Matrix & Vector Basic Linear Algebra & Calculus

Transcription

1 Mtrix & Vector Bsic Liner lgebr & lculus

2 Wht is mtrix? rectngulr rry of numbers (we will concentrte on rel numbers). nxm mtrix hs n rows n m columns M x4 M M M M M M M M M M M M First row Secon row Thir row First column M Secon column Thir Fourth column column Row number olumn number

3 Wht is vector? vector is n rry of n numbers row vector of length n is xn mtrix column vector of length m is mx mtrix 4

4 Specil mtrices Zero mtrix: mtrix ll of whose entries re zero Ientity mtrix: squre mtrix which hs s on the igonl n zeros everywhere else. 4 x I x 4

5 Mtrix opertions 5., 9, 7,,, 4,,, B B i h g f e c b i h g f e c b Equlity of mtrices If n B re two mtrices of the sme size, then they re equl if ech n every entry of one mtrix equls the corresponing entry of the other. 5

6 Mtrix opertions B B ition of two mtrices If n B re two mtrices of the sme size, then the sum of the mtrices is mtrix =+B whose entries re the sums of the corresponing entries of n B 6

7 Mtrix opertions ition of of mtrices Properties Properties of mtrix ition:. Mtrix ition is commuttive (orer of ition oes not mtter) B B. Mtrix ition is ssocitive. ition of the zero mtrix B B 7

8 Mtrix opertions Multipliction by sclr If is mtrix n c is sclr, then the prouct c is mtrix whose entries re obtine by multiplying ech of the entries of by c c c

9 Mtrix opertions Multipliction by sclr If is mtrix n c =- is sclr, then the prouct (-) =- is mtrix whose entries re obtine by multiplying ech of the entries of by c c Specil cse 9

10 Mtrix opertions Subtrction - n - - tht Note B B If n B re two squre mtrices of the sme size, then -B is efine s the sum +(-)B

11 Specil opertions Trnspose If is mxn mtrix, then the trnspose of is the nxm mtrix whose first column is the first row of, whose secon column is the secon column of n so on T

12 Specil opertions Trnspose If is squre mtrix (mxm), it is clle symmetric if T

13 Mtrix opertions ; b b b b Sclr (ot) prouct of two vectors If n b re two vectors of the sme size The sclr (ot) prouct of n b is sclr obtine by ing the proucts of corresponing entries of the two vectors T b b b b

14 Mtrix opertions Mtrix multipliction For prouct to be efine, the number of columns of must be equl to the number of rows of B. B = B m x r r x n m x n insie outsie 4

15 Mtrix opertions Mtrix multipliction If is mxr mtrix n B is rxn mtrix, then the prouct =B is mxn mtrix whose entries re obtine s follows. The entry corresponing to row i n column j of is the ot prouct of the vectors forme by the row i of n column j of B 4 x 7 B x x B 9 notice T 5

16 Mtrix opertions Multipliction of mtrices Properties Properties of mtrix multipliction:. Mtrix multipliction is noncommuttive (orer of ition oes mtter) B B in ge ne r l It my be tht the prouct B exists but B oes not (e.g. in the previous exmple =B is x mtrix, but B oes not exist) Even if the prouct exists, the proucts B n B re not generlly the sme 6

17 Mtrix opertions Multipliction of mtrices Properties. Mtrix multipliction is ssocitive. Distributive lw 4. Multipliction by ientity mtrix 5. Multipliction by zero mtrix 6. I B B B B B B ; I T T T B B ; 7

18 Mtrix opertions Miscellneous properties. If, B n re squre mtrices of the sme size, n then B oes not necessrily men tht B B. oes not necessrily imply tht either or B is zero 8

19 Inverse of mtrix Definition If is ny squre mtrix n B is nother squre mtrix stisfying the conitions B B I Then ()The mtrix is clle invertible, n (b) the mtrix B is the inverse of n is enote s -. The inverse of mtrix is unique 9

20 Inverse of mtrix Uniqueness The inverse of mtrix is unique ssume tht B n both re inverses of B (B) B() B B I BI I I B Hence mtrix cnnot hve two or more inverses.

21 Inverse of mtrix Some properties Property : If is ny invertible squre mtrix the inverse of its inverse is the mtrix itself - Property : If is ny invertible squre mtrix n is ny sclr then -

22 Inverse of mtrix Properties Property : If n B re invertible squre mtrices then (B) - BB (B) B B B I B B - B B Premultiplying both sies by - B Premultiplying both sies by B - -

23 Wht is eterminnt? The eterminnt of squre mtrix is number obtine in specific mnner from the mtrix. For x mtrix: For x mtrix: ; et( ) ; et( ) Prouct long re rrow minus prouct long blue rrow

24 Exmple onsier the mtrix 5 7 Notice () mtrix is n rry of numbers () mtrix is enclose by squre brcets et( ) Notice () The eterminnt of mtrix is number () The symbol for the eterminnt of mtrix is pir of prllel lines omputtion of lrger mtrices is more ifficult 4

25 Duplicte column metho for x mtrix For ONLY x mtrix write own the first two columns fter the thir column Sum of proucts long re rrow minus sum of proucts long blue rrow et( ) This technique wors only for x mtrices 5

26 Exmple Sum of re terms = + + = 5 Sum of blue terms = = Determinnt of mtrix = et() = 5 = 5 6

27 Fining eterminnt using inspection Specil cse. If two rows or two columns re proportionl (i.e. multiples of ech other), then the eterminnt of the mtrix is zero becuse rows n re proportionl to ech other If the eterminnt of mtrix is zero, it is clle singulr mtrix 7

28 ofctor metho Wht is cofctor? If is squre mtrix The minor, M ij, of entry ij is the eterminnt of the submtrix tht remins fter the i th row n j th column re elete from. The cofctor of entry ij is ij =(-) (i+j) M ij M M 8

29 Sign of cofctor Wht is cofctor? Fin the minor n cofctor of M Minor 4 M M ) ( ) ( ofctor

30 ofctor metho of obtining the eterminnt of mtrix The eterminnt of n x n mtrix cn be compute by multiplying LL the entries in NY row (or column) by their cofctors n ing the resulting proucts. Tht is, for ech i n n j n ofctor expnsion long the j th column et( ) j j j j nj nj ofctor expnsion long the i th row et( ) i i i i in in

31 Exmple : evlute et() for: - = et() = et()=() - (-) () = ()(5)-+()(6)-(-)()=98

32 Exmple 4: evlute 5 - et()= - By cofctor long the thir column et()= + + et()= -* (-) 4 +*(-) 5 5 +*(-) = et()= -(--)+(-) 5 (--5)+(-5)=5

33 Qurtic form The sclr U T vector squre mtrix Is nown s qurtic form If U>: Mtrix is nown s positive efinite If U : Mtrix is nown s positive semiefinite

34 Qurtic form Let Then ) ( ) ( U T Symmetric mtrix 4

35 Differentition of qurtic form Differentite U wrt U Differentite U wrt U 5

36 U U U Hence Differentition of qurtic form 6

37 Review: The Derivtives of Vector Functions The hin Rule for Vector Functions 7

38 . The Derivtives of Vector Functions 8

39 . Derivtive of Vector with Respect to Vector 9

40 . Derivtive of Sclr with Respect to Vector If y is sclr It is lso clle the grient of y with respect to vector vrible x, enote by. y. Derivtive of Vector with Respect to Sclr 4

41 Exmple 5 Given n 4

42 In Mtlb >> syms x x x rel; >> y=x^-x; >> y=x^+*x; >> J = jcobin([y;y], [x x x]) J = [ *x, -, ] [,, *x] Note: Mtlb efines the erivtives s the trnsposes of those given in this lecture. >> J' ns = [ *x, ] [ -, ] [, *x] 4

43 4 T T T x x x x x x x x n t nt t n t t t n t t t n nn n n n n x x x x x x T nn n n n n c c c c c c c c c x x Exercise: Some useful vector erivtive formuls

44 Importnt Property of Qurtic Form x T x Proof: T x x n n xi x jij i j n n n n x T i x j ij x x j j xi xi ( x x) i j j i x x x x n x j j i i j i n x T ( x x) T T x x x x T ( x x) x T x If is symmetric: n n T ( x x) x n n nn x x x x n n xt t n xt t n xt t t t nt 44

45 . The hin Rule for Vector Functions Let where z is function of y, which is in turn function of x, we cn write Ech entry of this mtrix my be expne s Fin5J Topic Fll Olin 45

46 The hin Rule for Vector Functions (ont.) Then On trnsposing both sies, we finlly obtin This is the chin rule for vectors (ifferent from the conventionl chin rule of clculus, the chin of mtrices buils towr the left) 46

47 Exmple 6 x, y re s in Exmple n z is function of y efine s z z y y z z y y z, n, we hve z z y y z 4 z4 y y z z z z4 z y y y y y y. y z z z z 4 y y y y y y Therefore, x 4x y x 4x y 4x z y z y y x x y y y y y y y x 4x 4x y 4x y x 47

48 In Mtlb >> z=y^-*y; >> z=y^-y; >> z=y^+y^; >> z4=*y+y; >> Jzx=jcobin([z; z; z; z4],[x x x]) Jzx = [ 4*(x^-x)*x, -*x^+*x-6, -4*x] [ -*x, 6*x^+8*x+, 4*(x^+*x)*x] [ 4*(x^-x)*x, -*x^+*x+6*x^, 4*(x^+*x)*x] [ 4*x,, *x] >> Jzx ns = [ 4*(x^-x)*x, -*x, 4*(x^-x)*x, 4*x] [ -*x^+*x-6, 6*x^+8*x+, -*x^+*x+6*x^, ] [ -4*x, 4*(x^+*x)*x, 4*(x^+*x)*x, *x] 48

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