矩阵习题课(一).pdf

SNJ K Home Page I K JJII JI 11 25 w 4 SSSKKK Zuo Liancui College of Mathematical Science Tianjin Normal University May 18 2011 SNJ K Home Page I K JJII JI 12 25 w 4 Main Contents 1 SNJ 2 K SNJ K Home Page I K JJII JI 13 25 w 4 1SNJ 1 A 1 5K A kA kaij mn 1 kD O kAnn kn A 2A3 3 23 A 2A3 3 8 A k A B kA kB k l A kA lA kl A k lA 1 A A kE X kE A A kE kA SNJ K Home Page I K JJII JI 14 25 w 4 A uB 1 A B 5K A aik mn B bkj ns K AB cij ms cij n X k 1 aikbkj AB C A BC m A B C AB AC B C D BD CD dd a SK AT A 1C C 1 Ams T AT sm AT i j A j i SNJ K Home Page I K JJII JI 15 25 w 4 AT T A A B T AT BT kA T kAT AB T BTAT A 1 T AT 1 A1 A2 As T AT 1 AT 2 AT s A1A2 As T AT s AT 2 AT 1 AT A A a11a12 a1n a12a22 a2n a1na2n ann AT A A 0a12 a1n a120 a2n a1n a2n 0 SNJ K Home Page I K JJII JI 16 25 w 4 2 1 AB A B AB min A B P Q PAQ PA AQ A 3 Ann e 3Bnn AB BA E K A B A A A 6 0 A z A n A A 1 1 A A SNJ K Home Page I K JJII JI 17 25 w 4 1 A 2 A A 1 A A 1p A 1 1 A 3 A B n AB AB 1 B 1A 1 4 A AT AT 1 A 1 T 5 eAnn Bnn n vAB E KA B B A 1 u kd X A 1 0 0 0 0 1 0 0 0 1 1 0 B 1 0 0 0 1 0 0 0 1 0 0 0 AB E A B 6 k 6 0 Ann kA 1 k 1A 1 SNJ K Home Page I K JJII JI 18 25 w 4 2 K 1 u 1 a1 a2 an b1b2 bn a1b1a1b2 a1bn a2b1a2b2 a2bn anb1anb2 anbn b1b2 bn a1 a2 an n P i 1 aibi x y 1 a11a12b1 a12a22b2 b1b2c x y 1 x y 1 a11x a12y b1 a12x a22y b2 b1xb2yc x a11x a12y b1 y a12x a22y b2 b1x b2y c a11x2 2a12xy a22y2 2b1x 2b2y c SNJ K Home Page I K JJII JI 19 25 w 4 2 O cos sin sin cos n A cos sin sin cos K A2 cos sin sin cos cos sin sin cos cos2 sin2 2sin cos 2cos sin cos2 sin2 cos2 sin2 sin2 cos2 A3 A2A cos2 sin2 sin2 cos2 cos sin sin cos cos2 cos sin2 sin cos2 sin sin2 cos sin2 cos cos2 sin sin2 sin cos2 cos cos3 sin3 sin3 cos3 An cosn sinn sinn cosn n 8B y n 1 b n k Kn k 1 An Ak 1 Ak A SNJ K Home Page I K JJII JI 110 25 w 4 cosk sink sink cosk cos sin sin cos cosk cos sink sin cosk cos sink cos sink cos cosk sin sink sin cosk cos cos k 1 sin k 1 sin k 1 cos k 1 d 8B n y 2 A A B 3 A a1 a2 an ai6 aj i 6 j A B B b11b12 b1n b21b22 b2n bn1bn2 bnn AB a1b11a1b12 a1b1n a2b21a2b22 a2b2n anbn1anbn2 anbnn BA a1b11a2b12 anb1n a1b21a2b22 anb2n a1bn1a2bn2 anbnn SNJ K Home Page I K JJII JI 111 25 w 4 AB BA a2b12 a1b12 b12 0 anb1n a1b1n b1n 0 a1b21 a2b21 b21 0 anb2n a2b2n b2n 0 a1bn1 anbn1 bn1 0 an 1bn n 1 anbn n 1 bn n 1 0 B b11 b22 bnn 4 A B 1 A 1 0 0 0 1 1 0 1 2 2 A 1 0 3 0 1 1 0 0 2 SNJ K Home Page I K JJII JI 112 25 w 4 3 5 A 1 0 0 1 1 0 0 1 1 A 1 A 1 6 0 A A 1 1 A A A 100 110 1 1 1 6 AX B XC B AXC B X A C X 1 1 21 X 1 0 1 0 1 0 X 1 1 21 1 1 0 1 0 1 0 11 2 1 1 0 1 0 1 0 111 2 1 2 1 1 21 X 1 2 3 0 1 2 0 0 1 1 0 1 0 1 0 SNJ K Home Page I K JJII JI 113 25 w 4 y K 1 y A B G y 7 y n n A7 X y A aij nn A n A B 1 2 n d 3 A a11 a22 ann SNJ K Home Page I K JJII JI 114 25 w 4 qA C 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 AC 0 a110 0 00a22 0 000 a n 1 n 1 000 0 CA 0 a220 0 00a33 0 000 ann 000 0 a11 a22 a33 ann A a11E 8 u Ann Bnn kAB BA 6 E y A aij nn B bij nn KAB BA cii n P k 1 aikbki bikaki AB BA SNJ K Home Page I K JJII JI 115 25 w 4 n X i 1 cii n X i 1 n X k 1 aikbki bikaki n X i 1 n X k 1 aikbki n X i 1 n X k 1 bikaki 0 E n AB BA 6 E 9 vA2 I A A B n y AB A B y AB AB 2 I ABAB I Am B dA B BA AB A B AB BA AB 2 ABAB A BA B A AB B A2B2 I AB 10 A B n n y XJA B AB KA B SNJ K Home Page I K JJII JI 116 25 w 4 y A B AB A AB B 0 A E B B 0 A E B E B E o A E E B E A E B E E A E 1 B E B E A E E l BA A B 0 A B BA oAB BA 2 11 A n n XJ u n X x1 x2 xn kAX 0 oA 0 y X O i 0 0 1 0 0 T i 1 2 n dK A i a1i a2i ani T 0 a1i a2i ani 0 i 1 2 n A 0 12 B r r C r n C r y 1 XJBC 0 oB 0 2 XJBC C oB E SNJ K Home Page I K JJII JI 117 25 w 4 y 1 B bij r r C cij r n C r 5 c11c12 c1r c21c22 c2r cr1cr2 crr 6 0 dBC 0 bi1c11 bi2c21 bircr1 0 bi1c21 bi2c22 bircr2 0 bi1c1r bi2c2r bircrr 0 i 1 2 r d g 5 X 1 0 k bi1 bi2 bir 0 i 1 2 r B 0 2 dBC C B E C 0 d 1 B E 0 B E 3 13 A B y AB B A y A B AB AB 1 B 1A 1 AB AB AB 1 A B B 1A 1 SNJ K Home Page I K JJII JI 118 25 w 4 B B 1 A A 1 B A 14 A aij nn e n XJi j i k j aik bkj 0 AB cij nn cij n P k 1 aikbkj i j cij i 1 X k 1 aikbkj n X k i aikbkj 0 l AB n 2 e n A aij nn Kk A a11a22 ann6 0 aii6 0 i 1 2 n A 1 xij nn Kk SNJ K Home Page I K JJII JI 119 25 w 4 A 1A x11x12 x1n x21x22 x2n xn1xn2 xnn a11a12 a1n 0a22 a2n 00 ann 1 0 0 0 1 0 0 0 1 1 x11a11 1 xi1a11 0 i 2 n x11 a 1 11 xi1 0 i 2 n 2 1 x21a12 x22a22 1 K x22 a 1 22 i 2 xi2a22 0 xi2 0 i 2 n y i j xij 0 A 1 n SNJ K Home Page I K JJII JI 120 25 w 4 15 A z 1 a y 1 A 1 z 1 a 1 2 Am z1 am m y 1 A z1 a A 1 1 1 a a a a 1 1 1 dd a 6 0 l k 1 1 1 aA 1 1 1 1 A 1 1 1 1 a 1 1 1 1 d L A 1 z1 a 1 SNJ K Home Page I K JJII JI 121 25 w 4 2 m 0 A0 I I z1 1 a0 b u K m 1 Am 1 1 1 1 a m 1 1 1 1 K Am 1 1 1 AAm 1 1 1 1 Aam 1 1 1 1 am 1A 1 1 1 a m 1 a 1 1 1 a m 1 1 1 K m m 0 A Am A 1 m u d 1 Am A 1 m z1 a 1 m am l u m SNJ K Home Page I K JJII JI 122 25 w 4 16 XJAk 0 o E A 1 E A A2 Ak 1 y y E A E A A2 Ak 1 E 4 1 17 e Ann i j aij Si j 2 i j 1 2 n Sk xk 1 xk 2 xk n k 0 1 2 y A aij Q i j x i xj 2 y SNJ K Home Page I K JJII JI 123 25 w 4 aij nn 11 1 x1x2 xn x2 1 x2 2 x2 n xn 1 1 xn 1 2 xn 1 n 1 x1x2 1 xn 1 1 1 x2x2 2 xn 1 2 1 xnx2 n xn 1 n A aij Q i j x i xj 2 18 A n n A 1 y 1 A a1 a2 an b 1 b2 bn 2 A2 kA 3 eAl 2 2 0 l 2 KA2 0 y 1 A 1 A 1 1 A 1 A b1b2 bn k2b1k2b2 k2bn knb1knb2 knbn 1 k2 kn b 1 b2 bn SNJ K Home Page I K JJII JI 124 25 w 4 2 A2 1 k2 kn