( CIP).:,3.7 ISBN 7 568 383 3.......... TB CIP (3) 334 3 37 ( ) 64536 www.hdlgpress.com.c 7879 6 9.75 479 3 7 3 7 45 ISBN 7 568 383 3O78 : 3.
995,.,.,.,. :,,,,.. :,,,,,,.,,,,.,,.
,,.,,,.,,,.,,,,.,.,,, ;,,.,,, ;,,,.,.,,,.,,.,,,,. : liujiapig6@63.com 3 6
( 3 ). ( 3 ). ( 5 ).3 ().4 ().5 (6) (3) (6). (6). (8).3 (9).4 (34).5 (38) (4) 3 (44) 3. (44) 3. (48) 3.3 (49) (53) 4 (56) 4. (56) 4. (6) 4.3 (67) 4.4 (7) 4.5 (75) (8) 5 (83) 5. (83) 5. (87) 5.3 (9) 5.4 (93) 5.5 (99) (3)
(6) (5). (5). (7).3 ().4 (3).5 (7).6 (3).7 (36).8 (39) (4) (44). (44). (45).3 (53).4 (55).5 (64) (7) 3 (73) 3. (73) 3. (78) 3.3 (8) 3.4 (86) 3.5 (87) (95) 4 (98) 4. (98) 4. (5) 4.3 (9) 4.4 () 4.5 (7) (5) 5 (9) 5. (9) 5. (33) (4)
3 6 (4) 6. (4) 6. (43) 6.3 (44) 6.4 (45) 6.5 (5) 6.6 (5) 6.7 (54) (57) 7 (6) 7. (6) 7. (63) 7.3 (7) 7.4 (77) 7.5 (8) (83) (86) (95) (37)
,.,.... m m a a a a a a a m a m a m m ( ). A, B,. a a a A = a a a a m a m a m A i j a ij A ( i, j).a ij i, j, A [ a ij ] m A = [ a ij ], m A A m.,.,,... A a a a a a a a a a a, a,, a A,. a a a B =. a a a
4 a a a A = a a. a = = diag[,,, ]. a. a ( ). a, a =,, I E. A = [ a, A = [ a, a,, a ]. a a ],. b B = b, m. b m, O....3 IT ( : ): WorkPad Tablet PC NC a a a 3 a a a 3 A = a a a 3 a a a 3 a ij i j.
5 b b B = b b b 3 b 3 b i i, b i i ( i =,, 3). a x + a x + + a x = b a x + a x + + a x = b a m x + a m x + + a m x = b m m a a a A = a a a a m a m a m., m( + ) a a a b A = a a a b a m a m a m b m., m x b x = x b = b x b m.,. m A = [ a ij ]B = [ b ij ],, A B, A = B... a i j = b ij ( i =,,, m, j =,,, ) A = [ aij ] m A A,
6 A = A= [ a ij ] m A. A = O; A = A (A m, ) : () () A = (A) ; () (+ ) A = A + A ; 3 m A = [ a ij ], B = [ b ij ], A B A + B,. A + B = [ a ij + b ij ] m (ABC m, ) : () A + B = B + A ; () ( A + B) + C = A + ( B + C) ; (3) ( A + B) = A + B.,. A - B = A + ( - ) B = [ a ij - b ij ] m,. A = 3 4 5 6, B = - 3 A = 3 B =.. 3 4 6 4 5 6 = 8 3 3 3( - ) 33 3 3 A - 3 B = - 6 4-6 - ( - 3) 8-9 - 3-6, A - 3 B. = 6-3 9 3 6 = - 4 4 9-7 6,, a b + a b + a 3 b 3, a i b + a i b + a i3 b 3 i ( i =, )., : C = a b + a b + a 3 b 3 a b + a b + a 3 b 3 a b + a b + a 3 b 3 a b + a b + a 3 b 3,. 4 A = [ a ij ] ms, B = [ b i j ] s, A B m C = [ c ij ], C = AB.
7, s c ij = a i b j + a i b j + + a is b sj = a ik b kj k = ( i =,,, m; j =,,, ), s s, b j [ ai, ai,, ais ] b j s = ai b j + ai b j + + ais bsj = aik bkj = cij k = b sj C ( i, j) ci j A i B j ( ).. A = 4-3 - - 3, B = - 3, AB BA. A 3, B 3, A B, AB, AB = 4-3 - - 3 B A, BA. 3 A = AB = - 3 =, B = BA = 4 + ( - ) + ( - 3) 3 ( - ) + ( - ) ( - ) + 33 = - 9 =, AB BA. = 4 A = - - B = - - AB BA. AB = BA = - - - - - - = - - = - -
8, AB BA ; 3, ABBA ; 4, ABBA,.,,, ABBA.AB A B, BA A B.AB = BA, AB. 4, AO, BO, AB = O,,, AB = O A = O B = O ;, AO AB = AC, B = C. () : () ( AB) C = A( BC ) ; () ( AB) = (A) B = A(B), ( ) ; (3) A( B + C) = AB + AC, ( B + C) A = BA + CA ; (4) Im Am = Am = Am I. A( B + C) = AB + AC. A m s = [ a ij ], B s = [ b ij ], C s = [ c ij ], A( B + C) = M = [ m ij ] m, AB + AC = N = [ ij ] m A( B + C) = AB + AC.., s m ij = a ik ( b kj k = s s + c kj ) = a ik b kj + a ik c kj = ij k = k = 5. A = [ a ij ], B = [ b ij ], i < j a ij = b i j =. C = AB = [ cij ], c ij = k = a ik b kj = j - k = a ik b kj + i < j, b kj =, a ik =, c ij =, C = AB. k = j a ik b kj,.a, k, A k A, A k = AAA ( A, k l ) : () A k A l = A k + l ; () ( A k ) l = A kl. k k, A B,, ( AB) k = A k B k. 6 cos- si = si cos cos - si si cos. =,. = k cos- si = si cos k cos k- si k si k cos k
= k +,. cos- si si cos k +, = = = = cos - si si cos k cos- si si cos cos k - si kcos- si si k cos k si cos cos kcos- si k si- cos k si- si k cos si k cos+ cos k si- si k si+ cos k cos cos( k + ) - si( k + ) si( k + ) cos ( k + ) x y = cos- si si cos ( x, y) ( x, y ). 6,,. A T..3 5 A, A, A. x y 9 A = a a a a a a, A T = a a a m a a a m a m a m a m a a a m () () ( A T ) T = A ; () ( A + B) T = A T + B T ; (3) (A) T = A T, ; (4) ( AB) T = B T A T. (4). A = [ aij ] ms, B = [ bij ] s, ( AB ) T ( i, j) AB ( j, i), A j B i, B T i A T j, B T A T ( i, j), ( AB) T = B T A T 7 A = - 3 AB =, B = - 4 7-3 - 4 7, ( AB) T. - = 7 5 4 9,
( AB) T = 4 ( AB ) T = B T A T = - 7 7 4-5 9-7 4 3 = - 5 9. A T = A, a ij = a ji ( i, j =,,, ), A. A T = - A, a ij = - a ji ( i, j =,,, ), A.. 8. A, B, C,, A = B + C, A T = B T + C T,, A T = B - C,, B = ( A + AT ), C = ( A - AT ), BC, A = ( A + AT ) + ( A - AT ).3.3. x, x,..., x y, y,..., y y = a x + a x + + a x y = a x + a x + + a x (.3.) y = a x + a x + + a x y = Ax y a a a x y = y, A = a a a, x = x y a a a x
(.3.) x = b y + b y + + b y x = b y + b y + + b y (.3.) x = b y + b y + + b y, y, y,, y x, x,, x, (.3.) (.3.), x = By, y = Ax = A( By) = ABy, AB, AB = I, I. x = By = B( Ax) = BAx BA, BA = I,. A, B, AB = BA = I, A, B A, A - = B. A =, A B, A - = B, B A, B - = A. - 3, A - =..3. - 3 A, A. BC A, AB = BA = I AC = CA = I, A. B = BI = B ( AC) = ( BA) C = IC = C A, AB = I, BA = I. A, AA - = A - A = I, AB = I, BA = I( BA) = ( A - A ) ( BA) = A - ( AB ) A = A - IA = A - A = I 3 A, A -, ( A - ) - = A. 4 A,, A, (A) - = A -.,,. 5 AB, AB, ( AB) - = B - A -. A B, A - B -, ( AB ) ( B - A - ) = A( BB - ) A - = AIA - = AA - = I ( B - A - ) ( AB ) = B( A - A) B - = BIB - = BB - = I AB, ( AB) - = B - A -..
, 5, A, A,, A k ( A A A k ) - = A - k A - k - A - 6 A, A T, ( A T ) - = ( A - ) T., ( A T ) ( A - ) T = ( A - A) T = I T = I = ( AA - ) T = ( A - ) T ( A T ), ( A T ) - = ( A - ) T. A -. A A + A - I = O, A, A -. A = A + I + A - I = O A( A + I) = I, ( A + I) A = I,. A A + I = A + I A = I AB A + B, A - + B -,, A - + B - = A - I + IB - = A - BB - + A - AB - = A - ( B + A) B - = A - ( A + B) B - 5, A - + B -, ( A - + B - ) - = B( A + B) - A A - + B - = B - + A - = B - I + IA - = B - ( A + B) A -, ( A - + B - ) - = A( A + B) - B,,. I, I..4.4. A, A, A,. 43 6-4 3 A = - 5-3, :
3 6-4 6-4 6-4 ( i) 3 -, (ii) 3 -, (iii) 3 -. 5-3 5-3 5-3 (i)a = A A A A, A = 6-4 3, A =, A = - 5, A =, : () A = A A r,, A = A s A sr A A r - 3 ; A s A sr () A B,, A A r A =, B = A s A sr A ij B ij, B B s B r B sr A = A + B A r + B r As + Bs Asr + Bsr (3) A ml, B l, A A t A =, B = A s A st B B r B t B tr A i, A i,, A it B j, B j,, B tj, AB = C C r C ij C s C sr t = A ik B kj ( i =,, s; j =,, r) k = A A r A T A T s (4) A =, A A s A sr T = A T r A T s r. A =, B = - -, AB A T. - -
4 A = -, B = A = - AB = A T = I A I A O I - - - = I A I B =.4. O I T =,, AB O I, B = I + OB A I + IB = I T A T O T I T = - - - I O = - = () A, A, Ai ( i =,,, s) ( ), A A I B A = A., A i ( i =,,, s), A - A s A - = A - A - s A A - s B = A, B - = A -. A s A - () m A m T A = T T m
5 T i = [ b i, b i,, b i ]A i. (3) m A A = [,,, ] a j j = a j A j. a mj A = [,,, ], AA T A T A. T AA T = [,,, ] T = T T T + T + + A T A = T T [,,, ] = T T T T T T T T T T A T A = I,,,,. 3 x + x + + x = b x + x + + x = b m x + m x + + m x Ax = b = b m x b A = [ a ij ] m, x = x, b = b A m, Ax = b Ax = T T x x = T m b b a i x + a i x + + a i x = b i b m b m
6 T i x = b i ( i =,,, m) A, Ax = b x [,,, ] x = b x x + x + + x = b Ax = b b a, a,, a. 4 A B, AB = BA = I.B I, A[ b, b,, b ] = [ e, e,, e ] Ab i = e i ( i =,,, ) e i I i, e i = [,,,,] T. A = O. A B b i Ax = e i ( i =,,, ). 5 : A m, x, Ax =.. x, x = ei = [,,,, ] T ( i =,,, ) ( ei i ), A A = [,, i,, ], Ax =, Ae i = [,, i,, ] = i = A i, i, A, A = O..5,.,. x + x - x 3 = 3 x + x = - - x - x - x 3 =
7 : () ; () ; (3) k ( k ).,,,.,,., A = - 3 - - - - ( A ).,..5. ( 3 ): () () (i, j ( ), r ij ( c ij ) ) ; () ( ) ( i () r i () ( c i ( ) ) ) ; (3) () k ( k ) () (i () k j (), r ij ( k) ( c ij ( k) ) ).. A B, A B, AB. : () AA ; () AB, BA ; (3) AB, BC, AC.. I, 3 ( )R ij ( C ij ), R i ( ) ( C i () ), R ij ( k) ( C ij ( k) ), R ij = C ij =
8 R i ( ) = C i () = R ij ( k) = C ji ( k) = k. i j m A, () B, m ( )()A.., A r i ()B, B m,, AB, A =, R i ( ) A = r ( ) i A B B = R i () A T T i T m, B = T T i T m T T i T m = T T i T m, A 3 c ij ( k)b, B m, c ij ( k) A B B = ACij ( k) = B, AB, A = [,,, ],
9, B = [,, i,, k i + j,, ] AC ij ( k) = [,,, ] k = [,, i,, k i + j,, ] = B,,, : r ij, R - i j r i ( ), R - i R ij ( - k).: = R ij ; r i () ( ) = R i ( ) ; r ij ( k)r ij ( - k), R - ij ( k) = C - ij = C ij, C - i () = C i ( ), C - ij ( k) = C ij ( - k).5.. m A,, m : N = Ir O O O, m A, R,, R l, C,, C s, R l R AC C s = I r O O O (.5.) r A mi ( m, ), r =, I. A. N = I r O O O.A = [ a ij ] m, A = O, (.5.),. AO, A., a. - a i a ; - a j a ; a ;,,, A
A ( m - )( - ). O T O A A, ( m - r)( - r)a r,. (.5.),, m A, m P Q, A = P I r O O O r A mi( m, ). (.5.) A. A = - 4-3 A = - 4-3 r ( 5 ) r ( ) r - 3 c 3 ( ),. r ( - ) - 3-4 5 - - c ( ) - 3 = [ I O] Q (.5.) R () R ( 5 ) R ( - ) R AC 3 () C 3 () = [ I O], A = R () R ( 5 ) R ( - ) R - [ I O] ( C 3 () C 3 () ) - = R - R - ( - ) R - ( 5 ) R - () [ I O] C - 3 () C - 3 (), = R R () R (5) R ( - ) [ I O] C 3 ( - ) C 3 ( - ) = P[ I O] Q P = R R () R (5) R ( - ) I = R R () R (5) = R R () - 5 Q = I 3 C 3 ( - ) C 3 ( - ) = = R - - = - C 3 ( - ) = - - - () : [ I r O], I r O, I r, O. (),,,
. A = - 4-3 = - 4-3,. = [ I O] Q 3 A A. A, A, A, R,, R l, C,, C s, R l R AC C s = Ir O O O (.5.) A,, r =., (.5.) A. A = R - R - l C - s C - (.5.3) m AB : m P Q, PAQ = B. A A r, A AI. c, AI.. A, [ AI ] [ IB], B = A -., R, RA = I, R = A -, B = R = A -. [ IB] = R[ AI] = [ RAR] A = [ AI] = - 3 - - -, A -. - 3 - - - r 3 ( 5 ) r 3 - - 3-5 r ( - 3 ) r 3 ( ) r 3 ( - ) r 3 ( ) r 3 ( 3) - - 5 3-3 - 3 5 6-6 - - 4 - - 5-4 - 5
r ( - ) - 6-6 A - = 5-4 - - 4-5 - 5 6-3 - 3 - - - 5 Ax = b, A, A -, A -, x = A - b,.,, A, b = [, -,] T - 5-3 x = A - b = 6-3 - 3 - = - - - 5-3,, A - A - [ AB] = [ IA - B], [ AB], A I, B A - B. 3 AX = B, A = 3 3 4 3 A, X = A - B. 3 5 [ AB] = 3 3 4 3 4 3 r ( ) r 3 ( - ) r ( - ) r 3 ( - ) r ( - ), B = r 3 ( - 3 ) 5 3 4 3 - - 4 - - 5 - - 9 - - - 3 3 - - 3 3 X = 3 - - 3 3 3 5 - - 5 - - 9 - - 6 - - r 3 ( - ) r 3 ( - 5 ) B. 3-4 6 - - - 3
3 X = A - B. YA = C Y = CA -, A C, c A C Y = CA -., YA = C, A T Y T = C T, [ A T C T ], Y T = ( A T ) - C T, Y. 4 AXB = D, A = I CA - [ A T C T r ][ I( A T ) - C T ], B =, D =, A B, AXB = D, R X = R - DC - 3 () = R DC 3 ( - ) = 3 3 3 3 3 3 XC 3 () = D, C 3 ( - ) = 3-3. A =. 3 - -.3. () [,,3] 3 x y u u z - 8, B = x ; () - 3 X + 3,x, y, z, u. - - 3 - = O,X. [, - 3] ; (3) - 3 4-3 - 3 5-4 ; a a a 3 x (4) [ x, x, x 3 ].4 A = a a a 3 a 3 a 3 a 33 x. x 3, A..5..6 A = 3, B = ;: () AB = BA? () ( A + B) = A + AB + B?
4 (3) ( A + B) ( A - B) = A - B?.7. () A = O,A = O ; () A = A,A = O A = I; (3) AX = AY,AO,X = Y..8. () A = (3) A = 3 [,, 3 ],A4 ; () A =,A ( ). 3-3,A 5 ;.9 f( x) = x 3-3 x + 3 x +,f( A),f( A) = A 3-3 A + 3 A + I, A = - -. A =, f( A). 3 - - 4 -, B = - 6,AB, BA, B T A A.. A (trace), tr( A),tr( A) = a ii i =, : tr( AB) = tr( BA) ; tr( AA T ) =. A = I - T T, I,, A A = I. i = j = a ij.b.3 A, B, : () A ; () AB - BA ; (3) AB AB = BA..4 : () A m, m x, x T A = A = O; () A m,x m, y,x T Ay = A = O..5 A = - - - - - - - - - - - -,A A - ( )..6 A A + A - 3 I = O,A -, ( A + I) -, ( A + 4 I) -..7 X. () A - XA = 6 A + XA, A = () AX - X + I = A, A = 3 4. 7 ;
5.8 AB A + B = AB. () A - I,AB = BA ; () B =.9 A = - 3 4,A -... () - 3 4 ; (). : () (3) X 5 4-6 3 X = - - 4 7 - ; () = 4 3-3. A = [ a ij ], P = ; (3). - 3 - - 4 6 - - X, P = =.,A. - 4 3 - -, P AP, P P A, AP P. r ij.3 A, AB.() B ;() AB - ; (3) A - i j B -..4 A = - 4 - - 8-6 3-3. ;
. (Cramer)....,.,,,.,43 4,435 5,, ( - )( - ) =!,, ;.,,,,. 43, 43 4 3, 4.435 7. j j j ( j j j ).,., 43 ;435... 3 A = a a a a a a, a a a a a a a a a a a a A ( ), A ( determiat), A det A,, a j a j a j, j j j,,,,!, : j j j, ; j j j,.
7 A = a a a a a a = ( - ) ( j j j ) j j j a j a j a j (..) a a a. j j j a = a,.,,,,. A = a a a ( ) A. a a a a,, j =,, a, a,, j,. j =, j =.,,, a a a,.,,,,. a a a A = = a a a (..) A a a a,, = diag[,,, ], =., I, I =. A = a a, - a a -, a -, - a -, A., (..), A a a a, - a A a a, - a, = ( - ) ( ( - ) ) a a, - a = ( - ) ( - ) + + + a a, - a = ( - ) ( - ) a a, - a 3 R i j (C ij )-. R ij a a a ij a ji a =, C ij = R ij = ( - ) ( ji) = ( - ) j - i - = - 3,
8 R ij = C ij = - ; R i ( ) = C i () = ; R ij ( k) = C ij ( k) = (..), a a a a = a a - a a (..3) a a a 3 a a a 3 = a a a 33 + a a 3 a 3 + a 3 a a 3 - a 3 a a 3 - a a a 33 - a a 3 a 3 a3 a3 a33 a a 3 a a 3 a a = a - a + a 3 (..4) a 3 a 33 a 3 a 33 a 3 a 3,,.!,,! (! = 3688),... a a j a a i a ij a i a a j a a ij i j, ( - ) - a ij, M ij.( - ) i + j M ij a i j, A ij, A ij = ( - ) i + j M ij, (..4) a a a 3 a a a 3 = a a 3 a 3 a 33 a a 3 a a 3 a a a 3 a 33 - a a 3 a 33 + a 3 a 3 a 3 : = a M - a M + a 3 M 3 = a A + a A + a 3 A 3, i, a a a a a a a a a = a i A i + a i A i + + a i A i = a ik A ik (..) k =
9 A ij a ij ( i, j =,,, )., j, a a a a a a = a j A j + a j A j + + a j A j = a kj A kj (..) k = a a a A ij a ij ( i, j =,,, )., (),. 4 A = 3 3 4 - - 5 6 - (..) (..) A = 3( - ) + 5 4-6 - 3 A =. = 3( - ) + 5 6 - = 35( - ) + ( - ) = - 3 3 5 -. (..) (..),, A = ( - ) + + 3 ( - + ) - + 5( - ) + 3 - A = ( - ) + + ( - ) 3 5 + + ( - )( - + 3 5 )3 A = ( - ) + 5 - = = + + = = + + =,,..3,.. A A T, A T = A. A T, A,.
3,, ;. ( ),...3, -,, i j, k ( ki, j), A = a k A k + a k A k + + a k A k = a ks ( - ) k + s M k s s = M ks ( s =,,, ) -,, M ks, A =. 3 ( ),, ( )( ). a a j a a i + b i a ij + b ij a i + b i a a j a = a a j a a i a ij a i + a i a j a a a j a b i b ij b i a a j a, i,. 4 ( )( ) : () A ( ) B, B = A. () A ( )k ( ) B, B = A. (3) A ( ) B, B = - A. () B ( ) B = A. () A = [ a,, a i,, a j,, a ], a j k a i ( ij) B, 3 () B = a,, a i + ka j,, a j,, a = a,, a i,, a j,, a + a,, ka j,, a j,, a = A + = A. (), 3,. (3) i, j, ()() c ji ( ) A = a,, a i,, a j,, a a,, a i + a j,, a j,, a c ij ( - ) a,, ai + j,, - ai,, a c ji ( ) a,, aj,, - ai,, a = - a,, a j,, a i,, a = - B.
3 ( ). ( ),. 3 A, ka = k A. 4 A r - r 3 ( - 4) r 4 ( ) - A = - 5 3-4 - 3 - - 5 3-3 - 5 3-4 - - 8 3-35 = - ( - 8) 5 = 4 - - 5 3-4 3 - - 5 3-3 r 3 ( - ) r 4 ( 5 ) - r 34 ( 3 8 ) - - 5 3-4 - 8-4 6-4 8-3 - 5 3-4 - - 8 5 4 A = a b c d a d c b c d a b c b a d 4 a b c d A r ( ) 3 r ( ) 4 a d c b a + c b + d a + c b + d a + c b + d a + c b + d = 4 a b c d a b c d A r ( - ) r ( - ) 34 d - b b - d c d a b = ( d - b) ( b - d) - c d a b = b - d d - b - 3. A,, A T = - A, A T = - A,, A T = A, 4 3, - A = ( - ) A, A = ( - ) A
3, A = - A, A =. 5 A, ik, jk, a ija kj = j = a i ja ik = i = k,, i, k, a kj = a ij ( j =,,, ki) a a a ija kj = j = a i a i a i a i k a a,,., ( ) ( ). (..)(..) 5 4 5 A = a ija kj = j = a ij A ik = i = A i = k ik A j = k jk 3 3 3 3 5 5 5-4 - 4 5 7 A 4 + A 4 + A 43 A 44 + A 45 (A 4 j ( j =,,5)a 4 j ). 4, = 7 5 A 4 + 5 A 4 + 5 A 43-4 A 44-4 A 45 = 7 4, 5, x = A 4 + A 4 + A 43, y = A 44 + A 45, 3 A 4 + 3 A 4 + 3 A 43 + A 44 + A 45 = 5 x - 4 y = 7 3 x + y = (.3.) (.3.)
33 A4 + A4 + A43 = x =, A44 + A45 = y = - 3 6 L ( + p): L = A O C B = a a O a a c c b b p c p c p b p b pp A r ij ( k), A, A = u L = A B (.3.3) O u u = u u u B c ij ( k), B, B = v O v p v pp = v v v pp, L r ij ( k), p c ij ( k), L L = A O C B = u u u c c v cp cp vp vpp L = u u u v v v pp = A B, A, B, A, B, U = A C O B = A B (.3.4) AB = A B (.3.5)
34 a a a b b b A = a a a, B = b b b, D = A O - I B, 4 a a a D = b b b A O - I B = O AB - I B = ( - ) - I B O AB = ( - ) ( - ) I B O AB = I AB = AB 6, D = A B, AB = A B. 5 4 D = a b c d - b a - d c - c d a - b - d - c b a A, D = A, A,. AA T = A = A A = A A T = AA T a b c d - b a - d c - c d a - b - d - c b a (, s = a + b + c + d ), A = AA T = s s s s a - b - c - d b a d - c c - d a b d c - b a. = s s s = s 4 = ( a + b + c + d ) 4 s D = A = ( a + b + c + d ),, a 4, a 4, D = ( a + b + c + d ).4,,,.. A = a a a ()
35 A ( - a) A = - a a a = a a a = - + ( - ) + = a - a - = a - ( a - ) a a - a a a = - a - ( - a ) = a - ( a - ) = a + ( - ) + ( - ) - + a 3,, A = a a a 3. a a = a a a a = ( a - ) a - a a b b b a b D = b b a,3,,, 4 a + ( - ) b b b b b a + ( - ) b a b a b D = = [ a + ( - ) b] a + ( - ) b b a b a b b a - b = [ a + ( - ) b] = [ a + ( - ) b] ( a - b) - a - b, ( - ), b b b b a b - a - b D = = b a + - a - b
36 ab a = b. ( + b a - b ) b b a - b. a - b 3, D = = b + ( a - b) b b b + a b = b + b a b b b a - b a - b b b b b a b + b b a = [ a + ( - ) b] ( a - b) - a - b b b a b b a + ( a - b) D - = b( a - b) - + ( a - b) D - D = ( a - b) D - + b( a - b) - D - = ( a - b) D - + b( a - b) - D = ( a - b) D + b( a - b) ( - ), ( a - b), ( a - b),, ( a - b) - D = ( a - b) - D + ( - ) b( a - b) - = ( a - b) - a + ( - ) b( a - b) - = [ a + ( - ) b] ( a - b) - 3. 3 (Vadermode) V = x x x x x x x - x - x -.. V = x x = x - x = j < i ( x i - x j ) (.4.) =.(.4.) -,., V :, x,
37 V =, i ( x i x - x x3 - x x - x x ( x - x ) x 3 ( x 3 - x ) x ( x - x ) x - ( x - x ) x - 3 ( x 3 - x ) x - ( x - x ) V = ( x - x ) ( x 3 - x )( x - x ) - x ) ( i =,3,, ), x x 3 x x x 3 x x - x - 3 x - -,, ( x i j < i. 4 + V = ( x - x ) ( x 3 - x )( x - x ) j < i ( x i - x j ) = j < i ( x i - x j ) D + = a ( a - ) ( a - ) a - ( a - ) - ( a - ) - a ( a - ) ( a - ) - x j ), D + +,, ;, D + +,,, D + = a - a - + a =! ( - )!! = ( i - j) j < i + ( a - ) ( a - + ) a 5 x + 4 3 x 8 6 x 3, - x 4 - x 8-7 x 3 =
38, x - x - x + = 4 3 x 4 - x 4 x 8 6 x 3 8-7 x 3 8 - x 3 = ( - ) ( - - ) ( - - ) ( x - ) ( x - ) ( x + ) = x = x = x = -.5, (Cramer)..5. A, AA * = A * A = A I (.5.) A A A A * = A A A = [ A ij ] T A A A A ( A ij a ij ), adj A. (.3.), (.3.) a a a A A A A a AA * a a A A A A = = a a a A A A A, A * A = A I. = A I A, A * = A -. (.5.) AA * = A I, A A * = AA * = A I = A I = A A, A * = A -, A =, A * = A - ( ). A -. A A, A - = A A * (.5.)
39 A, A -, AA - = I, A A - = I =, A. A - = A - (.5.), AA * = A * A = A I, A, A A A * =, (.5.). A A * A = I ( A * ) - = A A (.5.3) A, A ; A.. (.5.),. =, A =?,. 3 3 4 3, B = 3 4 6 7 8 A =, B =, A, B.A -. A = ( - ) + 4 3 =, A = ( - ) + 3 3 = - 3, A 3 = ( - ) + 3 3 4 = A = ( - ) + 3 4 3 = 6, A = ( - ) + 3 3 3 = - 6, A 3 = ( - ) + 3 3 4 = A 3 = ( - ) 3 + 3 = - 4, A 3 = ( - ) 3 + 3 = 5, A 33 = ( - ) 3 + 3 = - A - = A A * = - 3 6-6 - 4 5 - ( A * ) - = A A = T 3 = 3 - - 3 3 3-3 5 - A 3, A = 3, (3 A) - - A - A *.
4 A = 3 A - = 3, A - = A A *, A * = A A -,, (3 A) - - A - A * = 3 A - - 3 A * = 3 A - - 3 A A - = ( 3 - ) A - = ( - 3 )3 A - = - 8 8.5. (.5.)(.3.). 3 A, Ax = b x = [ x, x,, x ] T, A j x j = A j A b A j. ( j =,,, ) Ax = b, x = A - b = A A * b = A A A A A A A A A A b b, b x j = A ( b A j + b A j + + b A j ) = A j A ( j =,,, ),., +. 3 : A x = b,. Ax =.3. Ax = A, ( ) x = x x =, x i, A. x Ax = A =. =.
4 3,? ( - ) x - x + x 3 = - x + ( - - ) x + 4 x 3 = x + 4 x + ( - - ) x 3 =,, D =. D = - - - - - 4 4 - - = - - - - = 4 - - - - 4-4 - 6 - = ( - ) - 4 = ( - ) ( + 5-4) = ( - ) (- ) (+ 7) = - 6 - = = - 7 4 a, b, c. A = a a b b = c c x + ax + a x 3 = x + bx + b x 3 = x + cx + c x 3 =, A = a a b b = A ; A = c c a b c = ( b - a) ( c - a) ( c - b) a b c a b = ; A 3 = c a b c = x = A A =, x = A A =, x3 = A3 A =. : () 43; () 3695; (3) ( - )3. x 3 x x. 4 x 4 x 3. x 3 x 3 x.3 :
4 () (4) 3 4 3 4 5 3 4 5 6 4 5 6 7.4 : () (3) ; () x y x + y y x + y x x + y x y ; (5) + x.5 : () - x + y 7 3 6 9 4-5 8 3 7 5 ; () - y a - b b - c c - a b - c c - a a - b c - a a - b b - c a + a b + b (3) abcd =, c + c d + d 5 3 4 5 6 7 8 9. = ; () a b c d a b c d ; (3) a b c b c a c a b b + c c + a a + b. a + b x a x + b c - 3 3 3 5 9-5 - a + b x a x + b c = ( - x ) a 3 + b 3 x a 3 x + b 3 c 3 =..6 AA T = I,A = -, A + I =. ; ; a b c a b c a 3 b 3 c 3 ;.7 A = [,, 3, 4 ] ; B = [,, 3, 4 ] 4, i = i ( i,,3), 4 = 7 4, A + B = 64,A..8 k,a k = O, A. : () ; () I - A, ( I - A) - = I + A + A + + A k -..9 cos cos cos cos = cos
43. : () (3) 3 3 3 3 3 3 3 3 3 3 3 3 3 + + +. : () D = a c a c ; () a c b d ; + b d b d. ; () D = + a a a a + a a a a + a.. A 3,A = 3, ( 7 A) - - A *..3 3 A 3, A - A T O O A *..4 A = - 3 7 4-3 - 3 7, A * A -..5 f( x), f( ) =, f() = 3, f( - 3) = 8. x + y + z =.6 ax + by + cz = d,a, b, c,,. a x + b y + c z = d.7 3 A = [ a ij ],A ij = a ij a = -,: ( ) A ; () Ax = e, e = [,,] T. x + x + x 3 =.8,,x + x + x 3 = x + x + x 3 =?
,.??, ;. 3. k. 3.. m A, k k ( km, k),, A k, A k A r D, r + ( ), D A, r A, rak( A), r( A)., O r( O) =. B = - 4-5 - 4 8-3 6 - b 34 =, b b 4 = - 5 b 3 b 34 3, r( B) ;, B 4, r( B) 3., B,, B 3 B,,., B 3, r( B ) < 3., r( B) =. : () A, r( A ) = ; () A k, r( A) k ; A k +, (3) m A, (4) A m, r( A) k r( A) = r( A T ) (3..) r( A )mi( m, ) (3..), A, r( A ); A r ( A ) =.
3 45 (),. m A C k mc k k., m A, m,.,. 3.. 3 m : () ( k + ) ( ) k, ( k =,,, m - ). (),.,,. r( A). A = - 3-3 - 5 4-3 A, 3, A 4. 3-3 3-4,, r( A) = 3.,,,., : ( )? ( )?.. m A. m A = [ a ij ]. A,. A,, j j,, a j,, a i j,, A i, A, a j A, a, a,, a j -. j a j, A. A A, A, A., m, k ( km) A k.
46 A. AB, r( A) = r( B). : A B, r( A)r( B). r( A) = r, A r D r. r r ( k) ij i AB A B, B D r D r, D r = D r D r = - D r D r = kd r, D r, r( B)r. A r ji ( k) B, : Dr i ; Dr i j ; D r i j., B D r D r = D r, r( B) r;, ^D r, ^D r Dr = r i + kr j = r i + k r j = Dr + k^dr i A i r, r( B)r; ^D r =, D r = D r, r( B)r. A B, r ( A) r ( B). B A, r( B)r( A)., r( A ) = r( B ).,. A B, A T B T, r( A T ) = r( B T ), r( A) = r( A T ), r( B) = r( B T ), r( A) = r( B)., A B (AB), r( A) = r( B). A m, PQ m, r( PA ) = r( AQ) = r( PAQ) (3..3) () PQ,, PA( AQ PAQ) A ( ),.. : m A r( A) = r(i r ). A = PNQ = P I r O O O Q PQ, r( A) = r( N), r( N ) = r, r( A) = r,,,.
3 47 3 A = 3 4 3-5 4 7 5, r( A). A : A = r ( - 3 ) 3 4 3-5 4 7 5 r 3 ( - ) 5 - - 4 - -, r( A ) =.. A = 4 A = r 3 5 3-5 4 7 3 4 r ( ) r 3 ( - ) a b b b a b b b a 5 - - ( - ),3,,, a b b r b a b b - a a - b b b a a b b b - a a - b ab a + ( - ) b, r( A) = ; ab a + ( - ) b =, r( A) = - ; a = b, r( A) = ; a = b =, r( A) =. c a + ( - ) b b b a - b a - b 3 4-3 5 A =, B = 3, r( AB ). 5 3 AB = 6 9 3 6 6 9 3 6 6 9 3 6, r( AB) =. - 6-7 - 7 3 A = 8, r( A) = 3; r( B) =, (3..3 ) r( AB) = r( B) =.
48 3. m a x + a x + + a x = a x + a x + + a x = (3..) a m x + a m x + + a m x = Ax = (3..) m A = [ a ij ], x = [ x, x,, x ] T, m. (3.. ) x =,.,, ( ),,. A, Ax =, : A m x = r( A) <, - r( A). Ax =, r ( A ) <., r( A) =, A D, D ( ),, r( A ) =, r( A) <. r( A) = r <, A r, - r.,,. - r c, c,, c - r, - r c, c,, c -,,. - r,. r( A ) < ( m = ),. A m x = r( A) =.,,. x - x - x 3 + x 4 = x - x + x3-3 x4 = x - x - x 3 + 3 x 4 = A : r
3 49 r( ) - - - - 3 - - 3 x = c, x 4 = c, r ( - ) r 3 ( - ) - - - 4 - x - x - x 4 = x 3 - x 4 = r 3 ( ) r ( ) - - - x = c + c x = c x 3 = c x4 = c x x x 3 = c = c, ( c, c R) x 4 x + x - x 3 = 3 x + 7 x - 6 x 3 = 4 x + 8 x + x 3 =,., A r( A ) < 3. - A = 3 7-6 4 8 r ( - 3 ) r 3 ( - 4 ) - + 8 r( A) < 3, + 8 = = - 8. - 8, x =., A = - 8. = 3.3 m Ax = b (3.3.) m A = [ a ij ], m( + ) A = [ Ab ], x = [ x, x,, x ] T, b = [ b, b,, b m ] T m.
5,, : A m x = b A A = [ Ab]. Ax = b, r ( A ) = r ( A )., r( A) < r( A ), A =,. r( A ) = r( A). r( A) = r ( A ),, A, r( A) = r( A ) = r ( r), A r, r, - r, - r,. AX = B, r( A) = r( AB).,, r( A ) = r( A) =,,.r( A) = r( A) <, - r, - r c, c,, c - r, - r c, c,, c - r,,, ( ).,, :. A m x = b : () r( A) = r( A ),., r( A) = r( A ) =,. r( A ) = r( A ) <,, - r ( A ) () r( A) < r( A ),. A, A = - 3-3 - 5-3 - 3 x - x + 3 x3 - x4 = 3 x - x + 5 x 3-3 x 4 = x + x + x 3 - x 4 = 3 r ( - 3 ) r 3 ( - ) r ( - ) 3-3 - 5-4 - r( A) =, r( A) = 3,. - 3-5 - 4-5 - 4
3 5, A = r ( - 3 ) r 3 ( - ) 3-5 5-3 - 3 - - 3 - - 3 - - 4 - - 4-3 x - 5 x + 5 x3-3 x4 = x - x + 3 x 3 - x 4 = x - 3 x + x 3 - x 4 = r - 3-3 - 5 5-3 - 3 - r ( ) r 3 ( - ) x - 5 x 3 - x 4 = - x - 4 x 3 = - - 5 - - - 4 - x 3 = c, x 4 = c, x = 5 c + c - x = 4 c - x 3 = c x4 = c x x x 3 x 4 = c 5 4 + c, x = c, x = c, x x x 3 x 4 = c + c 4-5 4 + + - - 4-4, ( c, c R), ( c, c R),,.,, - r( A) ;,,.,.
5 3 x + x + kx 3 = 4 - x + kx + x 3 = k x - x + x3 = - 4 k, () ; ( ) ; ( 3)?., A = [ Ab], A = r 3 k - r ( - ) 3 k 4 - k k - - 4 r 3 r ( ) r 3 ( - ) - - 4 k - 3 k - 4 k - 8 - - 4 k - 8 - ( k - 4) ( k + ) k( k - 4 ) () k4 k-, r( A) = r( A ) = 3, ; () k = -, r( A) =, r( A ) = 3, ; (3) k = 4, r( A) = r( A) = < 3,., - - 4 A 8 3 4 x 3 = c, x = - 3 x 3 x = - x 3 + 4 x x x 3-3 = c - + 4, ( cr ),. k k A = - k = k + k + = ( k + ) (4 - k) - - - k (),.k4 k-, r ( A) = r( A ) = 3, ; () k = -, x + x - x 3 = 4 - x - x + x 3 = x - x + x 3 = - 4
3 53, - 4-4 A = - - - 3-8 - - 4 5 r( A) = < r( A ) = 3,. (3) k = 4, x + x + 4 x 3 = 4 - x + 4 x + x 3 = 6 x - x + x 3 = - 4, 4 4 4 4 4 4 3 A = - 4 6 5 5 4 4 - - 4 - - - 8 x 3 = c, x = - 3 x3 x = - x 3 + 4 x - 3 x = c - + 4, ( cr ) x 3 A, ( k - ) ( k - ), r ( -. )., ( k - ) = k -,,,, k4 k-, - x 3 = - x = ( k - 4) ( k + ) x 3 = k( k - 4), k k +, x + ( k - ) x 3 = 8, x = k + k + 4, x - x + x 3 = - 4, k + k( k + ) k +. 3. r, r? 3.,. - 5 4-3 - - ( ) A = ; ( ) B = 3-3 5 5-3 4.
54 3.3,. ( ) A = - 3 k - k - 3 k - 3-4 k - 6 ; () B = - 3-6 4-3 a 7 - - - 6 - b 3.4 A m, m,r( A) r( A )r( A) +, [ A ] A. a b a b a b. 3.5 A = a b a b a b a b a b a b,r( A) r( A ).. 3.6 A m, B m, ABx = Bx =., r( AB) = r( B). 3.7. 3 4 ( ) A = - - - ; ( ) B = 3-5 4 7 ; - - 5 - (3 ) C = 3 ; ( 4) D = 4-5. 4 3 3.8 ( )?,. x - x + x 3 = x - 3 x 3 = ( ) x + x + x 3 + + x = ; ( ) x ; - x + 5 x 3 = - x + x 3 = x - 4 x + 5 x 3 + 3 x 4 = x - x 3 = - x (3 ) 3 x - 6 x + 4 x 3 + x 4 = ; ( 4) x - 3 x 3 = - 5 x 4 x - 8 x + 7 x 3 + x 4 = x + 4 x = 3 x 3 3.9 a a a 3,. x + a x = a x + a x = a x + a 3 x = a 3 3.. x - x + x 3 + x 4 = ( ) x - x + x 3 - x 4 = - ; ( ) x - x + x 3 + 5 x 4 = 5 x - x + x 3 = - 5 (3 ) x + 5 x - 7 x 3 = 9 ; ( 4) 3 x + x - 6 x 3 = 3. : x + x + x 3 - x 4 = 3 x - x + 3 x 3 - x 4 = ; 3 x - x + 5 x 3-3 x 4 = x - x + 5 x 3 = 5 x + 3 x - x 3 = 4 x - 4 x + 6 x 3 =. 3 x + x + 4 x 3 = 9
3 55 a + a x - x = a x - x 3 = a x 3 - x 4 = a 3 x 4 - x 5 = a 4 x 5 - x = a 5 + a 3 + a 4 + a 5 =.,. 3.?. ( ) (3 ) x + x + x 3 = x + x + x 3 = ; ( ) x + x + x 3 = x + (4 - ) x + 7 = ( - ) x + x + 3 = ; ( 4) x + 5 x + 6 - = x + x + x 3 + x 4 = x - x 3 + x 4 = ; x + 3 x + ( a + ) x 3 + 4 x 4 = b + 3 3 x + 5 x + x 3 + ( a + 8 ) x 4 = 5 - x + x + x 3 = - x - x + x 3 = x + x - x 3 =. - 3.3 B - x =, ( ) ; ( ) B ; (3) B. 3 -
.,,,.,. 4. ( ). m A = [ a ij ], m a j j = a j, ( j =,,, ),,, A. a mj m A m T i = [ a i, a i,, a i ], ( i =,,, m) T, T,, T m A.,. m Ax = b, A = [ Ab]. ( A ), A. x Ax = [,,, ] x = x + x + + x = b A,,,, b. x 4.. () :,,, m, m k, k,, k m, (), k, k,, k m k + k + + k m m. b,,,, m, b = + + + m m, b ( ) ( ).
4 57 b = [, -, ] T, = [,,] T, = [,, ] T, 3 = [,, ] T, b = - + 3, b,, 3, b,, 3. = [ a, a,, a ] T e = [,,, ] T, e = [,,,] T,, e = [,,, ] T, = a e + a e + + a e., b (), x + x + + x m m = b.a m x = b.3.3, b ( ) :,,, m A = [,,, m ] A = [,,, m b]. () :,,, m () : b, b,, b s, ( ) (), ( ) ( ). () (),. b = - 3, = 3 - -, b =,,. - 5 6-5 9, [ x, x ] T, [, ] x x = b., [, b ]: b, b = r ( - 3-5 - 6 - - 5 3-9 ) r 3 ( - 5 ) r 4 ( - ) - - 5-3 x x = - 3 r 3 r 3 ( ) r 4 ( 3 ) r ( - ) r ( - ) - - 5-6 5-5 - 6-3 = x + x = - 3. b,. 3 () :,,, m, k, k,, k m, k + k + + k m m = (4..) () ;, k = k = = km = ( 4.. ),
58 (). e = [,,] T, e = [,,] T, e3 = [,,] T, e, e, e3, = [ a, a, a 3 ] T, k e + k e + k 3 e 3 = [ k, k, k 3 ] T =, k = k = k 3 = e, e, e 3 ; a e + a e + a 3 e 3 - =, e, e, e 3,.,,.,,,,.3. () :,,, m A = [,,, m ], ( ),.3., x + x + + x m m =, Ax = () :,,, m A = [,,, m ] m; r( A) = m. m m,,, m,,, m = ;,,, m,,, m.. = [, -, 7] T, = [, 4,] T, 3 = [ 3, - 6, 3] T,, A = [,, 3 ],, A = 3-4 - 6 7 3 r r ( ) r 3 ( 7 ) - 4-6 9-9 39-39 r ( - ) r ( 9 ) - 4 6-39 - 39 r ( 4 ) r 3 ( - 39 ) - (4..) r( A) = < 3( ),. k, k, k 3, k + k + k 3 3 =, Ak =, k 3 =, k = -, k =, k + k 3 = k - k 3 = k + k + k3 3 = - + + 3 =,, 3 =,, 3. 3-4 - 6 7 3 = 9-9 - 4-6 39-39, ( 4..),,,,. =
4 59., 3,, 3, 4, b b, b, b 3, b 4, = +, b = -, b3 = 3 + 4, b4 = 3-4. k b + k b + k 3 b 3 + k 4 b 4 = (4..3) k b + k b + k 3 b 3 + k 4 b 4 = k ( + ) + k ( - ) + k 3 ( 3 + 4 ) + k 4 ( 3-4 ) = ( k + k ) + ( k - k ) + ( k 3 + k 4 ) 3 + ( k 3 - k 4 ) 4 =,, 3, 4, k + k = k - k = k 3 + k 4 = k 3 - k 4 =, (4..3), b, b, b3, b4 k = k = k 3 = k 4 =.,., B = [ b, b, b 3, b 4 [ b, b, b 3, b 4 ] = [,, 3, 4 ] - - ], A = [,, 3, 4 ], C = - -,,, 3, 4, r( A ) = 4; C = 4, C, r( C ) = 4; 3. r( B) = r( AC) = r( A) = 4, b, b, b 3, b 4 4...... m,,, m, m >,. m,,, m m A = [,,, m ], r( A) mi(, m) = < m,,,, m. 3,,, m,,,, m,,.,,, m, k, k,, k m, k + k + + k m m =,, k + k + + k m m + m + + + =,
6,,,,.. 3 :,,, m,,,,,, m ( ). 4 j = [ a j, a j,, a rj ] T, j = [ a j, a j,, a rj, a r +, j ] T ( j =,,, m).,,, m,,,, m. A = [,,, m ], B = [,,, m ], r( A)r( B),,,, m, r( A) = m, r ( B ) r ( A) = m; B m, r ( B) m; r( B) = m,,,, m. 4 :,,, m,,,, m. 4 = [,,, 3, ] T, = [,,,, - ] T, 3 = [,,, - 3, ] T., k + k + k3 3 =, k k k 3 = 3 k + k - 3 k 3 k - k + k 3 k = k = k 3 =,,, 3. A = [,, 3 ] = r( A) = 3,,, 3. 3-3 - 3,, 3 45 e, e, e 3, e, e, e 3 =, e, e, e 3, 4,, 3. 4 4 = [,,,, ] T, 5 = [,,,,] T,,, 3, 4, 5 = 3-3 -,, 3, 4, 5, 3,, 3. =
4 6 5 a, a,, a k,,, k. k = +,, i = [, a i, a i,, a i ] T ( i =,,, k),,,, +,, a a a +,,, + = a a a + = j < i + ( a i - a j ),,, +. a a a + k > +,,,,, k. k < +,,,, k k,,, k,,,, k = ( a i j < ik - a j ),,, k, 4,,, k. 4..3. 3 () :,,, m ( m ) m -. () m -, s,,, s -, s +,, m,,,, s -, s +,, m, m = + + s - s - + s + s + + + s - s - + + m m, + ( - ) s + s + s + + + m m =,,, s -, ( - ), s +,, m m ( - ), ( ). (), k, k,, k m, k, k,, k m k + k + + k m m =, k s, s = - ( k k + + k s - s - + k s + s + + + k m m ) s s,,, s -, s +,, m., ( ), () m -. = [, ] T, = [, - ] T,, 3 + =,,, =,. 3 :,,, m ( m),,, m i m -. 4 ( ) :,,, m, ( ) :,,, m, b
6 b () :,,, m,. A = [,,, m ], B = [,,, m, b], r( A) r( B). (), r ( A ) = m; ( ), r( B ) < m +. m = r( A) r( B) < m +, r( B) = m. r( A) = r( B) = m, 3.3 Ax = [,,, m ] x = b, b () :,,, m,. b,,, m, 3.3 [,,, m ] x = b, r[,,, m, b ] = r[,,, m ] = m,,,, m,,,, m, b. 6 b ( ) :,,, m, ( ) :,,, m -. ( ) :,,, m -, b. : m (), ( ). m ()., k, k,, k m -, m = k + + k m - m - (4..4) b ( ) :,,, m,,,, m, (4..4) (4..5), b = + + m - m - b = ( + m k ) + + ( m - b ( ) :,,, m -,. + m m (4..5) + m k m - ) m - m ()., ( 4..5 ) m ( b () :,,, m - ),, (4..5 ), m = [ b - - - m m - m - ] 7,, 3,, 3, 4, ( ), 3?? () 4,, 3?? (), 3., 3, 4, 3, 3,,, 3, 4, 3. () 4,, 3., 4,, 3, ( ), 3, 4, 3, 3, 3, 4,, 3, 4, 4,, 3. 4.,,.. (), () r a, a,, ar,
4 63 () ( ) : a, a,, a r ; () () r + ( ( ) r + ), ( ) ( ) ( ). r ().,.., 4., ( ), ( ). A = [ a, a,, a m ]m, r( A ) = r, A r D r, A r. Dr r r B, Dr r( B) = r, 4. B r.a r + r( A ) < r +, 4. A r +. B r A, A r. D r r. r A, A. [ a, a, a 3 ] = 3-4 - 6 7 3 r[ a, a ] =, a, a ; r[ a, a, a 3 ] = a, a, a 3, a, a a, a, a3., r[ a, a 3 ] = r[ a, a 3 ] = a, a 3 a, a 3 a, a, a 3., :..,,?,. [ a,, a ] = A = T, [ b,, b ] = B = T m T, r A B, A B, A T,, T m B T m
64 T,, T m, Ax = Bx =, x a + + x a = x b + + x b =, a,, a b,, b. B A, B b,, b, bi. a,, a b,, b, a,, a a i.,. = [, -, 5, - 3] T, = [4, -, -, 3] T, 3 = [5, 4, - 9, 5] T, 4 = [ -, -, 6, - 5] T,,. 4 A = [,, 3, 4 ] = 4 5 - - - 4-5 - - 9 6-3 3 5-5 A B 4 5 - r 7 4 - A - - 44 66 5 3-45 4 5 - - 3-3 - 3 r( A) = r( B) =,,., b, b B, AB,,,, 3, 4., 3 = - 3 +, 4 = - 3 ( ) ( ), ( ) (). () s,,,, s ; () t,,,, t, st.,,, s (), ( )( ), (),,, t,,,, s,,, t, [,,, t,,, s ] [,,, t,,] r[,,, t,,, s ] = r[,,, t ] = t,, st. r[,,, t,, s ]r[,,, s ] = s c = B
4 65. () ( ) s r,,, rs sr, s = r..,,. ( ) () (), ( ), () (), ( ) ( ). () r, r. ( ) (), ( ) r, () r +, (). 3 () ( ), () ( ), ( )..,, 3 AB m, k, A m, B p, (4..).A r D r, r( ka) = r( A). r( ka ) = r( A ) (4..) r( A + B)r( A) + r( B) (4..) r( AB)mi( r( A), r( B ) ) (4..3) r( A) + r( B) - r( AB) (4..4) ka r k r D r, k (4..).A = [ a,, a ], B = [ b,, b ], r( A ) = s, r ( B ) = t, AB, a i,, a is b j,, b jt. A + B a + b,, a + b, a i,, a is, b j,, b j t.,, s + t, r( A + B)s + t = r( A) + r( B) (4..3).C = AB, C A C = [ c, c,, c p ], A = [ a, a,, a ], B = [ b ij ], C = [ c, c,, c p ] = [ a, a,, a ] b b p b b p
66 c j = b j a + b j a + + b j a C A,, r( C )r( A). C T = B T A T, r( C T )r( B T ), r( C)r( B). (4..3). (4..4), [ 3 ]. A A - A - 3 I = O, r( A - 3 I) + r( A + I) =. A - A - 3 I = O ( A - 3 I) ( A + I) = O, (4..) (4..) (4..4) = r( 4 I) r( 3 I - A) + r( A + I ) = r( A - 3 I ) + r( A + I) + r[ ( A - 3 I ) ( A + I) ] = r( A - 3 I) + r( A + I ) =. 3 ()(),, () (). ()(). r, () () ( ) : a,, a r, ( ) : b,, b r Kr (Kr. ( ) ( ), ( ) ( ), r ) [ b,, b r ] = [ a,, a r ] K r ( ), r[ b,, b r ] = r. 3 (4..3), r( K r ) r, r( K r ) = r. K r r( K r )r[ b,, b r ] = r, [ a,, a r ] = [ b,, b r ] K - r ( )( ), ( )( ). ()() r. ( ) (), () () (, ) (), ( ) (, ), ( ) (, ), ( ) (, ), (, ) r.() r, () ( ) r,, ( )(, ), (, ) ( ). ( ) (, ), (, ) ( ), ( ) (), ( ) ().,,. ( ) ( ) ; ( ) (, ).,, () (),,. = [, ] T = [, ] T,. 4 = [,,, ] T, = [,,, ] T = [,, -, - ] T, = [, -,3, 3] T, 3 = [,,, ] T,,,, 3.,, 3 = +,, 3,,,, 3,,,, 3.
4 67 [,,, ]: [,,, ] = - - 3-3 - - 3 = -, = - + 3, [, ] = [, ],, ; - - 3. [, ] = [, ] - - 3-3 - 3 - - 3 - - 3 =, - - 3 - - 3 -,,,,,,,,,,, 3 A = [, ], B = [,, 3 ],, AX = B BY = A, 3.3, AX = B BY = A r( A) = r( AB )r( B) = r ( BA ), r( A) = r ( B ) = r( AB ). [,,,, 3 ] = - - 3-3 - - 3 - r( A) = r( B) = r( AB) =,,,, 3. - - 3 - - 3-3 4.3 4.3. V, V ( ), V., V V.,
68 : av, bv, a + bv; av, R, av. R 3,. 3 3, 3 3, R 3. 3, R 3., R,. V = x = [, x,, x ] T x,, x R.a = [, a,, a ] T V, b = [, b,, b ] T V, a + b = [, a + b,, a + b ] T V, a = [,a,,a ] T V V = x = [, x,, x ] T x,, x R.a = [, a,, a ] T V, b = [, b, b,, b ] T V, a + b = [, a + b,, a + b ] V 3,.x V = x = +, R = +, x = +, x + x = ( + )+ ( + ) V kx = ( k )+ ( k ) V.,,,, m V = spa(,,, m ). x = + + + m m,,, m R (4.3.) V V, V V, V V. V, V R, R. 4.3. 3 V,,, r, (),,, r ; () V,,, r,,,, r = + + + r r (4.3.),,,, r V,,,, r, ( 4.3.),,, r, r V, dim V = r, V r. V, V,, V. 4 e = [,] T, e = [, ] T = [, ] T, = [, ] T
4 69 = [4,3] T.() e, e R ;, R ; () e, e,. () e, e, e, e, vr, 4. e, e, v, 4. 4 v e, e 3, e, e R., R. () = 4 3 = 4 e + 3 e e, e [ 4,3] T ;, = 4 3 = 3 +, [ 3,] T. R, ( ) ;. (), 4. 4.3 3, V, V.,,,, m V = x = + + + m m,,, m R,,, m,,,, m V,,,, m V., V, V,,, r ( V ), V. 4.3.3 V = spa(,,, r ),. 4,,,,,, V, [,,, ] = [,,, ] P (4.3.3) P, (4.3.3 ), P,,,,,,, P = [,,, ] - [,,, ]. V,,,,,, x = [ x, x,, x ] T, y = [ y, y,, y ] T,. x = Py y = P - = [,,, ] x = [,,, ] y, y = P - x. x = [,,, ] - [,,, ] y = Py 5 = [,, ] T, = [,, ] T, 3 = [,, ] T, x
7 = [6, 5,3] T, = [,, ] T, 3 = [,, ] T.( ),, 3,, 3 R 3 ; ( ),, 3,, 3 P ; (3),, 3 [, -,3],,, 3. (),, 3 =, 3,, 3 R 3.,, 3 = 6 5 3 = 3 3 = -,,, 3, R 3 =,,, 3 R 3. () [,, 3 ] = [,, 3 ] P P = [,, 3 ] - [,, 3 ], [,, 3 ] =, [,, 3 ] - = - - P = [,, 3 ] - [,, 3 ] = (3) = [,, 3 ] - 3 y = P - - 3, - - = [,, 3 ] y, = - - - - 3 6 5 3 = 3-7 = 3 4.4 3.,,, () A m x = r( A) <, - r( A). () A m x = b A A, r( A ) = r( A) = ; r( A ) = r ( A ) <, - r( A ).,.. 4.4.
4 7 a x + a x + + a x = a x + a x + + a x = a m x + a m x + + a m x = (4.4.) Ax = (4.4.) m A = [ a ij ], x = [ x, x,, x ] T m.. A = 3 4 - A A = 3 4 - x 3 = c, x 4 = c,, Ax =. - - - x = c - + c - 4 -, ( c, c R) = [, -,, ] T, = [ - 4,,, ] T, spa(, ). :, - 4 -,,, t Ax =, c + c + + c t t Ax =, c, c,, c t. A ( c + c + + c t t ) = c A + c A + + c t A t =, c + c + + ct t Ax =., Ax = xy, x + y x ( R).Ax = N( A), ( 4.4.)., N ( A ),,, t, N( A ) = x x = c + + c t t, c, c,, c t R. Ax =, r( A) = r <, Ax = dim N( A) = - r. A r, A r, A
7 b b, - r B = B, b r b r, - r x = - b x r + - - b, - r x (4.4.3) x r = - b r x r + - - b r, - r x A B, ( 4.4. ) (4.4.3 ). (4.4.3), x r +,, x, x,, x r, ( 4.4.3 ) ( ), ( 4.4.).x r +,, x (4.4.3) x x r = x r + x r + x = - b,, - b r - b,,,, - b r - r : - b, - r - b r, - r (4.4.3) ( ( 4.4. ) ) - r : - b - b - b, - r = - b r, = - b r,, - r =,,, - r N( A)., [ x r +, x r +,, x ] T - b r, - r - r - r,,,, r - r,,, - r., ( 4.4. ) = [,,, r, r +,, ] T,,, - r.,
4 73 = - r + - r + - - - r = [ d, d,, d r,,, ] T,,, - r, (4.4.), (4.4. ), (4.4.3) d = d = = d r =, =, = r + + r + + + - r (4.4.),,, - r. - r.,,, - r N( A). N( A),,, - r, ( ),,, - r ; ( ) Ax =,,, - r. (4.4.) k,, k - x = k + k + + k - r - r r.( 4.4. )., N( A) = x = k + k + + k - r - r k,, k - r R = spa(,,, - r ) (4.4.4) r( A ) =, = 4, - r( A) =. x 3 = x 4 x x =, x x 4 = - -, - x - ; x 3 x 4 = x = c x 3 = - - - + c ; x x - 4 x 4 = x = - 4, ( c, c R) x x 3 =, x = c - - + c - ( c, c R) -,,, - r( A),. -, - 4
74 4.4. m Ax = b (4.4.5) Ax = ( ). Ax = b :,,, t Ax = b, = c + c + + c t t,, Ax = ;, Ax = b. c + c + + c t = c + c + + c t = A= A( c + c + + c t t ) = c = ( c + c + + c t ) b A + c A + + c t A t c + c + + c t =, A= ; c + c + + c t =, A= b., Ax = b, - Ax =, + Ax = b. 3 Ax =, Ax = b, x = + Ax = b. Ax = A(+ ) = A+ A= + b = b, x = + Ax = b. 3. = c + c + + c - r - r, Ax = b x = c + c + + c - r - r + ( c, c,, c - r R).,, { x x = c + c + + c - r - r +, c, c,, c - r R},,,, - r Ax =, Ax = b. x g = x h + x p x g Ax = b, x h Ax =, x p Ax = b. A = 3 4 -, b = Ax = x h = c - 3 + c, Ax = b. - 4 A b Ax = b x p ( c, c R) = [,,, ] T, Ax = b
4 75 x g = x h + x p = c - + c - 4 + ( c, c R),.. 3 3,,, 3, = [,,3, 4] T, + 3 = [,3, 4,5] T,. A, 4, Ax = dim N( A) = - r( A) = 4-3 = N( A ), x p, Ax = b., x p =., + 3 -. + 3 - = 3 4 5 N( A ), x = + c( + 3 - ) = - 3 4 3 4 = + c - - - 3 - - - 3 ( cr) 3,,. 4.5 4.5. x = [ x, x,, x ] T, y = [ y, y,, y ] T, < x, y > = x y + x y + + x y x y.,, x y, < x, y > = x T y ( x, y, z, ) : () < x, y > = < y, x > = y T x ; () < x, y > = < x, y > ;
76 ). (3) < x + y, z > = < x, z > + < y, z >. x= < x, x > = x + x + + x, x x ( = ;, =, ( ). :. x, x> ; x =, x= ;. x= x; 3. x + yx+ y. < x, y > < x, x > < y, y >,, < x, y > = x ycos. x, y, x y. = arccos < x, y >, [, ] x y < x, y > =, x y, x y, xy., x =, x. r( A T A ) = r( A). A m, x. Ax = A T Ax =. x Ax =, A T A T ( Ax) =, ( A T A) x = ; x ( A T A) x =, x T Ax=,, Ax =. x T ( A T A ) x = x T =, ( Ax) T Ax =,, Ax = A T Ax =,, r( A T A) = r( A). 4.5....,,, r,,,, r,,, r + + + r r = T i ( i =,,, r),, T i i = i, T i i = i, i = ( i =,,, r).,,, r.
4 77 3 R 3 = [,, ] T, = [, -, ] T, 3,,, 3. A = Ax =, T T = -, 3 T 3 =, T 3 =, - x x x3 =, A = - - x - = - x, x 3 = x. 3 = -., :,,. 3,,, r V( V R ),,,, r,,,,, r V., R, e, e,, e R, ei I i ( i =,,, ), e, e,, e R. R 3, = 3., = -, 3 = 6,,, r V, V.,,, r,,,, r,,, r.,,,, r.. ( Schmidt ),,, r V.,,, r V. i =,,, r = - = <, > <, > r - < j, r > r = r - j = < j, j > j -
78 i =,,, r V., i i [,,, r ] = [,,, r ] P P, P,,,, r,,, r,,, r. 3 A = [,, 3 ] =, R 4. = = [,,, ] T, = - <, > <, > =, 3 = 3 - <, 3 > <, > - = R 4 =, = <, 3 > <, > = =, A - -, 3 = - - 4-3 = = 3 = - 3 4 3 4-3 4, 4 T 4 =, T 4 =, T 3 4 =, - - x + x 3 = - x + x + x 3 + x 4 = x + x - x 3 + x 4 = - - 4 3 4-4 = [, -,, ] T, 4 = [, -,, ] T, R 4
4 79,, 3, 4. 4 = [,,] T,, 3,,, 3.,, 3 T x =, =,. x + x + x 3 = - <, > =, <, > =, =, = - =, 3 = - <, > <, > -, 3 = = - - -, =,, :, x + x + x 3 = x = c + c = [ c, c, - c - c ] T ( c, c R). =. 3, x,, T x = x 3. T x =, c = - c, c =, x = [, -,] T, 3 = [, -,] T,.. 4 A A. A, T T - - A T A = I ( A - = A T ) [,,, ] = I = T T i j = i = j ij = - - ( i, j =,,, ) : A A.
8 A T A = I AA T = I, A., A () R 3 R 4. :.,, 3, 4 Q = [,, 3, 4 ] = () ; () ; (3) ; (4). - -,. - 4.. () = 3, = ; () = -, = 3, 3 = - ; (3) =, = 3 4, 3 = 5 6. 4. = [,,] T,,. () = [,3,] T, = [, -,] T, 3 = [7,5,] T ; () = [,,] T, = [,3, ] T, 3 = [,,] T. 4.3 = [,,3] T, = [,,] T, 3 = [3,4, a] T, a,,, 3 ; a,,, 3? 4.4 4,, 3, 4,3, : 4 k, k, k 3, k 4, k + k + k 3 3 + k 4 4 =. 4.5 = [,,] T, = [,-,] T, 3 = [,3,+ 3] T,= [,,- ] T, : (),, 3,? (),, 3,? (3),, 3? 4.6.
4 8 (),,, m,, 3, m. ( ) +, +,, m + m,,, m,,,, m. (3 ),,, m,,, m, +, +,, m + m. (4),,, m,,,, m+. 4.7, :,,, m i,,, i -. 4.8,. () =, =, 3 = 4.9., 4 = ; () = () = [, -,4] T, = [,,3 ] T, 3 = [,, - ] T ; () = [,,] T, = [, -,] T. 4. e, e,, e,,,,,,,, e i ( i =,,, ) i. 4.,, = -, = +, 3 = - + 3, :,, 3. 4. A m, B m, () m >, AB = ; ( ) m < AB = I, r( B) = m. 4.3,,,, = +, = + 3,, = +,,,,. -, = -, 3 = 4.4 A m, r( A) =,A = T. 4.5 A = I A, A, r( A - I ) + r( A + I) = 4.6 V = { x = [ x, x, x 3 ] T x + x + x 3 = }, V = { x = [ x, x, x 3 ] T x + x + x 3 = }, R 3, R 3,? 4.7,, 3 V = spa (,, 3 ) V dim V, = [, -,4] T, = [,,3] T, 3 = [,, - ] T. 4.8 = [,, 3, - ] T, = [ 3, -,, ] T, 3 = [, 3, 4, - ] T, 4 = [4, - 3,,] T, (),, 3, 4 ; () ; (3 )R 4. 4.9 = [5,,,] T, = [,,,] T, 3 = [,,8,5 ] T, 4 = [,, 3,] T R 4, = [,,,] T, = [,,,] T, 3 = [,,,] T, 4 = [,,, ] T, (),, 3, 4,, 3, 4 P ; () = 3 + + 3,, 3, 4. 4...
8 ( ) x + x + x 3 - x 4 = x + x + x 3 - x 4 = x + x + x 3 + x 4 = ; () x + ( - ) x + + x - + x =. 4. AB,AB = O, r( A) + r( B). 4. A lm, B m, r( AB) = r( B). 4.3 A, r( A ) = r( A + ). 4.4, = [,,,4 ] T, = [ -,,, - 3] T. 4.5 A -, A * A, A * x =,. 4.6. x + 3 x 3 + x 4 = x - 3 x + x 4 = - x + x + 7 x 3 + x 4 = 5 4 x + x + 4 x 3 = 6 4.7 3, x, x, x 3,, x = [4,3,,,] T, x = [,,,4,] T, x 3 = [,8,,,] T. 4.8 Ax = b r ( A 53 ) =,, + = [,3,] T, + 3 = [,5,] T,. 4.9,,, - r A m x = b - r +, r ( A ) = r, -, -,, - r - Ax =. 4.3 = [,,] T, = [,,] T, 3 = [,,] T. 4.3,, 3, = + + 3, = + + 3, :,,?,,. 4.3 AB,A = - B, : A + B =.
;., ;. 5. 5..,. A, x, Ax = x (5..), A ; x A ( ). (5..) ( A - I ) x = (5..), (5..) A - I = A - I f( ) = A - I = a - a a a a - a a a a - = ( - ) + a - + + a - + a A.A - I = A. A, A. :,. 5... A : () A - I,, A () i,, A - i I x =
84,, t, i x = c + c + + c t t ( c, c,, c t ) A i,, i, ( A - i I) x =, N( A - i I).N( A - i I) dim N( A - i I) = - r( A - i I) i, i, i f() =, m i. A, A - I =, A - I = A = - - - - 3 3 - A =, =, 3 = 3. =, ( A - I) x =, ( A - I) = - x = c[,,] T, ( c ). =, ( A - I ) = x = c[,,] T, ( c ). 3 = 3, ( A - 3 I) = x = c[,,] T, ( c ). = ( - ) ( - ) (3 - ) = - - - - - - - -.,,. A = 3-4 - 4 8 -.
5 85 A - I = = -,, 3 =. 3-3 - - 4 - - = ( - - ) - 4 - - 4 8 - - = - ( + ) ( - + ) = - ( + ) (- ) = = -, ( A - ( - ) I) x =, ( A + I ) = 5-4 4 8 5 9-9 9 = [,,] T, m - = - =.,3 =, ( A - I) x =, ( A - I) = x = c[,, ] T, ( c) - 4-4 8-3 6-3 - = [ -,, ] T ; m =, = ( > ). x = c[ -,,] T, ( c) 3 A = A - I = = 5,, 3 = -. x 3 - - - = (5 - ) - ( + ). = - ( + ) = 5, ( A - 5 I) x =, ( A - 5 I) = = c, - 4-4 - 4,3 = -, ( A + I) x =, 5-5 - 5 - - - = (5 - ) = (5 - ) ( + ) = - - 6 6 x = c[,, ] T, ( c) - - - -
86 ( A + I ) = x = c, x 3 = c, m 5 - x = c - + c, ( c, c ) = 5 = ; m - = - = ( = ). 5..3 A = [ a ij ],,,, i = i = i =, i = A (5..3) i = aii = tr( A) (5..4) A - I = f() = ( - ) ( - ) ( - ) =, ( 5..3). (5..4), ( - ) -., ( - ) - + + + ;, ( - ) - A - I ( a - ) ( a - )( a - ) ( - ) - a i i,, tr ( A)., i = i = i = i = a ii = tr( A) A x, A x., Ax = x, A A x = Ax, A x = x A x. A x, A ( A) = a I + a A + + a m A m ( ) = a + a + + a m m, x. 3 A A T A T. A =,. - I = A T - ( I) T = ( A - I ) T = A - I =, A A T,, =, x = c[,] T, ( c ) ; A T =,
5 87, =, x = c[, ] T, ( c),. 4 A, x A -, x., Ax = x, A - A - x = x ; A - x = x, A Ax = x. A, x A * A, x. 5,,, m A m, x, x,, x m,,, m, x, x,, x m..,. k, k +,,, k, k + x, x,, x k, x k +.. k +, t x + t x + + t k x k + t k + x k + = (5..5) t k + x + t k + x + + t k k + x k + t k + k + x k + = (5..6) (5..5) A A( t x + t x + + t k + x k + ) = (5..7) (5..6), t x + t x + + t k k x k + t k + k + x k + = (5..7) t ( - k + ) x, x, x,, x k + t ( - k + ) x, ti ( i - k + ) = ( i =,,, k) + + t k ( k - k + ) x k = ( i - k + ) ( i =,,, k), t i =., ( 5..5 ) t k + x k + =, x k +, t k + =. x, x,, x k, x k +. 5. A, B, P, B = P - AP (5..) A B, A B. : A A (). A B, B A (). 3 A B, B C, A C ( ).
88 4 A B, A T B T, A m B m ( m ). 5 A B, A - B -. 6 A B, A B,. 5, 6,. A B, P, P - P - A - P = B -, A - B -.5, 6. A B, P, B = P - AP, AP = B, B - I = P - AP - I = P - ( A - I) P = P - A - I P = A - I A B, A B. ( 5..3) ( 5..4) :.,. A B, A B. : A =, B =, A, P. A B (- ), P - AP = P - IP = P - P = I B A, B.B. A, A. A A. A, P, P - AP = P, P = [ x, x,, x ], x i AP = P A[ x, x,, x ] = [ x, x,, x ] Ax i = i x i ( i =,,, ) AP =,, i A, i x i. P, x, x,, x. A x, x,, x,,,, Ax i = i x i ( i =,,, ).P = [ x, x,, x ], AP = A[ x, x,, x ] = [ x, x,, x ] = [ x, x,, x ] P - AP = 5. 5 : = P
5 89 A, A. A m,,, m, r, r,, r m, r + r + + r m =, r i i r i ( i =,,, m), A. A,,, : A, m =., m, m =,, < m, A.,,,, P, P - () A = AP =. - 3 ; ( ) B = 3-4 - 4 8 - ; (3) C =,, 3, : () A,. 3 x = [,, ] T, x [,, ] T, x 3 [,, ] T P = [ x, x, x 3 ] = = 3, P - AP =.. () B, = < = m, B. (3) C 3 5, -, -, 3 x = [,, ] T, x = [ -,, ] T, x 3 = [ -,,],, C, P - CP =. P = [ x, x, x 3 ] = A B, A = - - a 3, B =, = b 5 - - () a, b ; () P, P - AP = B ; (3) A ( ). () (5..3) (5..4 ) tr ( A ) = tr ( B ) + a + 3 = + + b; A = B (3 a - 4 ) = b; b = a +, b = 3 a - 4 a = 3, b = 5.,
9 () =, ( A - I ) x = [,, ] T ; =, ( A - I ) x = [, -,] T ; = 5, ( A - 5 I ) x = [,, ] T. P = -, P - AP = B. (3) A = PBP -, A = PB P -, P = A = - 5 - - =, P - = - 5 + 5-5 - 5 +, 5.3,,,. A. A,, m =. 3, A, x, x, x x. 3. x = Ax, x = Ax,. A, x, x T = ( x ) T = ( Ax ) T = x T A T = x T A x T x = x T Ax = x T ( x ) = x T x ( - ) x T x =, ( - ), x T x =, x x., P P -, Q, Q -. AP =., AQ = Q T AQ =? A, 3,,, Q. A,,
5 9, Q.. A, Q, Q - () A = () A - I = ; ( ) A = - - - 4 4 - - - - = ( - ) (- 3)= AQ.. = ( - ) ( - 3+ ) - ( - ) =, =, 3 = 3. : x = [ -, -, ] T, x = [ -,, ] T, x 3 = [,, ] T,, = x x = [ - 3, - 3, 3 ] T, = Q = [,, 3 ] = () A - I =, =, 3 = - 7., =, A - I = - x = c + c = Q - - 3-3 3 x x = [ -,,] T, 3 = - 6 6 6 AQ = Q T AQ = = - - - - - 4 4 - - - - 4-4 - 6 - = ( - ) (- ) (+ 7) = - - - - 4 4 4-4, 3 = - - x 3 x 3 = [ 6, 6, 6 ] T, - - 4 - - = ( - ) ( + 5-4) -, ( c, c ), =, x = c, x 3 = c, -, =,
9 = = = 5 x - 3 = - 7, ( A + 7 I) = -, = 45 = c, x = c, = - <, > <, > = 4 5 8 - - 5 4 4 5 -. - 8-8 9 9 4 5, ( c). 3 = - ( - 4) 5 - - = 5 4 5 -, 3 =, 3 3-3, Q = [,, 3 ] = - 5 5 45 4 45 5 45 3 3-3, Q - AQ = = - 7. 633 A, 6 = [, -, ] T, ( ) 3 ; ( ) A. A,, 3 x T x = [, -, ] x =, x - x + x 3 =, x = c, x 3 = c, 3 x = c - + c, ( c, c ) () () = [,,] T, = [ -,, ] T, = = = [,, ] T, = -, = - 6 6 6, 3 = <, > <, > = -, T, = 3-3 3, Q = [,, 3 ],
5 93 Q - AQ = A = QQ - = QQ T = - 6, P = [,, ] = A = PP - = 6 6 - - 3-3 3-3 3 6 - P - AP = = 3 3 6-6 3, P - = 3 3 6 3-3 3 3 3-3 6-3 3-3 3 3 3 3 = 6 3 3 3 = - 3 4 - - 4 - - 4 3 3 3, 4 - - 4 - - 4 5.4,, ax + bxy + cy = (5.4.) x = x cos- y si y = x si+ y cos mx + y = m,. (5.4.),,,.,.,. 5.4.
94 f( x, x,, x ) = a x + a x x + + a x x + a x + + a x x + + a x (5.4.),. aij, f ; aij, f..,. a ij = a ji, a ij x i x j = a ij x i x j + a ji x j x i, (5.4.) f = a x + a x x + + a x x + a x x + a x + + a x x + + a x x + a x x + + a x f = a ij x i x j = a ij x i x j (5.4.3) i = j = i, j = f = x T Ax (5.4.4) a a a x = [ x, x,, x ] T, A = a a a, A T = A. a a a, f A, A f, f A.,. f( x, x, x 3 ) = x + x x + x x 3 + x - x 3. 3, a ii x i,, aij = aji, xi xj. A = - f( x, x, x 3, x 4 ) = x + x x + x x 3 + x - x 3, A = -
5 95,., ( 5.4.4), P, f y, y,, y (5.4.5) (5.4.4), P, D, (5.4.6). x = Py (5.4.5) f = d i y j = y T Dy (5.4.6) i = D = diag[ d, d,, d ] f( x) = x T Ax = ( Py) T A( Py) = y T ( P T AP) y P T AP = D (5.4.7) A B, P, A B. B = P T AP P, A B, B A. P, Q, A B, B C, B = P T AP, C = Q T BQ, C = ( PQ) T A( PQ), A C.., ( 5.4.7 ),.5.3 ( ),. 5.4. f( x) = x T Ax x = Qy, f = y T y. 5.3 A, Q Q T AQ =, x = Qy f( x)f( x) = x T Ax = ( Qy) T A( Qy) = y T Q T AQy = y T y = i y i, i =,,, A., ( 5.4.5 ), P.,,. (5.4.6), r( D), r( A), f A,. f = x T Ax x = Qy A. x = Qy, ( ), x = Qy, x = y
96, x = < x, x > = < Qy, Qy > = ( Qy) T ( Qy) = y T Q T Qy = y T y = < y, y > = y,,,.,, f =. f f( x, x, x 3 ) = x - 4 x x + x - 4 x x 3 A = A - I =, A, A - I = - - - - - - - - - - - - = ( - ) ( - ) + + ( - - - ) ( - - + ) - + ( - ) ( - + 3 - - ) - = 4- ( - ) ( - )+ 4(- ) = (- )(8 + - ) = (- ) (4 - )( + ) = = 4, =, 3 = -. = 4, ( A - 4 I) x =, ( A - 4 I) = - - - - 3 - - - 4 - - - - 4 = [, -, ] T, = 3 [, -, ] T. - =, ( A - I) x =, ( A - I) = - - - - - - - 4 - - - - = [,, - ] T, = 3 [,, - ] T. 3 = -, ( A + I) x =, ( A + I ) = 4 - - 3 - - 4 - - - - - - - 3 = [,, ] T, 3 = 3 [,, ] T.
5 97 3 3 3 Q = [,, 3 ] = - 3 3 3, x = Qy 3-3 3 f =,. f = x - 4 x x + x - 4 x x 3 = 4 y + y - y 3 3 f = ax + x x + x x 3 + bx x 3 x = Qy, f = y + y - y 3, a, b Q. f A = a b b f = y + y - y 3,,, - A, a + a + a 33 = + + 3 A = 3 a = + - b( - ab) = - a =, b = -, =, ( A - I) x =, ( A - I) = - - - - - = [,, ] T, = [,, ] T, = = [,, ] T, = - = [,,] T, = [, -, ]. 6 - - <, > <, > =, - T, 3 = -, ( A + I) x =, ( A + I) = - - - 3-3 3-3 - 3 = [ -,, ] T, 3 = 3 [ -,,] T, 6-3 Q = [,, 3 ] = - 6 3 6 3,.
98 5.4.3 () f = y + y + + y,,, f A,, x = y.,,. x = Py. f f = x - 4 x x + x - 4 x x 3 y = x - x f = ( x - x ) - x - 4 x x 3 = ( x - x ) - ( x + x 3 ) + 4 x 3 x = y + y - y 3 y = x + x 3, x = y - y 3, y 3 = x 3 x 3 = y 3 x x x 3 = - - y y y 3 f = y - y + 4 y 3 y = ( x - x ) x = y - y + y 3 y = x 3, x = - y + y 3 y 3 = x + x 3 x 3 = y, x x x 3 = - - y y y 3 f = y + y - y 3 x = Py f = x x + x x 3 - x x 3.,.
5 99 x = y + y x = y - y, x = x 3 = y 3 x 3 x - y y y 3 = P y y y 3, f = x x + x x 3 - x x 3 = y - y + ( y + y ) y 3 - ( y - y ) y 3 = y + y y 3 - y + 3 y y 3 = ( y + y 3 ) - y + 3 y y 3-4 y 3 = ( y + y 3 ) - ( y - 3 y 3 ) + y 3 z = y + y 3 z = y - 3 y 3 y = z - z 3, y = z + 3 z 3 z 3 = y 3 y 3 = z 3 y y y 3 =, f - 3 z z z 3 = P z z z 3 f = z - z + z 3 x y z - z z x x 3 = P y y 3 = P P z z 3 = - 3 z z 3 = - - z z 3 5.5 5.4,, x = Py,,,. f = x T Ax, r, x = Py x = Qz f = t y + t y + + t r y r ( t i ) f = k z + k z + + k r z r ( k i ) ( i =,,, r) t, t,, t r k, k,, k r.,,,,, r.
f f = y + y + + y p - y p + - - y r,,,,, -, -,, -,,,, (),.. ( ), ( ) f = x T Ax, x, x,, x, f >,,, A > ; f <,,, A < ; f,,, A ; f,,, A ; f,, f.,. f( x, x, x 3 ) = x + x + x 3, f( x, x, x 3 ) = x + x, f( x, x, x 3 ) = x - x.,,.,? : f = x T Ax. x = Py f = d i y i i = d, d,, d >, x, y = P - x (x ), f( x). f( x) = d i y i > i =. <, x = Py f = x T Ax f = d i y i, d s, y = e s i = x = Pe s, = [,,,,] T, f = x T Ax = e T s P T APe s = d s f = x T Ax. =. A A., : 3 f = x T Ax, : () x, f = x T Ax > ; () f ;
5 (3) P, A = P T P ; (4) f = x T ( - A) x. x = Py x = Qy,, A, ()() ; x = Cy,, A y T Iy, C T AC = I ( ), P = C - ()(3) ; f = x T ( - A) x = - x T Ax, x, x T Ax > x T ( - A) x <, ( )( 4)... A, a i i > ; A, a ii <. A, x = e i = [,,,,,] T, f = x T Ax = e T i Ae i = a ii > ( i =,,, ), A, x = e i = [,,,, ] T f = x T Ax = e T i Ae i = a ii < ( i =,,, ) A, 3 P A = P T P, P, P p, p,, p, A, a ii, A, A -, A * a ii = p T i p i = p i > <.. A -, A *. A, 3 A, A - A,, A -. A, A * = A A -, A *., A -, A *. A, P, A = P T P, A - = P - ( P T ) - = Q T Q Q = ( P T ) -, A -. A * = A A - = R T R, R = A ( P T ) -, A *. 3 t,. f = x - 4 x x - x + 4 x x3 - x 3 + 8 x x3 + t( x + x + x 3 ) q = x - 4 x x - x + 4 x x 3 - x 3 + 8 x x 3 A = - - - 4 4 - A - I =,,, - 7, q q = y + y - 7 y 3
,, f f = y + y - 7 y 3 + t( y + y + y 3 ) = ( + t) y + ( + t) y + ( t - 7) y 3 f, + t >, + t >, t - 7 >, t > 7 A k a, a,, a kk k A k (),,. 4 A A, D >, D >,, D > (5.5.) a a a k D k = a a a k, ( k =,,, ) a k a k a kk, ( 5.5.) D <, D <,, D < A., 4 : A ( - ) k D k > ( k =,,, ) (5.5.) D k A k. ( - f) = x T ( - A ) x, - A k ( k =,,, ) - a - a - a k - a - a - a k = ( - ) k D k > - a k - a k - a kk - A, A. 4 f = - 5 x - 6 y - 6 z + 4 xy + 4 xz. f a = - 5 <, f. a a a a = A = - 5-5 - 6-6 - 6 - A. = 6 >, A = - 8 <, 4