ממן12

ממן12 - 12" 1 A a...

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Unformatted text preview: 12 " 1 : A a a2 1 0 a 1 0 A2 = a 1 0 a 1 0 = 0 0 a 0 0 a 0 0 2 a 3 A = 0 0 3 a 4 A = 0 0 2a 1 a 2 2a 0 a2 2a 1 a 1 0 3 3a 2 3a a a 1 2 a 2a 0 = 0 a 3 3a 2 0 a 2 0 0 a 0 0 a3 3a 2 3a a 1 0 4 4a 3 6a 2 a a 1 3 2 a 3a 0 = 0 a 4 4a 3 0 a 3 0 0 a 0 0 a4 : An - , n a n 0 . n > 1 A = 0 na n -1 an 0 n2 - n n-2 n n a 0 a 2 n -1 na = 0 n a 0 ( ) ( )a ( )a n 1 n 0 n -1 n 0 ( )a ( )a ( )a n 2 n 1 n 0 n-2 n -1 n . , n = 2 : : n = m + 1 , n = m m a m 0 : A - . A = 0 ma m-1 am 0 m 2 - m m- 2 a 2 ma m -1 am Am+1 m a = 0 0 ma m -1 am 0 m +1 a = 0 0 ( m + 1)a m a m+1 0 m +1 m2 - m m-2 m2 - m m-1 a a a m + ma m ma m -1 + a a 1 0 2 2 ma m-1 0 a 1 0 = a m+1 a m + ma m = m m +1 0 0 a 0 a 0 a m2 + m m-1 m +1 (m + 1) 2 + (m + 1) ( m +1) - 2 a a (m + 1) a ( m +1) -1 a 2 2 (m + 1)a m 0 = a m +1 ( m + 1) a ( m +1) -1 m +1 m +1 0 a 0 a . . , n = m + 1 2 -( A - B ) = B - A , . , . ) An - I = ( An -1 + An -2 + K + A + I )( A - I ) ( x n - 1 = ( x - 1)( x n -1 + x n - 2 + K + x + 1) , , . : .1 ( An -1 + An -2 + K + A + I )( A - I ) = = An -1 + An - 2 + K + A + I - A A A A - An -1 - An -2 - K - A - I = I I I I = An + An -1 + K + A2 + A - - An -1 - An -2 - K - A - I = = An - I : . I - A - , A2004 = 0 : A2004 - I = ( A2003 + A2002 + K + I )( A - I ) 0 - I = ( A2003 + A2002 + K + I )( A - I ) - I = ( A2003 + A2002 + K + I )( A - I ) : - I - - I ( A2003 + A2002 + K + I )( A - I ) = - I I ) (- -( A2003 + A2002 + K + I )( A - I ) = I ( A2003 + A2002 + K + I )( I - A) = I , C ( I - A) = I - , C = ( An -1 + An -2 + K + A + I ) . I - A ,(III.48) . AB = A + B , B = 0 , - D( I - A) = I , C = - D , D = -C , . C ( I - A) = I , . A - I , D( A - I ) = I . B , B a 0 - : , AB = A + B ( A - I + I )B = A + B ( A - I )B + I = A + B B ( A - I )B + B = A + B ( A - I )B = A . A a 0 , , A A 44 K 4 A A 2 3 ") , 1 4A A2004 - , 2004 2004 2004 = 1 4A = A = A A 44 K 4 A A 2 A K A A .A 3 1 4 4 2 4 44 4 3 2004 2004 ( 2003 2004 , , A a 0 - 2004 a0 0 , A2004 = 0 , . A 2004 = 0 ,( ) , ! , A . , B = 0 , i j = ai b j , AB = [ i j ]m m b . B = a 1 a 1a L b m , A = M ma a .2 .i = i 1 L ai m , , a b b . AB - ) ABC = CAB = I - , m C .( , , , - , R m - , AB . ABx = 0 , R m - n - B - . , a1 , a 2 ,K , a n - , b1 , b 2 ,K , b n - . A . R m {a1 , a 2 ,K , a n } - {b1 , b 2 ,K , b n } , n < m - {b1 , b 2 ,K , b n } R m - u , v , R m - c1 , c 2 ,K , c m . {a1 , a 2 ,K , a n } . c i = a i bi , i . AB , R n - m , a1 , a 2 ,K , a m - A AB , R n - m , b1 , b 2 ,K , b m - B ,1 a i m . R m - m , c1 , c 2 ,K , c m - c i = a i [ b1 b2 K bm ] ) {c1 , c 2 ,K , c m } , , AB - . - ( i 1 , i 2 ,K , i n : a1 , a 2 ,K , a m c i , 1 a i m j - 1 j , 2 j ,K , n j , A i - ,1 a j m , AB i - i 1 , i 2 ,K , i m - , B i . = i k j k =1 n i = k j i =1 k n , i j = n k =1 i k k j , AB - =0 i =1 i j =1 m m , , - a 11 a21 , A = : n = 1 M a m1 , B = [ b11 b12 L b1m ] a b11 b1 11 L a11 m M AB = O M m1b11 L am1b1m a 3 : Dn 2 1 0 . Dn = . . 0 0 1 0 . . . 2 1 0 . . 1 2 1 . . . . . . . . . . . . . . . . . 0 . . . 1 0 . . . 0 0 0 0 . . . 2 1 0 2 0 1 0 0 . = 2 . . . 0 0 1 0 1 2 1 2 . . . 0 0 44 0 . 1 . . . . . . . . . . . 4 2 4 . . . . . 1 0 44 0 0 1 1 0 0 0 2 . . 0 1 . . - . . . 0 . . 2 1 0 0 1 2 0 0 44 3 144 0 . 1 . 2 . . . . . . . . . 4 2 4 . . . . . 1 0 44 (i) .1 ( n -a ( n -1) 1) ( n -1) ( n -1) 0 0 0 0 0 0 . . = . . 2 1 1 2 44 3 , Dn -1 : 2 1 . = 2 Dn -1 - . 0 0 1 1 . . 2 . . . . . . . . 0 . . 0 . . 4 44 2 4 0 0 0 0 . . . . 2 1 .1 2 44 3 . , Dn = 2 Dn -1 - Dn - 2 . Dn - 2 : D1 , D2 (ii) D2 = 2 1 =3 1 2 ( n - 2) ( n - 2) D1 = 2 = 2 . D3 = 2 D2 - D1 = 4 ,(i) : . Dn = n + 1 , n a 1 1 a k n , n a 3 . , 1, 2,3 (n , Dn +1 = 2 Dn - Dn -1 = 2 + 1) - n = n + 2 ,(i) , , Dk = k + 1 . a 1 1 a 1 1 A . v1 , v2 , v3 A = 1 1 a , . A a 0 , A R 3 , A = 0 R 3 , , .2 a 1 1 . 1 a 1 = 0 1 1 a : a 1 1 1 1 a A = a - + = a(a 2 - 1) - (a - 1) + (1 - a ) = 1 a 1 a 1 1 = a( a - 1)(a + 1) - 2(a - 1) = (a 2 + a - 2)( a - 1) = (a + 2)(a - 1) 2 , a = -2 a = 1 , a + 2 = 0 (a - 1) 2 = 0 , (a + 2)(a - 1) 2 = 0 . R 3 v1 , v2 , v3 a 4 i+ j B , [adj ( B )]i j = (-1) M ji 1 3 -2 -8 8 1 , B = , -2 4 3 .1 - 16 -8 8 20 10 -10 : B . adj ( B ) = 22 11 -11 1 3 -2 1 3 -2 I B , B I = 1 -8 8 I = 0 -11 10 I = 0 = 0 3 -2 4 0 -11 10 1 3 -2 -16 -8 8 -8 8 20 10 -10 0 = B I adj 1 = . , B ( B ) = : -2 4 22 11 -11 3 , ai i a 0 , 1 a i n , A = [ai j ] , , A .2 i+ j A . [adjA]i j = (-1) M ji , . ai j = 0 , 1, h i n, 1 j n, i j . adjA = [ci j ] : ci i a 0 i 2i A A i - A . ci i = (-1) M ii = M ii , M ii , , , , , . ci i a 0 : ci j = 0 i a j i+ j A , 1 A - (i, i ) - . ci j = (-1) M ji A i - , A - i - , M ji . A , M ji = 0 , i+ j A . ci j = (-1) M ji = 0 A . , adjA 5 a 1 a . 2 3 a 1 0 1 b 1 1 1 = b 1 1 1 2 , x = Ax = b , 3 0 1 0 b 0 a b 1 + 1 1 + 1 a = b , " . 2 2 , a3 3 b a 1 + 1 1 0 + 1 1 1 b = a + 1, b = a + 1, b = a a , B = 2 1 2 2 3 3 : 1 a3 0 1 . Bx = a .1 a1 + 1 1 0 a1 , Ax = b . B = a2 + 1 1 1 = a2 a3 0 1 a3 1 0 1 1 = A 0 1 . Bx = a , , B a 0 A a 0 : B , a1 . a2 a3 1 0 a1 1 1 = -a1 + a2 - a3 0 1 a3 1 0 1 0 0 0 = (- a1 + a2 - a3 ) = -a1 + a2 - a3 0 1 0 1 1 - : . 2 3 a1 1 1 = = 2 a B -a1 + a2 - a3 a3 B1 B2 1 0 -a + a2 - a3 1 1 = 1 =1 -a1 + a2 - a3 0 1 0 1 a1 1 1 = a2 1 - a1 + a2 - a3 1 0 a3 0 1 = 1 a1 + 1 a1 1 2 = = 2 + 1 a2 a B -a1 + a2 - a3 a3 a3 1 = -a1 + a2 - a3 a1 - a2 a 3 1 0 a - a2 + a3 1 1 1 = = -1 -a1 + a2 - a3 0 1 a1 + 1 1 a1 1 1 a1 B3 1 1 3 = = 2 + 1 1 a2 = a 1 a2 = 0 1 B -a1 + a2 - a3 -a1 + a2 - a3 a3 0 a3 0 0 a3 1 1 - . x = Bx = a , 0 1 a1 + 1 0 a 1 = 1 a2 + 1 :' " C x = 2 x 1 a 3 0 a3 1 a a 1 a . 2 3 a 1 b1 0 = 1 b2 x 1 0 b3 1 .2 : 1 a1 + 1 0 1 a2 + 1 1 0 a3 1 1 a1 + 1 0 - a1 + a2 - a3 0 a3 a 1 a 2 3 a R2 a R2 - R1 R2 a R2 - R3 a1 a + a - a - 2 3 1 a3 0 0 1 : -a1 + a2 - a3 0 , A = -a1 + a2 - a3 - A a 0 ,' + [ R2 a 1 R2 - a1 + a2 - a3 a 1 1 3 a 1 a1 + 1 0 0 1 0 0 a3 1 R3 R3 -* a3 R2 a 1 1 0 1 a1 + 1 0 0 1 0 D = 0 0 1 , D .Cx = a ...
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This note was uploaded on 07/23/2009 for the course MATH. 04101 taught by Professor ישראלפרידמן during the Summer '06 term at The Open University.

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