres The AB s orthogol Proof: QUADRATIC FORMS Def: Let be vetor d A squre mtr, the the produt qudrt form A s lled

11 Emple ) Let [ ] The A d A 8 ) Let [ ] The A d A The results re qudrt futos of the s; hee the me qudrt form Note: If [ ] d A { },,,,, ; the < ( + < ) + > A Thus, there s o uque mtr A for whh y prtulr qudrt form be epressed s A Emple A 8 s the sme s B The qudrt form s the sme eve though the ssoted mtr A the frst A d B hve the sme produt s dfferet from mtr B of the seod produt dgol elemets, d eh of them the sum of eh pr of symmetrlly pled offdgol elemets d re the sme

12 Remrk: For y prtulr qudrt form, there s uque symmetr mtr A for A whh the qudrt form be epressed s It be foud y prtulr se by rewrtg the qudrt A where A s ot symmetr s ( A + ), beuse A + s symmetr [ A ] ( A ) Emple A Hee, f A s symmetr, e,, we epress A A + < s Therefore, whe delg wth qudrt forms, we lwys tke A s symmetr Ths wll be oveet ot oly beuse the symmetr A s uque for y prtulr qudrt form, but lso beuse symmetr mtres hve my propertes tht re useful studyg qudrt forms, prtulrly those ssoted wth lyss of vre Herefter, wheever we del wth qudrt form ( or f ot, we epress A terms of ts symmetr outerprt ) A, we ssume A A NON NEGATIVE DEFINITE MATRICES All qudrt forms orrespodg qudrt form s zero oly for o EXAMPLE Let A A re zero for o For some mtres A the The A

13 Def: Whe A > for ll other th o the A s postve defte qudrt form, d A A s orrespodgly postve defte (pd) mtr There re lso symmetr mtres A for whh s well s for o A s zero for some o ull EXAMPLE Let A 7 7 The A Def: Whe A for ll d A for some o the A s postve semdefte qudrt form d hee A A s postve semdefte (psd) mtr Nottos: A > A s pd ; A A s psd Remrks: Postve defte d postve semdefte mtres re lled o egtve defte (d) mtres All symmetr dempotet mtres re psd (eept I, whh s the oly pd dempotet mtr) Verfy: ( I J ) s dempotet, hee psd ( ) the s re equl C s psd beuse t s postve eept for beg zero whe ll Redg Assgmet: Serle, Chpter : Determts, pp 8 8 Do the eerses for prte