An Introduction to Matrix Algebra

Transcription

1 Mdern Cntrl Systems, Eleventh Editin, by Richard C Drf and Rbert H. Bish. ISBN: Pearsn Educatin, Inc., Uer Saddle River, NJ. All rights reserved. APPENDIX E An Intrductin t Matrix Algebra E. DEFINITIONS In many situatins, we must deal with rectangular arrays f numbers r functins. The rectangular array f numbers (r functins) a a a n a A = D a a n T (E.) a m a m is knwn as a matrix. The numbers a ij are called elements f the matrix, with the subscrit i denting the rw and the subscrit j denting the clumn. A matrix with m rws and n clumns is said t be a matrix f rder (m, n) r alternatively called an m n (m-by-n) matrix.when the number f the clumns equals the number f rws (m n), the matrix is called a square matrix f rder n. It is cmmn t use bldfaced caital letters t dente an m n matrix. A matrix cmrising nly ne clumn, that is, an m matrix, is knwn as a clumn matrix r, mre cmmnly, a clumn vectr. We will reresent a clumn vectr with bldfaced lwercase letters as a mn y y = D y T (E.) Analgusly, a rw vectr is an rdered cllectin f numbers written in a rw that is, a n matrix. We will use bldfaced lwercase letters t reresent vectrs. Therefre a rw vectr will be written as z = z z zn 4, (E.) with n elements. A few matrices with distinctive characteristics are given secial names. A square matrix in which all the elements are zer excet thse n the rincial diagnal, a, a,...,a nn, is called a diagnal matrix. Then, fr examle, a diagnal matrix wuld be y m This material is rtected by Cyright and written ermissin shuld be btained frm the ublisher rir t any rhibited rerductin, strage in a retrieval system, r transmissin in any frm r by any means, electrnic, mechanical, htcying, recrding, r likewise. Fr infrmatin regarding ermissin(s), write t: Rights and Permissins Deartment, Pearsn Educatin, Inc., Uer Saddle River, NJ

2 Mdern Cntrl Systems, Eleventh Editin, by Richard C Drf and Rbert H. Bish. ISBN: Pearsn Educatin, Inc., Uer Saddle River, NJ. All rights reserved. 6 Aendix E An Intrductin t Matrix Algebra B = Cb (E.4) If all the elements f a diagnal matrix have the value, then the matrix is knwn as the identity matrix I, which is written as I = D (E.5) When all the elements f a matrix are equal t zer, the matrix is called the zer, r null matrix. When the elements f a matrix have a secial relatinshi s that a ij a ji, it is called a symmetrical matrix. Thus, fr examle, the matrix b b S. T. H = C S 8 (E.6) is a symmetrical matrix f rder (, ). E. ADDITION AND SUBTRACTION OF MATRICES The additin f tw matrices is ssible nly fr matrices f the same rder.the sum f tw matrices is btained by adding the crresnding elements. Thus if the elements f A are a ij and the elements f B are b ij, and if C A B, then the elements f C that are c ij are btained as c ij a ij b ij. Fr examle, the matrix additin fr tw matrices is as fllws: C = C S + C 4 (E.7) (E.8) (E.9) Frm the eratin used fr erfrming the eratin f additin, we nte that the rcess is cmmutative; that is, A B B A. Als we nte that the additin eratin is assciative, s that (A B) C A (B C). (E.) (E.) T erfrm the eratin f subtractin, we nte that if a matrix A is multilied by a cnstant a, then every element f the matrix is multilied by this cnstant.therefre we can write S = C 4 This material is rtected by Cyright and written ermissin shuld be btained frm the ublisher rir t any rhibited rerductin, strage in a retrieval system, r transmissin in any frm r by any means, electrnic, mechanical, htcying, recrding, r likewise. Fr infrmatin regarding ermissin(s), write t: Rights and Permissins Deartment, Pearsn Educatin, Inc., Uer Saddle River, NJ S.

3 Mdern Cntrl Systems, Eleventh Editin, by Richard C Drf and Rbert H. Bish. ISBN: Pearsn Educatin, Inc., Uer Saddle River, NJ. All rights reserved. Sectin E. Multilicatin f Matrices 7 aa aa aa = D aa m aa aa aa m (E.) Then t carry ut a subtractin eratin, we use a, and A is btained by multilying each element f A by. Fr examle, aa n aa n T. aa mn C = B - A = B 4 R - B6 R = B-4 R. (E.) E. MULTIPLICATION OF MATRICES The multilicatin f tw matrices AB requires that the number f clumns f A be equal t the number f rws f B. Thus if A is f rder m n and B is f rder n q, then the rduct is f rder m q. The elements f a rduct C AB (E.4) are fund by multilying the ith rw f A and the jth clumn f B and summing these rducts t give the element c ij. That is, c ij = a i b j + a i b j + + a iq b qj = a q (E.5) Thus we btain c, the first element f C, by multilying the first rw f A by the first clumn f B and summing the rducts f the elements.we shuld nte that, in general, matrix multilicatin is nt cmmutative; that is (E.6) Als we nte that the multilicatin f a matrix f m n by a clumn vectr (rder n ) results in a clumn vectr f rder m. A secific examle f multilicatin f a clumn vectr by a matrix is x = Ay = B a y a a a R Cy a a a S = B y + a y + a y a y y + a y + a y R. (E.7) Nte that A is f rder, and y is f rder. Therefre the resulting matrix x is f rder, which is a clumn vectr with tw rws. There are tw elements f x, and x (a y a y a y ) (E.8) is the first element btained by multilying the first rw f A by the first (and nly) clumn f y. Anther examle, which the reader shuld verify, is C = AB = B - AB BA. - R B - k= - R = B 7-5 a ik b kj. 6-6 R. (E.9) This material is rtected by Cyright and written ermissin shuld be btained frm the ublisher rir t any rhibited rerductin, strage in a retrieval system, r transmissin in any frm r by any means, electrnic, mechanical, htcying, recrding, r likewise. Fr infrmatin regarding ermissin(s), write t: Rights and Permissins Deartment, Pearsn Educatin, Inc., Uer Saddle River, NJ 7458.

4 Mdern Cntrl Systems, Eleventh Editin, by Richard C Drf and Rbert H. Bish. ISBN: Pearsn Educatin, Inc., Uer Saddle River, NJ. All rights reserved. 8 Aendix E An Intrductin t Matrix Algebra Fr examle, the element c is btained as c () ( ) 6. Nw we are able t use this definitin f multilicatin in reresenting a set f simultaneus linear algebraic equatins by a matrix equatin. Cnsider the fllwing set f algebraic equatins: We can identify tw clumn vectrs as x x x u, x x 6x u, 4x x x u. (E.) Then we can write the matrix equatin where x x = Cx S and u = C x Ax u, u u u S. (E.) (E.) A = C 4 We immediately nte the utility f the matrix equatin as a cmact frm f a set f simultaneus equatins. The multilicatin f a rw vectr and a clumn vectr can be written as y - 6S. xy = x x xn 4 D y T = x y + x y + + x n y n. (E.) y n Thus we nte that the multilicatin f a rw vectr and a clumn vectr results in a number that is a sum f a rduct f secific elements f each vectr. As a final item in this sectin, we nte that the multilicatin f any matrix by the identity matrix results in the riginal matrix, that is, AI A. E.4 OTHER USEFUL MATRIX OPERATIONS AND DEFINITIONS The transse f a matrix A is dented in this text as A T. One will ften find the ntatin A' fr A T in the literature. The transse f a matrix A is btained by interchanging the rws and clumns f A. Fr examle, if then 6 A = C - 4 S, - This material is rtected by Cyright and written ermissin shuld be btained frm the ublisher rir t any rhibited rerductin, strage in a retrieval system, r transmissin in any frm r by any means, electrnic, mechanical, htcying, recrding, r likewise. Fr infrmatin regarding ermissin(s), write t: Rights and Permissins Deartment, Pearsn Educatin, Inc., Uer Saddle River, NJ 7458.

5 Mdern Cntrl Systems, Eleventh Editin, by Richard C Drf and Rbert H. Bish. ISBN: Pearsn Educatin, Inc., Uer Saddle River, NJ. All rights reserved. Sectin E.4 Other Useful Matrix Oeratins and Definitins 9 6 A T = C (E.4) Therefre we are able t dente a rw vectr as the transse f a clumn vectr and write (E.5) Because x T is a rw vectr, we btain a matrix multilicatin f x T by x as fllws: 4 - S. - x T = x x xn 4. x x T x = x x xn 4 D T = x + x + + x n. x (E.6) Thus the multilicatin x T x results in the sum f the squares f each element f x. The transse f the rduct f tw matrices is the rduct in reverse rder f their transses, s that (AB) T B T A T. (E.7) The sum f the main diagnal elements f a square matrix A is called the trace f A, written as tr A a a a nn. (E.8) The determinant f a square matrix is btained by enclsing the elements f the matrix A within vertical bars; fr examle, det A = a a a a = a a - a a. (E.9) If the determinant f A is equal t zer, then the determinant is said t be singular. The value f a determinant is determined by btaining the minrs and cfactrs f the determinants. The minr f an element a ij f a determinant f rder n is a determinant f rder (n ) btained by remving the rw i and the clumn j f the riginal determinant.the cfactr f a given element f a determinant is the minr f the element with either a lus r minus sign attached; hence cfactr f a ij a ij ( ) i j M ij, where M ij is the minr f a ij. Fr examle, the cfactr f the element a f x n a det A = a a a a a a a a (E.) is a = - 5 M =- a a a a. The value f a determinant f secnd rder ( ) is (E.) This material is rtected by Cyright and written ermissin shuld be btained frm the ublisher rir t any rhibited rerductin, strage in a retrieval system, r transmissin in any frm r by any means, electrnic, mechanical, htcying, recrding, r likewise. Fr infrmatin regarding ermissin(s), write t: Rights and Permissins Deartment, Pearsn Educatin, Inc., Uer Saddle River, NJ 7458.

6 Mdern Cntrl Systems, Eleventh Editin, by Richard C Drf and Rbert H. Bish. ISBN: Pearsn Educatin, Inc., Uer Saddle River, NJ. All rights reserved. Aendix E An Intrductin t Matrix Algebra a a = a a a a - a a. The general nth-rder determinant has a value given by (E.) det A = a n j= a ij a ij with i chsen fr ne rw, (E.) r det A = a n i= a ij a ij with j chsen fr ne clumn. (E.) That is, the elements a ij are chsen fr a secific rw (r clumn), and that entire rw (r clumn) is exanded accrding t Eq. (E.). Fr examle, the value f a secific determinant is (E.4) where we have exanded in the first clumn. The adjint matrix f a square matrix A is frmed by relacing each element a ij by the cfactr a ij and transsing. Therefre a a adjint A = D det A = det C a n a a a n 5 S = = = 9, a n a n T a nn T a a = D a n a a a n 5 a n a n T. a nn (E.5) E.5 MATRIX INVERSION The inverse f a square matrix A is written as A and is defined as satisfying the relatinshi The inverse f a matrix A is A A AA I. (E.6) adjint f A A - = det A (E.7) This material is rtected by Cyright and written ermissin shuld be btained frm the ublisher rir t any rhibited rerductin, strage in a retrieval system, r transmissin in any frm r by any means, electrnic, mechanical, htcying, recrding, r likewise. Fr infrmatin regarding ermissin(s), write t: Rights and Permissins Deartment, Pearsn Educatin, Inc., Uer Saddle River, NJ 7458.

7 Mdern Cntrl Systems, Eleventh Editin, by Richard C Drf and Rbert H. Bish. ISBN: Pearsn Educatin, Inc., Uer Saddle River, NJ. All rights reserved. Sectin E.6 Matrices and Characteristic Rts when the det A is nt equal t zer. Fr a matrix we have the adjint matrix and the det A a a a a. Cnsider the matrix The determinant has a value det A 7. The cfactr a is In a similar manner we btain adjint A = B a -a -a a R, A = C - - a = S. 4 =. (E.8) (E.9) (E.4) A - = adjint A det A = a- 7 b C S. -5 (E.4) E.6 MATRICES AND CHARACTERISTIC ROOTS A set f simultaneus linear algebraic equatins can be reresented by the matrix equatin y Ax, (E.4) where the y vectr can be cnsidered as a transfrmatin f the vectr x. We might ask whether it may haen that a vectr y may be a scalar multile f x. Trying y lx, where l is a scalar, we have lx Ax. Alternatively Eq. (E.4) can be written as lx Ax (li A)x, where I identity matrix. Thus the slutin fr x exists if and nly if (E.4) (E.44) det (li A). (E.45) This determinant is called the characteristic determinant f A. Exansin f the determinant f Eq. (E.45) results in the characteristic equatin. The characteristic equatin is an nth-rder lynmial in l. The n rts f this characteristic equatin are called the characteristic rts. Fr every ssible value l i (i,,...,n) f the nthrder characteristic equatin, we can write (l i I A)x i. (E.46) The vectr x i is the characteristic vectr fr the ith rt. Let us cnsider the matrix This material is rtected by Cyright and written ermissin shuld be btained frm the ublisher rir t any rhibited rerductin, strage in a retrieval system, r transmissin in any frm r by any means, electrnic, mechanical, htcying, recrding, r likewise. Fr infrmatin regarding ermissin(s), write t: Rights and Permissins Deartment, Pearsn Educatin, Inc., Uer Saddle River, NJ 7458.

8 Mdern Cntrl Systems, Eleventh Editin, by Richard C Drf and Rbert H. Bish. ISBN: Pearsn Educatin, Inc., Uer Saddle River, NJ. All rights reserved. Aendix E An Intrductin t Matrix Algebra A = C - - The characteristic equatin is fund as fllws: 4S. - (E.47) (E.48) The rts f the characteristic equatin are l, l, l. When l l, we find the first characteristic vectr frm the equatin Ax l x, (E.49) T and we have x k - 4, where k is an arbitrary cnstant usually chsen equal t. Similarly, we find and l - det C - E.7 THE CALCULUS OF MATRICES - l - The derivative f a matrix A A(t) is defined as da d t>dt dt At = C da n t>dt - -4S = -l + l + l - =. l + x T -4, x T = -4. da t>dt da n t>dt (E.5) (E.5) That is, the derivative f a matrix is simly the derivative f each element a ij (t) f the matrix. The matrix exnential functin is defined as the wer series ex A = e A = I + A! + A! + + Ak k! + = a (E.5) where A AA, and, similarly, A k imlies A multilied k times. This series can be shwn t be cnvergent fr all square matrices. Als a matrix exnential that is a functin f time is defined as da n t>dt S. da nn t>dt k= A k k!, e At = a k= A k t k k!. If we differentiate with resect t time, then we have d dt eat = Ae At. (E.5) (E.54) This material is rtected by Cyright and written ermissin shuld be btained frm the ublisher rir t any rhibited rerductin, strage in a retrieval system, r transmissin in any frm r by any means, electrnic, mechanical, htcying, recrding, r likewise. Fr infrmatin regarding ermissin(s), write t: Rights and Permissins Deartment, Pearsn Educatin, Inc., Uer Saddle River, NJ 7458.

9 Mdern Cntrl Systems, Eleventh Editin, by Richard C Drf and Rbert H. Bish. ISBN: Pearsn Educatin, Inc., Uer Saddle River, NJ. All rights reserved. Sectin E.7 The Calculus f Matrices Therefre fr a differential equatin (E.55) we might stulate a slutin x e At c fc, where the matrix f is f e At, and c is an unknwn clumn vectr. Then we have r dx dt = Ax, dx dt = Ax, Ae At Ae At, (E.56) (E.57) and we have in fact satisfied the relatinshi, Eq. (E.55). Then the value f c is simly x(), the initial value f x, because when t, we have x() c. Therefre the slutin t Eq. (E.55) is x(t) e At x(). (E.58) This material is rtected by Cyright and written ermissin shuld be btained frm the ublisher rir t any rhibited rerductin, strage in a retrieval system, r transmissin in any frm r by any means, electrnic, mechanical, htcying, recrding, r likewise. Fr infrmatin regarding ermissin(s), write t: Rights and Permissins Deartment, Pearsn Educatin, Inc., Uer Saddle River, NJ 7458.

10 Mdern Cntrl Systems, Eleventh Editin, by Richard C Drf and Rbert H. Bish. ISBN: Pearsn Educatin, Inc., Uer Saddle River, NJ. All rights reserved. This material is rtected by Cyright and written ermissin shuld be btained frm the ublisher rir t any rhibited rerductin, strage in a retrieval system, r transmissin in any frm r by any means, electrnic, mechanical, htcying, recrding, r likewise. Fr infrmatin regarding ermissin(s), write t: Rights and Permissins Deartment, Pearsn Educatin, Inc., Uer Saddle River, NJ 7458.