1. 2. 3. Householder 4. 5. Exponential (e At ) 6. e At 7. (a) Euler [1] (b) [2,3,4,5,6,7] (c) [8] (d) [9] 8. -2-
(1) A (A T = A) Cayley-Hamilton [10] [11] e At t e At -3-
(2) 1. Householder n R n n n H H=I 2(nn T /n n) Householder Householder (Householder reflection) (H=H T ) (H T H=I) (H 2 Householder H ν (u)=u 2(nn T /n n)u H ν =I 2(nn T /n T n) I H ν = I 2(nn T /n T n) Householder H ν (u)=u 2(nn T /n T n){u} n H ν (a) H ν (u)= u 2 (nn T /n n)u H ν = I 2(nn T /n n) (b) (H ν ) 2 =I H ν -1 =H ν (c) H (A) H T AH=D (2.1.1) D -4-
(d) n=2 ν R 2 cosα n= sin α H ν =I- 2 ν T ν cos2α sin2α ν 2 = sin2α cos2α (2.1.2) u H n n H n (u) 1 X X 2 u n H n (u) X 1 X 1 H n (n) (1) ν (X 1 cos+x 2 sin=0) p q r n=3 n= (p 2 +q 2 +r 2 =1) H= 1 0 0 0 1 0 0 0 1 p r -2 q { p q r} = 2 1 2 p 2 pq 2 pr 2 pq 1 2q 2 2qr 2 pr 2qr 2 1 2r -5-
2. e At x & = Ax, x(0)=x o (2.2.1) x x& x(t) o A n n A (2.2.1) x=e At x o (2.2.2) A P n n + P n-1 n-1 + + P 1 + P o =0 (2.2.3) Cayley-Hamilton [5] P n n + P n-1 n-1 + + P 1 + P o I=0 (2.2.4) -6-
3. A R n (or C n ) A T = A [10] 4. A AC=CD D C A A=CDC -1 (2.4.1) f(x) (x-a) f(a) f(x)=(x a)q(x)+f(a) (2.4.2) Q(x) f(a) f(x)=(ax 2 +bx+c)q(x)+px+q (2.4.3) Q(x) px+q x f(a)=(aa 2 +ba+ci)q(a)+pa+qi (2.4.4) a,b,c,p,q I [11] -7-
5. A 3 3 A= 0 c b c 0 a b a 0 (2.5.1) 2 2 2 A 0a + b + c i 2 2 a + b + f(a)=e At c 2 i f(a) =[a(t)a 3 +b(t)a 2 +c(t)a+d(t)i]s(t)+ [p(t)a 2 +q(t)a+r(t)i] (2.5.2) s(t) p(t)a 2 +q(t)a+ri e At ( (similar transform)[5] Q B=Q -1 AQ B A ) e At 1 cos ω t = ω 2 A 2 1 cos ω t + ω 2 A+I (2.5.3) ω= a + 2 2 2 + b c 6. (2.5.2) e At t Q=e At QQ T =I (2.6.1) -8-
a. t=0 e At I( ) b. t=π/2 e At?? c. t=π e At?? ( Householder Givens QR Jacobi ) e At t 2 e At, t R t =? Householder ( H n =I-2(nn T /n n) ) ν n=(p,q,r)=? A A T =A n(2) -9-
Cayley-Hamilton [10] t (1) A 2 2 3 3,, n n (2) 2 2 4 4 (3) (Euler) (4) t (mapping) (5) [9] -10-
1. 2. Cayley-Hamilton 3. t 4. t 5. e At 6. -11-
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15. R. Bellman, Introduction to Matrix Analysis, Society Industrial and Applied Mathematics, Philadelphia, 1995. 16. M. M. Mehrabadi, S. C. Cowin and J. Jaric, Six-Dimensional Orthogonal Tensor Representation of The Rotation about an Axis in Three Dimensions, Int. J. Solids Structures, 32(1995), pp.439-449. 17. Y. Saad, Iterative Methods for Sparse Linear Systems, PWS, 1996. 18. P. Lancaster and M. Tismenetsky, The Theory of Matrices, 2 nd ed., Academic Press, 1985. 19. C. Morler and C. V. Loan, Nineteen Dubious Ways to Compute the Exponential of a Matrix, SIAM Review, Vol.20, No.4, pp.801-836, 1978. 20. M. Vidyasagar, A Novel Method of Evaluating e At in Closed Form, IEEE Transactions on Automatic Control, pp.600-601, 1970. 21. R. C. Ward, Numerical Computation of the Matrix Exponential with Accuracy Estimate, SIAM J. Numer Anal., Vol.14, No.4, pp.600-610, 1977. -14-
-15- ( NSC-88-2211-E-019-005) (1) ( ) (2) (Mathematica Matlab) (3)