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1 6 19,,,

2 à A si A s n + a n s n a 2 s + a 1 à a 1 a 2 a n 1 a n à ( 1, λ i, λ i 2,, λ i n 1 ) T ( λ i, λ 2 i,, λ n 1 i, a 1 a 2 λ i a n λ ) n 1 T i ( ) λ i 1, λ i,, λ n 2 a T 1 n 2 i, a 2 a n λ i (1) λ i λ i A λ i n + a n λ i n a 2 λ i + a 1 0 à ( 1, λ i, λ i 2,, λ i n 1 ) T λi ( 1, λi, λ i 2,, λ i n 1 ) T (2) ( 1, λi, λ 2 i,, λ ) n 1 T i 0 à v i ( 1, λ i, λ i 2,, λ i n 1 ) T T ( v 1, v 2,, v n ) T 1 ÃT diag ( λ 1, λ 2, λ n ) (3) 1

3 2 1 λ 1 λ λ 2 λ 2 V 2 1 λ n λ n n 1 λ 1 n 1 λ 2 ( ) T v 1, v 2,, v n n 1 λ n T V T 2

4 A Ã Λ T 1 AT diag(λ 1, λ 2,, λ n ) (1) B T 1 B β 1 β n (2) C CT θ 1, θ 2,, θ n ) ( β i β1, i, βm i, θ i θ1, i, θl i (3) T A 1 T 0 si, B Λ si, 0 I B m p rank A si, B rank Λ si, B (4) (5) rank Λ si, B λ 1 s β 1 rank λ n s β n 1 (6)

5 s n β i 0( i) ( i β i 0 s λ i i rank 1 ) β i 0( i) ( 45) (A, B) s rank A si, B n T I l A si C T Λ si C (7) A si Λ si rank rank C C (8) rank Λ si C λ 1 s rank λ n s θ 1 θ n s n θ i 0( i) ( i θ i 0 λ i s i rank 1 ) θ i 0( i) (C, A) s A si rank n C (9) 2

6 (4-90) (4-90) A 11 A 12 à (1) 0 A 22 B 1 B (2) 0 C C 1 C 2 à B (1)(2) 0 a 11 a 1n b 11 b 1n A, B a n1 a nn b n1 b nn ã 11 ã 1r ã 1(r+1) ã 1n à ã r1 ã rr ã r(r+1) ã rn ã (r+1)(r+1) ã (r+1)n 0 B ã n(r+1) ã nn b11 b1m br1 brm 0 t 11 t 1r w 1(r+1) w 1n T t n1 t nr w n(r+1) w nn 1

7 AT T Ã (3) B T B (4) (3) a 1k t k1 a 1k t kr a nk t k1 a nk t kr r r t 1k ã k1 t 1k ã kr r r t nk ã k1 t nk ã kr n n n n t 1k ã k(r+1) + t nk ã k(r+1) + a 1k w k(r+1) a 1k t kn a nk w k(r+1) a nk t kn kr+1 kr+1 w 1k ã k(r+1) t 1k ã kn + w nk ã k(r+1) t nk ã kn + 1 r A t 1, t 2,, t r t 1, t 2,, t r A 11 (5) kr+1 kr+1 w 1k ã kn w nk ã kn A 11 (4) B t 1, t 2,, t r B 1 (6) B 1 U c AU c U c B, AB, A 2 B,, A n 1 B AU c AB, A 2 B,, A n 1 B, A n B AB A n 1 B A n A n 1 A 0 B U c Range(AU c ) Range(U c ) (9) Range(B) Range(U c ) (10) t 1 t r U c rank(u c ) r (9) ( ) Range A t 1, t 2,, t r Range( ) t 1, t 2,, t r (11) (7) (8) 2

8 At 1, t 2,, t r t 1 t r (5) A 11 (10) (6) B 1 (4-90) (A 11, B 1 ) Ũ c B, Ã B,, Ã n 1 B rank Ũ c rank B 1 A 11 B 1 A n 1 11 B r (12) (12) rank B 1 A 11 B 1 A n 1 11 B 1 r (13) A 11 (r r) A r 11 A n 1 11 A r 1 11 A 0 11 rank B 1, A 11 B 1,, A r 1 11 B 1 r (14) (A 11, B 1 ) CsI A 1 B CsI Ã 1 B (15) 1 si A 11 A 12 B 1 C 1 C 2 (16) 0 si A 22 0 C 1 C 2 si A 11 1 si A 11 1 A 12 si A si A 22 1 C 1 si A 11 1 C 1 si A 11 1 A 12 si A C 2 si A 22 1 B 1 0 C 1 si A 11 1 B 1 (18) (4-91) (2-33) A B 0 D 1 A 1 A 1 BD 1 0 D 1 B 1 0 (19) (17) 3

9 S 1 : x 1 A 1 x 1 + b 1 u 1, y 1 c 1 x 1, S 2 : x 2 A 2 x 2 + b 2 u 2, y 2 c 2 x 2 S 2 S 1 S 2 u 2 y 1 x1 A 1 0 x 1 b 1 + u 1 (1) x 2 b 2 c 1 A 2 x 2 0 x 1 y 2 0 c 2 (2) x 2 S 1, S 2 I A 1 s 2 + a 2 s + a 1, I A 2 s 2 + á 2 s + á 1, c 2 adj(si A 2 )b 2 b 2 s + b 1 A 1 si 0 A si b 2 c 1 A 2 si C 0 c 2 s a s a b 1 s á 1 0 b 2 1 s á s 2 + a 2 s + a b A si 2 s + b C S 1 (3) (4) 1

10 4-3 rank A si C n (5) (A, B) 4 4 S 2 S 1 s 4 3 I A 1 s 2 + a 2 s + a 1, I A 2 s 2 + á 2 s + á 1, c 1 adj(si A 1 )b 1 c 2 s + c 1 A 1 si 0 b 1 A si B b 2 c 1 A 2 si 0 s a 1 s a s 1 0 c 1 c 2 á 1 s á s 2 + á 2 s + á 1 c 2 s + c A si B (6) (7) 4-3 ranka si, B n (8) (A, B) 7 4 S 2 S 1 s 7 3 2

11 Av i λ i v i, A T u i λ i u i, u i, v i u T i v i 1, λ i λ j ( i, j), v 1, v 2,, v n 1 u 1, u 2,, u n T,, z i (0) u T i x(0), x(t) v 1 e λ1t z 1 (0) + + v n e λnt z n (0) (1), u T i Av j u T i λ j v j λ j u T i v j ( ) T u T i Av j A T u i vj λ i u T i v j λ j u T i v j λ i u T i v j (λ i λ j )u T i v j 0, λ i λ j ( i, j), u T i v j 0 (i j), u T i v i 1, u T 1 u T 1 v 1 u T 1 v 2 u T 1 v n u T 1 v 1 0 u T 2 ( ) u T 2 v 1 u T 2 v 2 u T 2 v n v 1 v 2 v n u T 2 v 2 I ( ) u T n u T n v 1 u T n v 2 u T n v n 0 u T n v n, (u 1, u 2,, u n ) T (v 1, v 2,, v n ) 1, 1 t 0, x(0) v 1 z 1 (0) + + v n z n (0) u T i, u T i x(0), { u T i v j z j (0) j1 i j u T i v j 1 i j u T i v j 0 u T i x(0) z i (0) j1 v j z j (0), 1

12 (i) (4-30) (4-37) cadj(si A)b 0 (1) adj(si A) s n 1 I + (A + α 1 I)s n 2 + (A 2 + α 1 A + α 2 I)s n (A n 1 + α 1 A n 2 + α 2 A n α n 1 I) (2) c b cb cab ca p 2 b 0, ca p 1 b 0 cadj(si A)b ca p 1 bs n p + (ca p b + α 1 ca p 1 b)s n p (ca n b + α 1 ca n 1 b + α n p+1 ca p 1 b) (3) (1) (3) 0 s n p n p (ii) (i) p n rankb, Ab,, A n 1 b n (4) rankb, Ab,, A n 2 b n 1 (5) (5) n n (5) i 0 (4) A n 1 b i 0 x i (t) e i x(t) e i b, Ab,, A n 2 b 0 (6) 1 1 (i) x i 1 1

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