(x, y, z f(x, y, z (x, y, z (.,. 5.. F = R C, V F. V ( : ( : x V, x x = x = ; (2 : k F, x V, (3 : x, y V, kx = k x ; x + y x + y, V, (V, ( V,, x V x.

?,,,. : 5.. α R n, β R m, U R m n, Uα β (5... 5.. U J(U = Uα β U T U = I UU T = I (, (., J(U = Uα β U, ( U, Uα β mn,,!.,,.?,,.,,.,,,. ( m n,,.,? : 5..2 x, x < ( x = + x + x 2 + + x m + (5..2 :, A n, (I A = I + A + A 2 + + A m + (5..3? x <? ρ(a <.,?, ρ(a < A <, A,!,, (.,, f(x, y, z f(x = ( f x, f y, f z (5..4 29

(x, y, z f(x, y, z (x, y, z (.,. 5.. F = R C, V F. V ( : ( : x V, x x = x = ; (2 : k F, x V, (3 : x, y V, kx = k x ; x + y x + y, V, (V, ( V,, x V x. : =, x = x. 5.. R, V, a (a >. R. : R? C. 5..2 V = C n R n, C n ( R n, V : ( x = max x j ( l - ; j n (2 x = n x j ( l - ; j= ( /2 n (3 x 2 = x j 2 ( l 2 ; (4 x p = ( j= /p n x j p, p (Hölder l p p-. j=, n =, 5..2 C R (., 2 Hölder p = p = 2, Hölder p. ( 3, 5.. 3, V. l p (p ( (2. (3, Minkowski ( 4 : n x j + y j p j= /p n x j p j= 3 /p n + y j p j= /p, p. (5..

. 5.. l ( Lebesgue 49., F l, l 2, l, l p., L, L 2, L p, L. 2. < p <, l p,, 5. 3. C R n V, V, 5..2 V. 4. - Manhattan ( -,, d(x, y = x y.., -? -? 5..3 ( -, ( - ( : x = x 2 = 5.. x =, - -!, V,.,. - -.(?, - -??,. 5..4 F n n F n, A = (a ij n n, A F = n a ij 2 i,j= A = max j n A = max i n /2 = tr(a A ( Frobenius F- n a ij i= n a ij j= ( 6, F n n, F n n. 49 Henri Léon Lebesgue(875-94,, Lebesgue. 3

. - - 5..2 l - l -,.??, 7. 5.. ( α F m (F = C R, A F m n. x = (x, x 2,, x n T F n, x β = Ax α, β F n. 5..5 (x, y R 2, (x, y = x 2 + 2y 2 2xy, 5.. R 2 (?. x 2 + 2y 2 2xy =!,!,,,,. (,,,. (. 5..6 - -, E ij, (. F-,, n 2 n,., F ( F = C R n V.,?,. 5..2 C >, x V, V (, α β V. C x α x β C (5..2 α β. 5..7 R 2, -, -, (/2 x x x 2 x 2 x. (5..2 C 2.. 32

5..2 V (, α β V. α β C C 2, x V, x α C x β, x β C 2 x α, (5..3,, 8. 5..3 x, x 2,, x n, V ( (, v V. α β V. α β lim x n v α = n lim x n v β =. n ( {x n } v.,. 5... V n, α, α 2,, α n V, x V, x x = x α + x 2 α 2 + + x n α n. V x, x 2,, x n, x = ϕ(x, x 2,, x n. y = y α + y 2 α 2 + + y n α n V, x i y i, ϕ(x, x 2,, x n ϕ(y, y 2,, y n. ϕ(x, x 2,, x n ϕ(y, y 2,, y n = n = x y x y = (x i y i α i i= n x i y i α i n α i 2 n x i y i 2 i= i= i= x x, x 2,, x n. 5... α β V. x =, (5..2. x, x β. 5.., x α, x β x, x 2,, x n, f(x, x 2,, x n = x α x β x, x 2,, x n. β S = {x V x β = }. S, S, f S., f S C 2 C >. x, x/ x β S, x C α = x α C 2. x β x β 33

C x β x α C 2 x β..,,. 5..8 V = C[, ] [, ]. f = f = max f(x (5..4 x f(x dx (5..5 V.? V ( f n, f =, n 2, 2nx, x 2n ; f n = 2nx + 2, 2n < x n ;, x > n. ( 9 lim f n = n lim f n =. n, 5..3.,,.,,. 5..4 m n mn.,.. 5..3 F n n, F n n, A, B F n n, ( AB A B (5..6 F n n.. (, AB A B,,. 2. ( A <, A k (k,. 5..9 5..4 (5..6,.. F-,. 34

A = (a ij n n, B = (b ij n n. n n AB F = a ik b kj 2 n n n ( a ik 2 ( b kj 2 = i,j= k= i,j= k= k= n n a ij 2 b ij 2 = A F B F, i,j= i,j= (5..6. -, F- -, F,. 5.., A k A k., A <, { A k }. 5.. (, I. I = I 2 I 2.,,!(? 5..3,.,.. 5..2 ( A n., Ax x A., Ax α = A x α. (5..7, A A = Ax α x α. (5..8, ai a,., A, (5..8, x., A, x Ax =, A =,. (5..8, Ax α A = sup. (5..9 x x α 5..4 x F n, F n n F n. A F n n Ax A x (5... 35

5..2,. 5..2 (V, n, A n. A = sup x F n Ax x (5..,... A, A >, V, Ax =, A =.., (, 5 A = max Ax = Ax sup Ax = sup x = x F n x x x. (5..2 (F n,, Ax x,, (5..2 sup max. A, B F n n, (A + Bx Ax + Bx A + B = sup sup x F n x x F n x Ax Bx sup + sup = A + B. x F n x x F n x. ABx y, AB = ABy. By =, By =, AB = A B, ; By, AB = ABy By By max x F n Bx A(Bx Bx Ax sup B = A B. x F n x max x = Bx.. ( 3, 5..9 - -, - - ( - -, 2 5..4. 2. - - - -, - -,, -, 4. 5.. ( F n n ; (2 F n, F n n ; (3 F n n, F n. 36

(3. F n n, x F n, x = xj T (5..3 J = n i= e i ( xj T = (x, x,, x x. (5..3, 2. 5..3 (5..3. -! : - -? F- 2-?,.,,. 5..5 U, V, σ Hom(U, V. σ(x σ = sup x U x (5..4 Hom(U, V U, σ(x σ x., U = V,, στ σ τ (!. (5..4, U V.. U, (5..4 σ (?, σ (. U,, σ. 5..4 σ : (x, y T (x, T. σ =.,. 5..6 U, V (, σ Hom(U, V. ε >, δ >, x, y U, x y < δ, σ(x σ(y < ε, σ. 5..3. ( 24.,,,., 5..3., V = R[x] (5..4 37

( 5..8. σ : n a nx n n na n. σ Hom(V, R (?. 5.. R 2,?? 2.?? 3.? ( 8. 4.. p-, (x y, x y = x y 2?? 5. A A??? 6.?? 7. (5..3? 8.? - -,?,,. 5.2. (V, n, x, x 2,, x k, V, α V. lim x k α = k x, x 2,, x k,, α,. lim x k = α x k α. k 5.2. V = R ( =, 5.2.. V = F n, 5.2. ( 25: 5.2. {x (k } F n, x (k = (x (k, x(k 2,, x(k n, k =, 2,. x = (x, x 2,, x n F n. lim k x (k = x lim k x(k i = x i, i =, 2,, n.,, ( n.,.,. 5.2. {x (k } F n, F n, lim k x(k = x lim k x(k x =. 38

,,., 24.. 5.2. x x. n n 2.,. 5.2.2 A m = (a (m ij n n F n n, m =, 2,. i, j, lim m a(m ij = a ij, i, j n, {A m } A = (a ij n n, {A m }.. 5.2.,, lim A m = A., m 5.2. {A m } F n n, F n n, lim A m = A m.. 5.2.2 A m = ( m sin m ( m 2m m 2 ( cos m lim A m = m lim A m A =. m, ( e 2 2, 25.. 5.2. ( A m A, B m B, {a m } a, {b m } b, a m, b m, a, b F, lim (a ma m + b m B m = aa + bb, m lim A mb m = AB. m, P, lim P A m P = P AP. m (2 lim A m = A, F n n, A m. m (3 A m A, A m A, lim m A m = A. 5.2.3 A F n n, A : (5.2., A. I, A, A 2, A 3, (5.2. 39

( 29. 5.2. A B, A B., A Jordan. 5.2.3 ( ; (2 J = J n (λ λ n Jordan, (, s > k, C s k = J m = J λ < λ = n =. λ m Cmλ m Cmλ 2 m 2 Cm n λ m n+ λ m Cmλ m Cm n 2 λ m n+2 λ m Cm n 3 λ m n+3.... λ m. (5.2.2. 5.2.2 A C n n. A A λ : λ,, λ =, λ = Jordan.. Jordan., x. 5.2.2 S m = m k= Ak, {S m } lim k Ak =.., 5.2.3 A C n n, C n n, ρ(a A. 9.,. ε. B = B =, lim m Bm =, 5.2.,. A + ε A, A <. A + ε lim m Bm =. 5.2.2, B λ, A + ε λ A., λ A + ε. ε, λ A. 4

5.2.3 ( Neumann 5 A, A, I A (I A = I + A + A 2 + + A m + = A k. (5.2.3 k= 5.2.4,., -,.. 5.2.4 {A k k =,, 2, }. n, S n = n k= A k. lim S n = S, n k= A k S, k= A k = S., k= A k.. 5.2.2 ( k= A k, lim A k = ; k (2 k= A k = A, k= B k = B, a F, (A k + B k = A + B, k= aa k = aa. k= A k= a ka k., k= a ka k k= a kt k A t. k= a ka k k= a kt k. 5.2.2 J λ n Jordan, f(t = k= a kt k r. λ < r, k= a kj k, f(λ f (λ 2 f (λ (n! f (n (λ f(λ f (λ (n 2! f (n 2 (λ.......... f (λ f(λ. 5 Carl Gottfried Neumann (832-925,, (5.2.3 Neumann. 4

S m = m k= a kj k, S m (λ = m k= a kλ k. (5.2.2, J k = λ k Ck λk Ck 2λk 2 C n k λ k n+ λ k Ck λk C n 2 k....... λ k n+2... C k λk λ k,, s > k, Ck s =. m a k λ k k= S m = m a k Ck λk k= m a k λ k k= m a k Ck 2λk 2 k= m a k Ck λk k= m k= m k= a k C n k λ k n+ a k C n 2 k λ k n+2.......... m a k Ck λk k= m a k λ k k= = S m (λ S m(λ 2! S m(λ S m (λ S m(λ (n! S(n m (n 2! S(n 2 m.......... S m(λ S m (λ (λ (λ. S m (t= m k= a kt k r, λ <r, S m (λ, S m(λ, S m(λ,, S m (n (λ, lim S m(λ = f(λ, lim m m S(k m (λ = f (k (λ, k =, 2,, n., lim S m = m a k J k = k= f(λ f (λ 2! f (λ (n! f (n (λ f(λ f (λ (n 2! f (n 2 (λ.......... f (λ f(λ. 5.2.5 f(t = k= ( t 3 k, f(j, J = 2 2 2 2. 42

f(t = ( t 3, 3. J 2 f(t, f(j = k= ( J 3 k f(j = f(2 f (2 2! f (2 3! f (2 f(2 f (2 2! f (2 f(2 f (2 f(2. f (t = 3 ( t 3 2, f (t = 2 9 ( t 3 3, f (t = 2 9 ( t 3 4, f(2 = 3, f (2 = 3, f (2 = 6, f (2 = 8. f(j = 3 3 3 3 3 3 3 3 3 3. 5.2.2. 5.2.4 (Lagrange 5 -Sylvester f(t = k= a kt k, r. A Jordan λ i J J 2 λ i J =..., J i =......, i s,... Js P, A = P JP. i =, 2,, s, λ i < r, k= a ka k, f(j f(a = a k A k f(j 2 = P... P, k= f(js λ i f(j i = f(λ i f (λ i 2! f (λ i (n i! f (n i (λ i f(λ i f (λ i (n i 2! f (ni 2 (λ i.......... f (λ i f(λ i, i =, 2,, s. 5 Joseph Louis Lagrange(736-83,, Turin (,. 43

5.2.6 f(t = 2 t + 2t 3, f(a, A = 2 2 2. f(t,., 5.2.4. f( = 3, f ( = 5, f(2 = 6, f (2 = 23, f (2 = 24, f(a = 3 5 3 6 23 2 6 23 6. lim A nb n, lim A n, lim B n?? n n n 2. A, B, A + B, AB?. Lagrange-Sylvester,,. (z C: sin z = e z = m= z m m!, ( m z 2m (2m!, m= cos z = + ( m z2m (2m!. m=. Lagrange-Sylvester, A m m!, ( m A 2m (2m!, I + m= m= m= ( m A2m (2m!,. ( e A, sin A, cos A, A,., ( + z a = + ln( + z = m= ( m zm, z <, m m= a(a (a m + z m, z <, a, m! 44

ln(i + A = ( m= m Am, ρ(a <, m (I + A a = I + ρ(a A. m= a(a (a m + A m, ρ(a <, m!, e λi = e λ I, sin (λi = ( sin λi, cos (λi = ( cos λi. Lagrange-Sylvester, e A sin A n Jordan (,,,. 5.3. J λ n Jordan, sin A = 2! 3! 2! e A = e λ.......... 2! sin λ cos λ sin λ 2! cos λ 3! sin λ cos λ sin λ 2! (n! (n 2! sin [(2π/(n +λ] (n! sin [(2π/(n 2+λ] (n 2!.......... sin λ cos λ sin λ 2! sin λ cos λ sin λ (5.3. (5.3.2 5.3. A = ( 2, e A, sin A, cos A. A λ =, λ 2 = 2. A ( 2. λ, λ 2 α = ( (, α 2 =. 45

( P = (, A = P 2 ( e A e = P e 2 ( sin sin A = P sin 2 ( P = P = ( sin sin + sin 2 = sin 2 ( cos cos A = P cos 2 P. ( ( ( e e e + e 2 e 2 = e 2 ( ( ( sin sin 2, P = ( cos cos + cos 2 cos 2., 5.3.2 n A λ, λ 2,, λ n. e A, sin A, cos A e λ, e λ 2,, e λ n ; sin λ, sin λ 2,, sin λ n ; cos λ, cos λ 2,, cos λ n., ( 33 5.3. A A = s λ i P i. i= f(t r > ρ(a. f(a = s f(λ i P i (5.3.3 i= 5.3.3 4.2.3 A = 8 4 4 7 4 4 8 A = 9P 9P 2, sin A = ( sin 9(P P 2. e A. 5.3.2 ( AB = BA, e A e B = e A+B = e B e A ; (2 (e A = e A ; (3 e A = e tr A. ( AB = BA,, m (A + B m = CmA k m k B k. k= 46

e A+B m = C m! ma k m k B k m= k= m = k!(m k! Am k B k m= k= ( A k = B j = e A e B. k! j! k= j= (2 ( B = A, e A e A = I, (e A = e A. (3 A λ, λ 2,..., λ n, e A e λ, e λ 2,..., e λn, e A = e λ e λ2 e λ n = e λ +λ 2 + +λ n = e tr A., ( 34. 5.3.3 ( ( Euler 52 e ia = cos A + i sin A, cos A = 2 (eia + e ia, sin A = 2i (eia e ia, cos ( A = cos A, sin ( A = sin A. (2 AB = BA, cos (A + B = cos A cos B sin A sin B, sin (A + B = sin A cos B + cos A sin B..,.. 5.3. m n A a ij t, A, A(t = (a ij (t m n.,,,.. 5.3.2 A(t = (a ij (t m n. a ij (t, i m, j n, lim a ij (t = t t a ij ( a ij C, A(t t A = (a ij m n. 5.3. lim A(t = A, lim B(t = B. t t t t ( A(t, B(t, lim (A(t + B(t = A + B. t t 52 Leonhard Euler(77-783,,,,,,, 22. e ia = cos A + i sin A, e iπ + =. 47

(2 A(t, B(t m n, n s, lim (A(tB(t = lim A(t lim B(t = AB. t t t t t t (3 k, lim (ka(t = ka = k lim A(t. t t t t 5.3.2, A(t = (a ij (t m n, A(t,. A(t, : A (t = (a ij(t m n. A(t, : b a ( b A(t d t = a a ij (t d t. m n 5.3.2 ( (aa(t + bb(t = aa (t + bb (t; (2 (A(tB(t = A (tb(t + A(tB (t; ( t (3 A(sd s = A(t; (4 (5 a t a t a A (sd s = A(t A(a; BA(sd s = B t a A(sd s, t a A(sBd s = ( t a A(sd sb, B. 5.3.4 ( deat dt = Ae At ; d cos At (2 = A cos At, = A sin At. d sin At dt, deat dt t= = A; dt d sin At dt t= = A. (,, 34. A Jordan J n (λ = λi + N. 5.3.2(, e Jt = e λt e Nt, de Jt dt = λe λt e Nt λt dent + e dt. dent dt = Ne Nt. 5.3. A t t Ae As ds = e At e At (5.3.4 48

, A, t t A sin Asds = cos At cos At (5.3.5 t t A cos Asds = sin At sin At (5.3.6 t t e As ds = A (e At e At. 5.3.4 x = x(t = (x (t, x 2 (t,, x n (t T A = A(t = (A ij (t n n, x T Ax t. (x T Ax = (x T Ax + x T (Ax = (x T Ax + x T A x + x T Ax. ((x T Ax T = x T A T x = x T Ax. (x T Ax, (x T Ax = x T Ax., (x T Ax = x T A x + 2x T Ax... e A e B = e B e A?, f(x, f(af(b = f(bf(a?, A ( A? 2. e A e B, e B e A e A+B., e A e B = e B e A, e A e B = e A+B? 3., cos 2 A + sin 2 A = I? 4. (A(t 2 = 2A(tA (t? 5. A(t, (A(t? A (t? 6. A(t, A (t? 7. 5.4.2, A, R t t e As ds.? Lagrange-Sylvester. 5.4. J = ( λ λ......... λ s s, e Jt = e λt 49 t t 2 2! t s (s! t t s 2 (s 2!.......... t.

(2 A = P JP A Jordan J = J J 2 J s, e At = P (e J t e J 2t e J st P. (, (2 (. J = λi + N, N =......,... e Jt = e λti+tn = e λt e tn = e λt (tn k s = e λt k= t k N k. k! N k k,,. k= k! 5.4. e Jt, J = ( J J 2, J = a a a, J 2 = ( b b. 5.4., e Jt = e at t t2 2! t, e J 2t = e bt ( t., ( e Jt e J t = e J 2t = e at te at t2 e at 2! e at te at e at e bt te bt e bt. 5.4.2 N 3 Jordan, t e Ns ds = t t 2 2! t 3 3! t t 2 2! t. 5

A n. A m(λ = λ m + a λ m + + a s λ + a m, A I, A, A 2,, A m., f(a, m A g(a. g(t? n n +, g(a,. Lagrange-Sylvester, g(t g(t A.. 5.4. A m(λ = (λ λ k (λ λ 2 k2 (λ λ s k s, (5.4. λ, λ 2,, λ s. f(t = k= a kt k, { f(λ, f (λ,, f (k (λ, f(λ 2, f (λ 2,, f (k 2 (λ 2,, f(λ s, f (λ s,, f (k s (λ s } (5.4.2 f(t A. 5.4.2 (Sylvester f(t = k= a kt k, g(t = k= b kt k. f(a = g(a f(t g(t A,, f(λ i = g(λ i, f (λ i = g (λ i,, f (ki (λ i = g (k i (λ i, i =, 2,, s. A Jordan P AP = J = J J s, f(a = g(a f(j = g(j f(j i = g(j i, i, Lagrange-Sylvester, 4. 5.4. A, f(a = g(a f(λ = g(λ, λ σ(a. 5.4.3 A m(λ = λ 2 (λ π. sin A = A π A2. f(t = sin t, g(t = t π t2. 5.4.2, f(t g(t A, f(a = g(a., f( = sin = = g(, f ( = cos = = g (, f(π = sin π = = g(π. sin A = f(a = g(a = A π A2. 5.4.4 e At, A = 2 4 2 3. 5

A λi A = λ 2 4 λ 2 3 λ = (λ (λ 22., (x (x 2 A, A m(x = (x (x 2 2. f(λ = e λt ( t, e At = f(a. g(λ = a + a λ + a 2 λ 2 ( a, a, a 2. f(λ g(λ A, e t = f( = g( = a + a + a 2, e 2t = f(2 = g(2 = a + 2a + 4a 2, te 2t = f (2 = g (2 = a + 4a 2. a = 4e t 3e 2t + 2te 2t, a = 4e t + 4e 2t 3te 2t, a 2 = e t e 2t + te 2t. e At = e 2t 2e t 2e 2t + 3te 2t 4e t + 4e 2t e 2t 3e t + 3e 2t e t.. Lagrange. ( 4. 5.4. ( Lagrange f(x n, a i, i =, 2,, n +. i =, 2,, n +, L i (x = (x a (x a i (x a i+ (x a n+ (a i a (a i a i (a i a i+ (a i a n+ (5.4.3 n+ f(x = f(a i L i (x. i= (5.4.3 n L i (x Lagrange. 5.4. A m(λ = (λ λ (λ λ 2 (λ λ m, f(t r > ρ(a. f(a = m f(λ i L i (A, i= L i (x, i =, 2,, m Lagrange. 52

( 2 5.4.5 A = 2, e At. λi A = λ 2 2 λ = (λ+(λ 3, A λ =, λ 2 = 3, f(λ = e λt, Lagrange, m(x = (x + (x 3. f(a = f(λ L (A + f(λ 2 L 2 (A, L (A = A λ 2I = ( 2 2 λ λ 2 4 2 2, L 2 (A = A λ I = ( 2 2 λ 2 λ 4 2 2. f(λ = e t, f(λ 2 = e 3t. e At = f(a = e t ( ( 2 2 4 + e 3t ( 2 2 2 2 4 2 2 = ( e t + e 3t e t + e 3t 2 e t + e 3t e t + e 3t. A, Lagrange Lagrange-Sylvester,,, 44. 5.4.2 ( Lagrange-Sylvester n A s i= k i = m n. m(λ = (λ λ k (λ λ 2 k2 (λ λ s ks, f(a = s i= ( ϕ i (A a i I + a i2 (A λ i I + + a iki (A λ i I k i, ϕ i (A = (A λ I k (A λ i I k i (A λ i+ I k i+ (A λ s I k s, i s, d j ( a ij = (j! d λ j (λ λ i k f(λ i, i s; j =, 2,, k i. m(λ λ=λ i,.,, 5.3.4 x A, x T Ax?,. 53

5.5. A n, x T Ax x T x =. Lagrange L = x T Ax + λ( x T x, (L i 2 n a ij x j 2λx i. j= L 2 n a ij x j 2λx i =, i n, Ax = λx., j= x T Ax x T x =,!, Lagrange λ. L = f(x = f(x, x 2,, x n (f = ( f x, f x 2,, f x n (5.5., f(x = f(x, x 2,, x n n, f x,. 5.5. f( f(x = f(x, x 2,, x n n (. f (f,. f x = ( f x, f x 2,, f x n (5.5.2, (., (. 5.5.2 f = (f, f 2,, f m T : F n F m n (, n f i, i m, f m n f f f x x 2 x n (f, f 2,, f m f 2 f 2 f (x, x 2,, x n = x x 2 2 x n (5.5.3 f m x 2 (f,f 2,,f m (x,x 2,,x n ( f Jacobian, J(f. J(f, Jacobian., f (f, f 2,, f m, (5.5.3. (,,,.,., f J(f, (f f. f m x 54 f m x n

5.5.2 x n (, A m n, (J(x = x x = I n = xt x = J(xT ; (2J(Ax = Ax x = A, J(xT A T = xt A T x = A T. 5.5.3 f : (x, y T (a x + b y, a 2 x + b 2 y T F 2, ( a b J(f =. a 2 b 2 f!, f = (f, f 2,, f m T : F n F m, f - A m n, f ( A! ( 54. 5.5.4 f(x = f(x, x 2,, x n n, (= J(f = f x J(J(f f Hessian 53, H(f, H(f = J(J(f = 2 f x 2 2 f x x 2 2 f x 2 x 2 f x 2 2 2 f x n x 2 f x n x 2 2 f x x n 2 f x 2 x n 2 f x 2 n (5.5.4, f, Hessian H(f. f(x Hessian H(f f(x. 5.5.5 f(x, y = x 2 + 2xy + y 2 x, J(f = (2x + 2y, 2x + 2y, Hessian ( 2 2 H(f =. 2 2 5.5.6 f = f(x, y = (x 2 + 2xy + y 2 x, x 2 + 2xy + y 2 + y T ( f f, f 2, Jacobian J(f J(f = ( 2x + 2y 2x + 2y, 2x + 2y 2x + 2y + Jacobian J(f =. Jacobian ( 939 C n f Jacobian J(f =, f (., Jacobian ( 48. Jacobian S.Wang 54 98.,, 48. 53 Ludwig Otto Hesse(8-874,, Carl Gustav Jacob Jacobi. 54 Stuart Sui-Sheng Wang(, 946-,, Oakland. 55

( (,. 5.5. f, g n x, h(x n, p(x, q(x m, a, b F, ( ( J(af + bg = aj(f + bj(g; (2 ( J(pq T = pj(q + qj(p; (3 ( J(f/g = (gj(f fj(g/g 2 ; (4 ( J(f(h = f(h h J(hT. (4. x = (x, x 2,, x n T, h = (h, h 2,, h n. f(h(x x = (f(h, h 2,, h n = ( n i= f h i h i x, n i= f h i h i x 2,, n i= f h i h i x n., f(h h T h x = ( f, f,, f J(h T = ( h h 2 h n n i= f h i h i x, n i= f h i h i x 2,, n i= f h i h i x n... f(h h h = (h, h 2,, h n, h T h. 5.5.7 f = f(u, v, g(x, y = (x 2, ax + y 2. f(g = f(x 2, ax + y 2, J(f(g = f(g gt = J(f (x, y (x, y = ( f u, f ( 2x v = (2x f a 2y. u + a f v, 2y f v,. f = (f, f 2,, f m T g = (g, g 2,, g m T m, J(f T g 5.5.(2, 49. x T x (, 5 : x T x x x xt = xt + xt x x = 2xT (5.5.5 5.5.. Lagrange L x J(L = J(x T Ax + J(λ( x T x = x T J(Ax + (Ax T J(x T λj(x T x = 2x T A 2λx T, x T A = λx T, x T A x A T. 56

,,,, ( x, x T,. 5.5.8 ( x = (x, x 2,, x n T, y = (x, x + x 2,, x + x n T? f(g, f n, y = g(x = (x, x + x 2,, x + x n n (. 5.,. F n n F (n 2,. F n n X = (x ij, X D(X, tr(x trx,. D(X tr(x x ij (D(X = J(D(X tr(x n 2,, J(D(X = D(X X = J(tr(X = tr(x X == D(X x D(X x 2 D(X D(X x n x 2 D(X x 22 D(X x n tr(x x tr(x x 2 D(X x n2 tr(x D(X x 2n D(X x nn tr(x x n x 2 tr(x x 22 tr(x x n tr(x x n2 tr(x x 2n tr(x x nn (5.5.6 (5.5.7 (5.5.6 (5.5.7., ( 52. 5.5.2 X n,,, ( x x 5.5.9 X = 2 x 2 x 22 J(D(X = (adjx T, J(tr(X = I n (5.5.8, D(X = x x 22 x 2 x 2, ( x22 x J(D(X = 2 = (adjx T. x 2 x tr(, AXB(, A, B, J(tr(AXB = tr(axb X, 53. 57

5.5. x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ. J = (x, y, z (r, θ, φ = sin θ cos φ sin θ sin φ cos θ r cos θ cos φ r cos θ sin φ r sin θ r sin θ sin φ r sin θ cos φ, J = r 2 sin θ. n f(x = f(x, x 2,, x n J(f, f(x Hessian H(f. H(f (x, x 2,, x n f(x (x, x 2,, x n., n H(f (x, x 2,, x n?, f(x p q X m n, f(x X?,, vec, (. 5.5.3 f : F p q F m n p q, f(x = f (X f 2 (X f n (X f 2 (X f 22 (X f 2n (X f m (X f m2 (X f mn (X, X = (x ij p q, f(x mp nq J(f = f(x X = vec(f(x vecx (5.5.9 5.5. f : F 2 2 F 2 2 2 2, f(x = X, J(f = vec(f(x vecx = = I 4!, ( x ( n f(x, x 2,, x n, f(x x = (x, x 2,, x n T, f(x R n R, vecx. (, 53. 5.5.3,, ( f = f(x = (f ij (x m p g = g(x = (g ij (x p q n J(f(xg(x = (g(x T I m J(f(x + (I q f(xj(g(x (5.5. J(xx T = x I n + I n x (5.5. 58

(2 A C m n, B C p q, X C n p, J(AXB = AXB X = BT A; J((AXB T = BT X T A T X = B A T = (J(AXB T. (3 A C m n, X C n m, tr(ax X = tr(xa X = AT. (4 A C n n, X C m n, tr(xax T X = X(A + A T. (5 α, β C n, X C m n, (α T X T Xβ X = X(αβ T + βα T. (6 α, β C m, X C m n, (α T XX T β X = (αβ T + βα T X. (7 X C m n, f = f(x = (f ij (X p q g = g(x = (g ij (X m n, (f(g(x X = (f(g (g g X., (, [6] : 5.5.. ( X R m n, f(x X ( X f(x, J(f(X = f(x X X=X = ;, f(x J(f = f(x X =. (2 f(x X, f(x. 5.5.2 ( X R m n, f(x X. X f(x c(x =, L(X, λ = f(x λc(x Lagrange. λ R, J(L X (X, λ = L(X,λ X (X,λ =, c(x =. 59

5..,. K CDMA, k A k, b k (= ±. : y = RAb + n (5.5.2 y, K R (, b = (b,, b k T, K n σ 2 ( n Gauss. : k m k ( ˆb k = sgn(m T k y k, K? M M = (m,, m K (5.5.3 J(M = E{ b My 2 2} (5.5.4,. ( J(M J(M = tr(i + tr(m(ra 2 R + σ 2 RM T tr(arm T tr(mra (5.5.5 5.5., J(M M 5.5.3 ( 46, R (5.5.6 M, R, J(M M = (5.5.6 J(M M = 2M(RA2 R + σ 2 R 2AR (5.5.7 M(RA 2 R + σ 2 R = AR (5.5.8 M = A(RA 2 + σ 2 I. ( BMUD,.. α C m, β C n, J(α T Xβ = αt Xβ X =? 2. α C m, β C n, J(α T X T β = αt X T β X =? 3. X, J(X 2 =? 4. J(X = X,?? rvecx 5. Jacobian. I. -, : 6

u (t u m (t. x (t. x n (t. y (t y s (t 5.6. u (t,, u m (t m, y (t,, y s (t s, x (t,, x n (t n. x = x(t = (x (t,, x n (t T, u = u(t = (u (t,, u m (t T, y = y(t = (y (t,, y s (t T,, (, (. { x (t = Ax(t + Bu(t (5.6. y(t = Cx(t + Du(t A C n n, B C n m (, C C s n (, D C s m. D,, D =. A, B, C,, (A, B, C. (5.6.,., t, t t., x(t = (x (t,, x n (t, (5.6..., (5.6., (5.6.. 5.6. n =, x (t = ax, x(t t=t = x(t. x = x(t e a(t t. x (t = ax + f(t, x(t t=t = x(t, f(t. t x = x(t e a(t t + e a(t s f(s d s. t, { x (t = Ax x(t t=t = x(t, (5.6.2 x = x(t = (x (t, x 2 (t,, x n (t T, x(t = (x (t, x 2 (t,, x n (t T n, A n. n 2,, (5.6.2 5.6., Taylor, : x (t = Ax x (t = A 2 x,, x (k (t = A k x, (5.6.3 6

x (t = Ax(t, x (t = A 2 x(t,, x (k (t = A k x(t, (5.6.4 x(t t = t Taylor x = x(t + (t t x (t + (t t 2 2! x (t + + (t t k k! x (k (t + = x(t + (t t Ax(t + (t t 2 2! A 2 x(t + + (t t k k! A k x(t + = e A(t t x(t (5.6.5 (5.6.5 ( 55, 5.6. (5.6.2, x(t = e A(t t x(t. 5.6.2 { x (t = Ax(t, x(t t= = x(,, A = 2 2 2 2, x( =. 5.6., x(t = e At x(. e At. A Jordan J =, P =., x(t = e At x( = P e Jt P x( e t = e t te t e t = 2t + 2t + 2t + e t. 2 A = (a ij n n B = (b ij n m, u(t = (u (t, u 2 (t,..., u m (t T, x(t = (x (t, x 2 (t,..., x n (t T. { x (t = Ax(t + Bu(t (5.6.6 x(t t=t = x(t 62

, (5.6.6, Lagrange., (5.6.6 e At, At d x(t e e At Ax(t = e At Bu(t. (5.6.7 d t (e At x(t, ( 58 5.6.2 (5.6.6, t x(t = e A(t t x(t + e A(t s Bu(s d s. t. x = e At y y = e At x. 5.6.3 { x (t = Ax(t + u(t x(t t= = (,, T, A = 3 2 2, u(t = (,, e 2t T. 5.6.2, x(t = e At x( + t e A(t s u(s d s. e At. λi A = λ 3 2 λ λ 2 = λ(λ 2(λ 3. A 3, A. λ =, λ 2 = 2, λ 3 = 3 α = (, 5, 2 T, α 2 = (,, T, α 3 = (2,, T. P = 2 5 2, P = 6 3 3 9 2 2 4. 63

e At x( = P = e 2t 2 5 2 = 6 e 3t P x( e 2t + 3e 2t 8e 3t 5 + 3e 2t 4e 3t 2 4e 3t e 3t. ( 6 3 3 9 2 2 4 t ea(t s u(s d s. t e A(t s u(s = 6 e A(t s u(s d s = 6 e 2s + 9e 2t 8e 3t s 5e 2s + 9e 2t 4e 3t s 2e 2s 4e 3t s. 2 + (9t + 5 2 e2t 8e 3t 5 2 + (9t + 3 2 e2t 4e 3t + 3e 2t 4e 3t. x(t = 6 2 + (9t + 2 2 e2t 6e 3t 5 2 + (9t + 9 2 e2t 8e 3t + 3e 2t 8e 3t.. : { y (n + a y (n + a 2 y (n 2 + + a n y =, y (i (t t= = y (i, i n. (5.6.8 x = y, x 2 = y = x, x 3 = y (2 = x 2, x n = y (n = x n, x = x 2, x 2 = x 3, x n = x n, x n = a n x a n x 2 a x n. x(t = (x (t, x 2 (t,, x n (t T, x( = (x (, x 2 (,, x n ( T = (y, y,, y (n T. 64

(5.6.8 { x (t = Ax(t x(t t= = x( (5.6.9 A = a n a n a n 2 a. (5.6., A (5.6.8 r n + a r n + a 2 r n 2 + + a n. 5.6., (5.6.9 x(t = e At x(. (5.6.8 (5.6.9, (5.6.8 y = (,,, x(t = (,,, e At x( y = (,,, e At y.. y (n (5.6.8 n : { y (n + a y (n + a 2 y (n 2 + + a n y = u(t, y (i (t t= = y (i, i n, (5.6.. (5.6., { x (t = Ax(t + Bu(t, x(t t= = x(, x(t = (x (t, x 2 (t,, x n (t T = (y, y,, y (n T, x( = (x (, x 2 (,, x n ( T = (y, y,, y(n T, A =, B =. a n a n a n 2 a 5.6.2, (5.6. ( y(t = (,,, e At x( + 65 t e A(t s Bu(s d s..

5.6.4 y(t: { y (3 3y (2 6y + 8y = u(t, (y(, y (, y ( = (,,. x(t = (x (t, x 2 (t, x 3 (t T, x (t = y(t, x 2 (t = y (t, x 3 (t = y (t. x( = (,, T., ( y(t = (,, e At x( + t e A(t s Bu(s d s, A = 8 6 3, B =. A λi A = λ 3 3λ 2 6λ + 8 = (λ (λ + 2(λ 4. A Jordan J = 2, P = 2 4. 4 4 6 e At = P = e t e 2t 2 4 4 6 = 8 e 4t P e t e 2t e 4t 8 6 4 2 4 5 2 6e t + 4e 2t 2e 4t 4e t 5e 2t + e 4t 2e t + e 2t + e 4t 6e t 8e 2t 8e 4t 4e t + e 2t + 4e 4t 2e t 2e 2t + 4e 4t 6e t + 6e 2t 32e 4t 4e t 2e 2t + 6e 4t 2e t + 4e 2t + 6e 4t e A(t s Bu(s = 8 u(s 2e t s + e 2(t s + e 4(t s 2e t s 2e 2(t s + 4e 4(t s 2e t s + 4e 2(t s + 6e 4(t s., ( y(t = (,, e At x( + t e A(t s Bu(s d s = 8 (4et + 5e 2t e 4t t 9 et e s u(s d s + t 8 e 2t e 2s u(s d s + 2 t 9 e4t e 4s u(s d s.. n (5.6.8,,,,, 66

(5.6.8. 2. (5.6. Laplace. II (5.6.. (5.6., x(t., (, y(t x(t. y(t,.,,. 5.7., [, t ] u(t ( t t, x( x(t =, x(. x(,..,, x(, ( x(t,., ( x(t =, 62. 5.7. n =, x (t = ax(t + bu(t. b. 5.7.2 ( x λ (t = λ 2 ( a x(t + b u(t, ab. 5.7. (A, B, C n Hermite W (, t = t e ta BB e ta d t (5.7.. : W (, t. 5.6.2, t x(t = e At x( + e A(t t Bu(t d t. (5.7.2 u(t = B e A t W (, t x(. (5.7.3 67

u(t (5.7.2 x(t = e At x( t ea(t t BB e A t W (, t x( d t ( = e At x( e At t e At BB e A t d t W (, t x( = e At x( e At W (, t W (, t x( =. u(t, x( x(t =. x(,. : α = (a, a 2,, a n T W (, t α =. (5.7.4 α W (, t α =. t α (e At BB e A t α d t =. α e At B =, t t. (5.7.5, u(t x(t =. (5.7.2, t e At x( + e A(t t Bu(t d t =. (5.7.5, x( = e At t t e A(t t Bu(t d t = e At Bu(t d t. t α x( = α e At Bu(t d t =. x( α =, α. W (, t. 5.7.,. 5.7.2 (A, B, C W = (B, AB, A 2 B,, A n B n. n nm W. (A, B, C x(, u(t (5.7.2( x(t =, t e A(t t Bu(t d t = e At x( (5.7.6 x(. e At, (5.7.6 t e At Bu(t d t = x( (5.7.7 68

Cayley-Hamilton, n e At n e At = a i (ta i, (5.7.7 n (A i B i= z i (t T = t t i= (a i (tu(t d t = x( (5.7.8 a i (tu(t d t, i n. z = (z (t, z (t,, z n (t T, (5.7.8 (B, AB, A 2 B,, A n Bz = x( (5.7.9 (A, B, C (5.7.9 n..? z,, (5.7.9? 5.7.2, B,. (,. (5.6., ( : { x(k + = Ax(k + Bu(k, k (5.7. y(k = Cx(k, k x(k, u(k, y(k,, (5.7.. (5.7. (A, B, C., (5.7. k x(k = A k x( + A k i Bu(i, k (5.7., i= x(n = A n x( + (B, AB,, A n B(u(n, u(n 2,, u( T (5.7.2 5.7.2., (A, B, C (A, B, C. 5.7.3 A = 2 2, B = 2. (A, B, C. 69

, (A, B, C,. 5.7.2. W = 2 5 3 4 2 2 5 3 4 3 4 2 3 2 9 3 W 3,.. 5 3 4 3 4 2 2 4 5.7.4, B n, W n. 5.7.2, W. :,,. B, W, W,, B, (B, AB,, A.. 5.7.2, [, t ] y(t x(,. y(t [, t ], ( ( (., C,. 5.7.5 C,. 5.7.6 ( x λ (t = λ 2 y(t = (a, bx(t x(t + Bu(t, C = (a, b λ λ 2. 65. 5.7.3 (A, B, C n Hermite M(, t = t e A t C Ce At d t. 7

: y(t = Cx(t = Ce At x( + C t e A(t s Bu(s d s. Ce At x( = y(t C e A t C, t, M(, t x( = M(, t, t x( = M(, t t t ( y(t C ( y(t C e A(t s Bu(s d s. t x(,. :. y(t = Ce At x( + C t t e A(t s Bu(s d s d t. e A(t s Bu(s d s d t. x(. M(, t, α C n M(, t α =. e A(t s Bu(s d s (5.7.3 α M(, t α =. t α e A t C Ce At α d t =. y(t = Ce At (x( + α + C t e A(t s Bu(s d s. x( (5.7.3, x( + α (5.7.3., α, x(,. M(, t., (A, B, C (A, B, C, 66. 5.7.4 (A, B, C M = C CA. CA n n. M. 7

5.7.7 ( x (t = x(t + ( y(t = x(t. ( u(t,. B,. ( C M = = CA 2. 2 M 2,.,.,.,.. A Jordan J = P AP, P. z(t = P x(t x(t = P z(t, (A, B, C { z (t = Jz(t + P Bu(t y(t = CP z(t (5.7.4 (5.7.4,. 5.7.8 (A, B, C, A, P B, CP, 63 66.. 5.7.8, A A? 2. (A, B, C (5.7.4?. C n, : ( ; (2 x : x x = ; (3 x = x ; (4 x y x y. 2. : x C n, ( x 2 x n x 2 ; (2 x x n x ; (3 x x 2 n x. 3. ( R 2, p- ; (2 ; 72

(3 R 3 ( (2, - - ; (4 < p <, l p,. p = /2, 3/2, p < p. 4. Minkowski (5... 5. ( (2 ; x + y 2 + x y 2 = 2( x 2 + y 2. (3 :, ; (4 (3 l p,. 6. 5..4. 7. 5... 8. 5..3. 9. 5..8.., x, y, αx + βy, α, β α + β =. 5,..??.. 2. A F- a, U, AU UA F-?. 3. -, 2- - -, 2- - ( 4. : ( -, -, ; (2 -. 5. (5..2. 6. ( A 2 = (ρ(a A /2, A ; (2 ; (3 A, A ρ(a; (4 V Hermite, V. 7. A 2 /3. 8. ( : ; (2 α β, p >. [( α p + ( β p ] /p? (3 (? 9. A A.? 2. (5..3. 2. T, A R n n. : ( T 2 = ; (2 A 2 = T A 2 ; (3. 73

22. A, B n, A B,. A Cond(A = A A. : A B / A.. 23. ( A C m n, σ, σ 2,, σ n A. : ( Cond(A = σ (A/σ n (A, σ (A σ n (A A.( 4.5.5. (2 A F = `P r i= σ2 i /2 = (tr(a A /2 ; (3 A 2 = σ max(a. 24. ( 5..3; (2 U, V (, σ Hom(U, V. : σ σ. 25. 5.2. 5.2..! k 2 +k k 26. A k = 2 k 2 +, lim 2 ( A k. 2 k k k 27. lim k A k = A. ( A k, A? (2 A k, A? (3 A k, A? 28. lim n An = B, B. 29. 5.2.. 3. A = 3. A = @ 2 2 2 «, X k=.6.8.2.6.8 32. ( A = 2 2 (2 J = B @ 33. ( 5.3.; A k 2 k. A. A. «, e A, sin A, cos A; 2 C A, ej, sin J, cos J. (2 (,. 34. 5.3.3 5.3.4(2. 35. A, e At : ( A = @ 2 2 3 3 A, (2 A = @ 8 2 6 A, (3 A = @ 2 3 3 2 2 A. 36. : cos A, sin A, e A : (A ; (2A ( A 2 = I. 37. A(t = B @ sin t cos t t sin t t e t t 2 t 3 C A, t. lim t A(t, 74 d d t A(t, d 2 d t A(t. 2

38. A(t = @ e 2t te t t 2 e t 2e 2t 3t A, Z 39. : ( A, e A ; (2 A Hermite, e ia. A(t d t d d t Z t 2 A(s d s. 4. 5.4.2. 4. 5.4. Lagrange. 42. ( J n (λ n Jordan, sin Jt, cos Jt; (2 n A, sin At cos At. 43. ( A = @ 2 A, e At, sin At, cos At; 3 «(2 A =, e At, sin At, cos At. 44. 5.4.2 Lagrange-Sylvester.( : f(x m A (x Lagrange. 45. N n, dent dt 46. ( A = (2 A = «, R t eas ds; «, e A e At ; (3 A 2 = A, e At R t eas ds. 47. A 2 A + 2I =, e At R t eas ds. 48. ( Jacobian ; (2 Jacobian ; = Ne Nt R t ens ds. (3 5.5.6. 2 ( 2 Jacobian. n? 49. f = (f (x,, f n (x T g = (g (x,, g n (x T n, J(f T g ( 5.5.(2. 5. x = (x, x 2,, x n T, A n, n ( x Ax 2 2 Hessian. 5. 5.5.8. 52. 5.5.2. 53. ( 5.5.3, J(tr(AX J(tr(XB; (2 (5.5.7. 54., 5.5.3. 55. (5.6.5 (5.6.2. 56. ( x (t = 2 4 3 «x(t; (2 x (t = @ 75 Ax(t.

57. x (t = Ax(t x( ( A = 2 3 «, x( = «; (2 A = @ 58. 5.6.2. 2 2 A, x( = @ A. 59. ( x (t = @ 2 2 3 2 Ax(t + @ Ae 2t. (2 x (t = Ax(t + Bu(t x(, A = @ 6 6 A, B = @ 2 6 2 A, u(t =, x( = @ 6. y + 6y + y + 6y = e t y( = y ( = y ( =. 6. ( x (t = Ax(t + γe at x(t = βe at (αi Aβ = γ, β, γ n, a C; (2 x (t = Ax(t + e 2t C, A = 3 2 2 «, C = «, x( = «. A. 62. 5.7..( : 5.7.. 63. ( A, ; (2 A Jordan, ; (3 A Frobenius ( (5.6., B e n,.. 64. 63,. ««2 2 (A =, B = (, T ; (2A =, B = (, T ; ««a a (3A =, B = (c, d T ; (4A =, B = (c, d T. b a 65. 5.7.6. 66. 5.7.4. 67. ( A, ; (2 A Jordan, ; (3 A Frobenius, C e T n,.. 68. 67,. «a a (A =, C = (c, d T ; (2A = @ a A, C = (c, d, f T. a a 76