高等代数 (2)

/r!;<("� 'P IP )Z� 59.77.1.116 Ju� gdjpkc.xmu.edu.cn �x}~ �� ������~{z�� (|y�x}) u�smr 1. $ t=( ) Æ�0�6 f(x1, x2, x3) = x 2 1+x 2 2+5x 2 3+2tx1x2−2x1x3+4x2x3 U*� 2. �0�6 f(x, y, z) = xy + xz + yz, N�.D&_R A = ( ), A &�SY ) ( ), 0�6&U�9[� p = ( ); :�9[� q = ( ) ; 8D� s = ( ); � &A36) ( ) . 3. ?W0�6 2x2+3y2+3z2+2ayz(a > 0)\CUS HG� x′ 2 +2y′ 2 +bz′ 2 , N a = ( ), b = ( ). 4. 5� n V�.�℄R A )U*R& 5 =�<"Q (a� d*B). 5. �0�6 f(x1, x2, x3) = X ′AX = ax21 + 2x 2 2 − 2x 2 3 + 2bx1x3(b > 0), ~^0 �6&℄R A &�SYXE) 1, �SYXK) −12, N a = ( ), b = ( ). 6. ?W n V�.�℄R A &�SY^F m = 0, t =U���N A &℄� U�9[��:�9[�M8D�6�� ( ). 7. h= n O℄R A H B &�K AB �U*℄R&�6�<"Q� ( ). 8. (5 6) �.�℄R   0 1 11 0 0 1 0 0   &℄� ( ), 8D�� ( ). (�[ 3 : 2002 |) k�tvr 1. � A ) n VU*℄R�N −2A �� ( ). (A) :*℄R (B) :*℄R (C) U*℄R (D) U*9HVF? 1 2. �.�℄R A =   1 t 0t 1 t 0 t 1   �U*&�6�<"Q� ( ). (A) −1 ≤ t ≤ 1 (B) −1 2 < t < 1 2 (C) t ≤ −1 2 I t ≥ 1 2 (D) t ≤ −1 I t ≥ 1 o�lqr 1. ?W0�6 f(x1, x2, x3) = 2x1x2 + 2x1x3 − 2x2x3, (1) 5�0�6 f &℄R A; (2) �� A &�SYH�S3i� (3) �>US09 H�R f G)Æ 7� 2. �F n K�0�6 f(x1, x2, · · ·, xn) = (x1 + a1x2) 2 + (x2 + a2x3) 2 + · · ·+ (xn−1+an−1xn) 2+(xn+anx1) 2,~^ ai(i = 1, · · · , n))����,�$ a1, a2, · · · , an pfF_"QÆ�0�6 f(x1, x2, · · · , xn) )U*0�6� 3. �0�6 f(x1, x2, · · · , xn) = n ∑n i=1 x 2 i − ( ∑n i=1 x 2 i )(n ≥ 2) &℄EU�: �9[�� 4. � λ&Y�� f(x1, x2, x3, x4) = λ(x 2 1+x 2 2+x 2 3)+2x1x2−2x2x3+2x1x3+x 2 4 �U*&0�6� 5. ?W0��s4� x2 + ky2 + z2 + 2xy + 2xz + 2yz = 4 a#CUS H  xy z   = P   x ′ y′ z′   G)&Las4� y′ 2 + 4z′ 2 = 4, � k &YEUS℄R P . 6. �0�6 f(x, y, z) = x2 + 2y2 + 4z2 + 2xy + 4xz + 2yz, E�' H41 �5%G09 HG�0�6)A37��Z>�}-�0�6�7)U*0� 6� 7. � f = a ∑n i=1 x 2 i + b ∑n i=1 xixn−i+1, ~^ a, b ����, a, b pf�q" QÆ� f U*� 8. �US H�0�6 2x21 − 4x1x2 + x 2 2 − 4x2x3 G)Æ 7��}*;0� 6�7U*� 2 9. ��0�6 f(x1, x2, x3) = ax 2 1 + ax 2 2 + ax 2 3 + 2x1x2 + 2x1x3 − 2x2x3 (1) �US H X = QY � f G�Æ 7� (2) , a )FYÆ� f &℄) 2? �Æ�� f(x1, x2, x3) = 0 &X� 10. �0�6 f(x1, x2, x3) = x 2 1 + ax 2 2 + x 2 3 +2bx1x2 +2x1x3 +2x2x3 \US0 9!H (x1, x2, x3) T = P (y1, y2, y3) G)Æ 6 y 2 1 + 4y 2 2. (1) � a, b MUS℄R P ; (2) ,0�6 f �U*&o�)�q� 11. � f(x1, x2, x3, x4) = 2x1x2 +2x1x3 +4x1x4 +2x2x3, �6�M��I E 9�I ��G)A36��5�1D&a{09 H� 12. ��n a ��qYÆ� n K0�6 aΣni=1x 2 i − ( ∑n i=1 xi) 2 �U*&� 13. � f(x1, x2, x3, x4) = X ′AX )�-�0�6� A &�SY) λ1 = 1(0 `) E λ2 = −1(0`), �W ε1 = (1, 1, 0, 0) E ε1 = (1, 1, 0, 1) ��G λ1 = 1 &� S3i��0�6 f(x1, x2, x3, x4). 14. R0�6 f(x1, x2, x3) = x 2 1 + 2x 2 2 + 3x 2 3 − 4x1x3 G)Æ 6��5��E &07!H� 15. ��0�6 f(x1, x2, x3) = 2x 2 1 + x 2 2 − 4x1x2 − 4x2x3 (1) � f &�SYE�S3i� (2) �US℄R T , i H X = TY , � f G)Æ 6� (3) 5� f &U:�9[�E8D�� 16. EUS09!HG.s0�6)Æ 6 f(x1, x2, x3) = 2x 2 1 + x 2 2 − 4x1x2 − 4x2x3. 17. (15 6) EUS HG.s0�6)Æ 7 f(x1, x2, x3) = x 2 1 + x 2 2 + x 2 3 − 4x1x2 − 4x1x3 − 4x2x3. (<�5�US H&℄RE1D&Æ 7). (�[ : 2002 |) 18. (25 6) � 3 K�0�6 f(x1, x2, x3) = 4x 2 1 + x 2 2 + 4x 2 3 + 4x1x2 + 8x1x3 + 4x2x3. 3 (1) �US09 HR f(x1, x2, x3) G)Æ 7� (2) � X0 = (c1, c2, c3) � 3 *#+3i�pf f(c1, c2, c3) = 9, ��#+3i X0. (�� : 1999 |) 19. (12 6) EUS09 HR�0�6 f(x1, x2, x3) = 2x 2 1 + 5x 2 2 + 5x 2 3 + 4x1x2− 4x1x3− 8x2x3 G)Æ 7��5�1D&US HEÆ 7� (�� : 2001 |) 20. (35 6) ?Wv=�.�℄R A &�S/2� f(λ) = λ5 + 3λ4 − 6λ3 − 10λ2 + 21λ− 9. (1) � A &8j�EL4/2�� (2) � VA = {g(A)|g(x) ∈ R[x]}. Vt� VA �09bP��� dimVA; (3) t ��q��Æ� tE + A �U*℄R�~^ E )#+℄R� (4) >�>=^ &���.TR&�.�℄R A, �%�&�S/2�) f(λ). (�� : 2006 |) 21. (156)Vt n+1V�℄R A�U*℄R�A =   2 2 2 2 · · · 2 n+1 n+1 22 2 23 3 · · · 2 n+2 n+2 ... ... . . . ... 2n+1 n+1 2n+2 n+2 · · · 2 2n+1 2n+1  . (�� : 2007 |) 22. (5 6) Vt� n V�.�℄R&�SY>*���� (�� : 2008 |) p�wnr 1. Q) nV.�U*4R�x) n*�3i�Vt�0 ≤ xT (Q+xxT )−1x < 1, Qf xT �� x &b\� 2. A ) n VU*℄R� I ) n V#+℄R�Vt |A+ I| > 1. 3. � n V℄R An =   2 −1 0 · · · 0 0 −1 2 −1 · · · 0 0 0 −1 2 · · · 0 0 · · · · · · · · · · · · · · · · · · 0 0 0 · · · 2 −1 0 0 0 · · · −1 2   . �℄R A3 &8j 4 ���Vt An �U*&� 4. � A = (aij)n×n �a{&.��℄R�Vt�0�6 f(x1, x2, · · · , xn) =∣∣∣∣∣∣∣∣ 0 x1 · · · xn −x1 a11 · · · a1n · · · · · · · · · · · · −xn an1 · · · ann ∣∣∣∣∣∣∣∣ &℄R� A &��℄R A∗. 5. ��0�6 f(X) = XTAX,X ∈ Rn, λ � A &�SY�Vt�M5l3i α =   k1 k2 ... kn  , �% f(α) = λ(k21 + k22 + · · ·+ k2n). 6. � A,B )$V.�U*R� A > B(N A−B )U*R), �,�7>* F A2 − B2? )�q� 7. Vt� S ) n V.�U*℄R�N (1) �M(>&.�U*℄R S1, �% S = S 2 1 ; (2) A � n V�.�℄R�N AS &�SY���� 8. � A,B +� n VU*℄R�Vt� (1) AB &�SY� Gl� (2) AB = BA, N AB �U*℄R� 9. � A ) U*℄R�Vt�.�AU�� ε, εI + A )U*℄R� 10. � f(x1, · · · , xn) = X ′AX �>=�0�6� F� n *j3i X1, X2 � X ′1AX1 > 0, X ′ 2AX2 < 0. Vt���M� n *j3i X0 6= 0, � X ′ 0AX0 = 0. 11. Vt�B A )>=2.� (AT = −A) �℄R�N B = (I −A)(I + A)−1 )US℄R� 12. Vt� (1) .�>�U*.�℄R B, �M�a{R C, � B = CCT ; (2) � B1, B2 `)�.�℄R�� B2 )U*&�N℄R A = B1B2 &�SY `)����a1�G.TR� 13. A � n VU*℄R� B � n V�.�℄R�?W A H B 1��Vt B 5 =�U*℄R� 14. � A )U*℄R�Vt A a��� n = U*℄RXE� 15. � AT = A, Vt A a{$�Y$�M℄R B, �% AB +BTA U*� 16. � A � n V5C&�.�R�Vt� A2 ��℄R (A2)∗ �U*R� 17. � A � n V�4R�Vt� (1) BT = B, N AB &�SY)��� (2) B U*�N AB &�SYU G 0; (3) B U*�� AB = BA, N AB U*� 18. �6 ℄R D = ( A B BT C ) �U*℄R�Vt C − BTA−1B =�U* ℄R�~^ BT �� B &b\℄R� 19. � A � n OU*℄R� B � n O U*℄R���pf A2 = B2, V t� B �U*℄R��� A H B 1�� 20. A � n V4R��.�A&5l3i α, +F αTAα > 0. Vt� �MU*℄R B M2.�℄R C, �% A = B + C, ��.�A3i α, +F αTAα = αTBα, αTCα = 0. 21. � A )>= n V�.�℄R�� |A| < 0. Vt���M� n *3i X 6= 0, � X ′AX = 0. 22. � f(x1, · · · , xn)�>=℄) n&0�6�Vt�F R n &>= 1 2 (n−|s|)*d bP V1 �M (~^ s )8D�), �.�> (x1, · · · , xn) ∈ V1, F f(x1, · · · , xn) = 0. 23. � A � n V�.�℄R�B = (b1, · · · , bm), bi +� n *j3i�m < n, b′ibj = { 1 (i = j) 0 (i 6= j) . Vt�M (cij)n×m, �% ∑n k=1 ckrcks = { 1 (r = s) 0 (r 6= s) , � ∑n i=1 λicij(j = 1, · · · , m) ) B ′AB &�SY (λ1, · · · , λn � A &�SY). 24. (12 6) (1) � M = ( A B B′ D ) )U*R��V� A,D,D − B′A−1B U*� 6 (2) � A U*��V��M C U*�� A = C2. (�[ : 2005 |) 25. (1) � A,C ) n,m V�.�℄R� B � n×m �℄R� ( A B B′ C ) U *�N ∣∣∣∣ A BB′ C ∣∣∣∣ ≤ |A| · |C|, �'D$�Y$ B = 0 Æ�g� (2) � A = (aij)n×n � nV�℄R� |aij | ≤ 1, �V |A| 2 ≤ nn. (�[ : 2008 |) 26. (13 6) � A � n VU*℄R�3ig β1, β2, · · · , βs pf β ′ iAβj = 0(1 ≤ i < j ≤ n). ,3ig β1, β2, · · · , βs &℄ay�/��Vtz&Wn� (�[ : 2010 |) 27. (20 6) m�0�6 f(x1, x2, · · · , xn) = X TAX , ~^ A = AT = (aij)n×n, X = (x1, x2, · · · , xn) T . � λ H µ 6�� A &h Hh4�SY�N.G�A& n *83i β ∈ R1×n `F µββT ≤ βAβT ≤ λββT . (�[ 3 : 1999 |) 28. (10 6) � A,B � n V�4R�� A �U*℄R� B �2.�℄R�V t 2|rank(BTAB). (�[ 3 : 2000 |) 29. (20 6) � A = (aij), B = (bij) `� n VU*℄R�Vt C = (aijbij) = � n VU*℄R� (�[ 3 : 2001 |) 30. (20 6) m�0�6 f(x1, x2, · · · , xn) = X TAX , ~^ A = AT = (aij)n×n, X = (x1, x2, · · · , xn) T . � λ H µ 6�� A &h Hh4�SY�N.G�A& n *83i β ∈ R1×n `F µββT ≤ βAβT ≤ λβAβT . (�[ 3 : 2004 |) 31. (15 6) � A �>= n V�℄R�Vt� (1) �B A a{�wq AAT �>=U*℄R� (2) �B A �.�&�wqe�M>=�� s, �% En + sA �>=U*℄ R� (�[ 3 : 2006 |) 32. (12 6) � A,B �h= n V�.�U*℄R�Vt� AB ��.�℄R $�Y$ AB ��.�U*℄R� (�� : 2000 |) 33. (10 6) � A � n× n �U*℄R�Vt�.�FUT� k, �MU*℄R B, �% A = Bk. (�� : 2001 |) 7 34. (10 6) � A,B ���I R & n V4R� AB + BA = 0. Vt��B A �.�℄R� U*�NF AB = BA = 0. (�� : 2004 |) 35. (10 6) � A =   4 4 22 4 2 2 2 4  . Vt� A �>=U*℄R����F&� -�/2� f(x), �% f(A) =�U*&� (�� : 2005 |) 36. � A ) n VU*℄R� (1) (5 6) Vt�.�AT� k, Ak =�U*℄R� (2) (5 6) Vt��M Rn &>=xK (·, ·), �% (·, ·) M Rn &v=J.& ,i℄R� A; (3) (10 6) .�A n×m &�℄R B, Vt� rank(B) = rank(BTAB). (�� : 2008 |) 37. (5 6) � A � n V�.�℄R�Vt� A �U*℄R&�6�<"Q� .�AT� k, Ak �U*&� (�� : 2009 |) (kO��kj Te) 8