,,,,,,., Penrose i,, i j X A {i,, i j }-, X A {, 3}-, A,3 ; A Moore- Penrose A = A,2,3,4., A 5,, Moore-Penrose A {}- A, A. m n Moore-Penrose A, {}- A,

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1 , Ax = b A m n m = n, x = A b., A, A A = UR : x = R U b 6.. A Ax = A b, A A. A = R U, A A = I n,, A, A A. n < m, AA = In m m 6..2 A n < m, AA = I m,, A = R U A. A? A, B, AB BA,., A m n F n F m. A A F m F n. A A AA F n F n F m F m. A A F n, A A x F n, ImA A RA. A A A RA. AA A RA. A A,?,., Moore m n. A C m n, Moore AX = P RA XA = P RX 6..3 X C n m, P L L. 6..3, Moore. 955, Penrose 56 : X C n m AXA = A 2 XAX = X 3 AX = AX 4 XA = XA 6..4 X A C m n Moore Penrose., Moore,, A, P RA A, Penrose,! Penrose., Penrose 55 Eliakim Hastings Moore ,,,. 56 Sir Roger Penrose93-,,,. 988 Wolf. Stephen Hawking. 77

2 ,,,,,,., Penrose i,, i j X A {i,, i j }-, X A {, 3}-, A,3 ; A Moore- Penrose A = A,2,3,4., A 5,, Moore-Penrose A {}- A, A. m n Moore-Penrose A, {}- A, {, 3}- {, 4}-,. Moore-Penrose Moore Penrose, Moore Penrose. Moore P RA P RX., C n A A 2 = A, C n = RA NA, A C n, A RA, A NA. A C n NA RA., RA = NA, A, A, A x Ax. RA = NA, A 2 = A A = A, A.,, A A 2 = A, A = A., L M C n, C n = L M, C n M L P L,M. M = L, P L,M P L. P L,M P L,M. 6.. A =,, P L,M x y x + y. L = ImP L,M = RA = {x, y T C 2 y = }, M = KerP L,M = NA = {x, y T C 2 x + y = }. P L,M. dim L = r, dim M = n r. {α, α 2,..., α r } {α r+, α r+2,..., α n } L M. P L,M α i = α i, i r, P L,M α j =, r + j n. X = α, α 2,..., α r, Y = α r+, α r+2,..., α n, 78

3 X, Y n r n n r, P L,M X = X, P L,M Y =. P L,M X, Y = X,. X, Y n, P L,M = X, X, Y L, T R 2, M, T R 2, R 2 M L P L,M = = =. 6..2,. 6.., A, Ax = b Ab., b RA Ab.,, A Ax = A b, Ax = Ab, x = Ab. P L. L α, α 2,, α r, L {α r+, α r+2,, α n }. X = α, α 2,, α r, Y = α r+, α r+2,, α n. X, Y n r n n r X Y = Y X =, 6.. P L = X, X, Y = X, X, Y X, Y X, Y X = X, X, Y X, Y Y X = X, X Y Y X Y = XX X X. P L = XX X X 6..2, X L., α, α 2,, α r L, X X = I, P L = XX R 3 L α =,, T α 2 =,, T, P L α =,, T L. 79

4 6..2 X = α, α 2 = X X = X X = 2,. = 2, P L = XX X X = 2 =. 6..3, α L P L α = P L α,? =. 6.. A C m n, X C n m Penrose 6..4, X A Penrose., A, A A Penrose. m n Penrose n m Penrose A C m n C n C m, X C n m C m C n. Penrose 2 AX XA, C m RA C n RX., RA RX. Penrose 3 4 AX XA Hermite, Penrose AX XA RA RX, Moore 6..3! 6..2 A C m n. X C n m Moore 6..3, X A Moore., I n C n, A, Moore Penrose., Penrose A C m n, A Penrose, A. 8

5 A, X. A. A : σ... A = U σ r V, m n U, V n m, r A. X = V σ... σ r... n m U, 6..5 X Penrose. A. X Y A Penrose, X Y Penrose X = XAX = XX A = XX A Y A = XAY = XAA Y Y = A Y Y = Y AY = Y. Penrose.. Penrose A Penrose,, A A Penrose,. x x Penrose x x x., x Penrose 6..7 A = Penrose B T. Penrose. B = 6..8 n Jordan J n Penrose J T n. Penrose, Penrose A A AA, RA = RA, NA = NA. 8

6 6..3 Moore Penrose A Moore Penrose. Moore Penrose. A Moore-Penrose. X A Moore, AX 6..4, n α, AXAα = Aα, AXA = A., XAX = X. 6.., AX = AX, XA = XA. X A Penrose., X A Penrose, AXA = A, AX RA. RA = NA 6.., NA = NX, α RA, Xα =. AXα =, AX RA. AX = P RA. XA = P RX. X 6..3, X A Moore , Proj RA b, 2.3.2? 6..3 Moore A RA AA, Proj RA b = AA b Moore-Penrose, Moore-Penrose, A =,, B =, T, AB = B A = /2, AB B A. A, λ = { λ, λ, λ = Moore-Penrose A,, A, A = A; 2 A = A ; A T = A T ; 3 λa = λ A ; 4 diag λ,, λ n = diag λ,, λ n; 5 A AA = A, A AA = A, AA A = A, A A A = A ; 6 A A = A A ; 7 A = A A A ; 8 A = B + C, B C = BC =, A = B + C ; 9 ra = ra = ra A = tra A., Moore-Penrose, Penrose Moore. HomU, V Moore-Penrose HomV, U. 82

7 6.. R 2 σ : x, y T y, T, Moore- Penrose σ : x, y T, x T, σσ : x, y T x, T Imσ, σ σ : x, y T, y T Imσ. 6.. V = F[x] n, V : fx f x,, V, Moore-Penrose.,, x, x 2,, x n V, Moore-Penrose 3 { x i x = i /i, i n, i = n.. 2 2? 2. n A B AB = BA =? 3. P, Q, P AQ = Q A P? Moore-Penrose Moore-Penrose A., , A A SVD A A = UDV, A = V D U A, A =. A A = UDV = , A = = 4. A,, A A, A = A A A ; 2 A, A = A AA ; 3 A = LR C m n r, L, R. A = R L = R RR L L L

8 6.2.2 A = αβ, A = A α 2 2 β , A =,, T,, A = /4A T A, A = 2 3. A A = LR = 2, RR = RR = 3 L L = 2 L L = , 2. = = ,, A = R RR L L L = = A, A A A QR A A = QR, Q, R. A = R Q

9 6..27, A A A A. Hermite Moore-Penrose. A σ,, σ s, s >, Hermite A A A A = σ 2 P + + σ 2 sp s, P i = P i, P i P j = δ ij P i s A A = σ 2 P + + σ 2 s P s s φx = σi i=x 2, φ i x = x σj j i i= P i = I, 7 φ j A A = φ j σ 2 j P j, j s P j = φ ja A φ j σ 2 j, j s A Lagrange-Sylvester Lagrange-Sylvester A = s i= σ 2 i A A σj 2I j i σi 2 σ2 j A j i A =, A., A T A = 2, λi A T A = λλ λ 3. A σ = 3, σ 2 = A = [ 3 AT A 3I 3 + AT A I 3 ]AT = 4 3 I 3 AT AA T = A, AX = I 2, A,,. 85

10 . 6.2.? 2. Moore-Penrose? 3. Moore-Penrose? {}- Moore., A m n. n m G A {}-, A, m b, Ax = b, x = Gb.. {} A =, A =, y, y. =,. 6.3., A AA b = b, Penrose {} n m G m n A {}- AGA = A. : z C n, b = Az m, z Ax = b. x = Gb, AGb = b. AGAz = Az. z, AGA = A. : AGA = A, Ax = b, AGb = AGAx = Ax = b. x = Gb., G A {}-. A A {} ,, A. A{} A {}-,, m n {} = C n m A{} = {X AXA = A} 6.3. A, AXA = A A, A {}-, A{} = {A = A } , n m m n {}-!, m n = n m. {}-, G A {}-, A G {}-. A = A. 86

11 6.3.4 A C m n. AX = I A., XA = I A.,,. 6.3., A {}-. 9. {}-, {}-, P, Q, Q A P P AQ{}, Q A P P AQ {} P AQ = Q A P,,,., A A, A,2 {}- {, 2}-., 6.3., A, AB = B A, A C m n, P C m m Q C n n Ir P AQ =, Ir X M = Q Y Z P, A {}-, A {} , X C r m r, Y C n r r, Z C n r m r. AMA = P Ir = P Ir X = P Ir Q Ir X Q Y Z Ir Q = A. Q P P Ir Q M A {}-., A A {}-. P, Q, A W X = Q P. Y Z A = AA A, P Ir Q = P Ir = P W X = P W W X Q Q Y Z Ir Q W Q = P 87 P P Ir Q. Q

12 W = I r, A {}-,, A C m n, A A {}, A A {}- ; 2 ra ra ; 3 A A{}, A{} = {A }; 4 λ A λa{}, λ A λa {}- ; 5 AA A A ; 6 RAA = RA, NA A = NA., A {}- ra, A, A = 4. 5 A I3 I 2 = P = 4, Q =. A {}- A = Q I 2 X P, X C 2. {} , A {}- AXA = A,, AXA =., AXA = A, AXA =. 88

13 6.3.3 A C m n, ra = r. AXA = C n m mn r 2 ; 2 AXA = X = Y A AY AA, Y C n m C n m C m n σ : X AXA, X C n m AXA = Kerσ., σ A T A, dim Kerσ = dim NA T A = mn r 2, ra B = rarb AXA =. Y AXA =. AY A =, A AY AA =. Y = Y = Y A AY AA 6.3.3, AXA = , A A {}-, AXA = X = Y A AY AA, Y C n m A =, A = /2 X = Y A AY AA a b = a c a, b, c. AXA = 3., AXA =, 6.3.4, Rao 57 A C m n, A A {}-, A = a, b, c. A{} = {A + Y A AY AA Y C n m } X = A x y + x z, , AXA = A = a b + a c 57 Calyampudi Radhakrishna Rao92-,,,, Pennsylvania., 89

14 AXA = A AXA I n = AX I m A =. Y = XA I n, Z = AX I m, AY =, ZA =. AXA = A, A C m n, A C n m A {}-, AXA = A X = A + Y I m AA + I n A AZ, Y, Z C n m {} A C m n, A A A = I n ; A {}- A ; 2 A AA = I m ; A {}- A., 2. : A, P C m m, Q C n n In P AQ =. 6.3., A A = Q[I n, X]P. A A = Q[I n, X]P P In Q = QI n Q = I n. : A A = I n, A.,, A A? A C m n, ra = r, A A {}-, ri n A A = n r. P AQ = Ir P, Q m n, A Ir X = Q Y Z, m n P. n m I n A Ir X A = I n Q Y Z Ir = I n Q Y P P Ir n m Q = Q n n Y Q m n I n r Q. 9

15 ri n A A = n r , A, A A,, A, A A.. {}-, {}-? 2. {}-? 3. A A AA? 4. A {}-? {3}- {4} X Penrose, AXA = A, XAX = X X A {, 2} {, 2}-,. {, 2}-,., {, 2}-, x T, x. {, 2}-, A{, 2} A {, 2}-., A{, 2} A{} A = B =,, C =, D = /6A T A {}-, {, 2}- Moore-Penrose, B, C A Moore-Penrose, B A {, 2}-. A{, 2} A{}. {, 2}-, A C m n, ra = r, P C m m, Q C n n P AQ = Ir, Ir B A{, 2} = {Q C CB P B C r n r, C C m r r }

16 {, 2}-, A C m n, A{, 2} = {X AX 2 X, X 2 A{}}; 2 X A{, 2}, rx = ra; 3 P, Q, P AQ,2 = Q A,2 P X Penrose, AXA = A, AX = AX X A {, 3}-. A{, 3} A {, 3}-. A{, 3} A{} A C n m m n {, 3}-. {, 3}-,., {, 3}-, x T, x, A{, 3} = A{, 2}., T {, 3}-?, =, T B A C m n {, 3}-, A{, 3} = {B + I n BAY Y C n m } A{, 3} A{}, X = B + I n BAY AXA = A. A[B + I n BAY ] = AB B A{, 3} AX Hermite A = αβ, A A{, 3} = { α α β β ββ + I β Y Y } β {, 3}-, A,3 A C m n {, 3}-, A{, 3} = {X AX = AA,3 }; 2 AA,3 = AA = P RA ;, AA,3 ; 3 A,3 A ; 4 I m A,3 I m A{, 3} X Penrose, AXA = A, XA = XA X A {, 4}-. 92

17 , A{, 4} A {, 4}-. A{, 4} A{}., A {, 4}- {, 3}- XA = XA AX = AX,. A {, 4}-, 3-32, {, 3} A C n m m n {, 4}-. {, 4}-,., {, 4}-?,, T, A,4 = A., T {, 4}-, x T, x, A{, 4} = A{, 2} B A C m n {, 4}-, A{, 4} = {B + Y I m AB Y C n m } A = αβ, A A{, 4} = { α α β β + Y I αα α α Y } {, 4}-, G A C m n {, 4}-, A{, 4} = {X XA = GA}; 2 GA = A A = P RA ;, GA ; 3 AG ; 4 G I n A I n {, 4}.., {, 2}-? 2. Hermite {, 2}- Hermite? 3. {, 3}- {, 4}-? 4. {, 2}-, {, 3}-, {, 4}-? 5. A{, i} = A{, j}, i j 4?, AXB = C. Ax = b,,..,.,,. {}-. 93

18 6.5. A m n A {}-, Ax = x = I n A Az, 6.5. z n. AA A = A I n A Az Ax =. A r, Ax = n r. L = {I n A Az z }, L I n A A, ri n A A = n r.. Ax = b, - A b Ax = b A m n A {}-, Ax = b, x = A b + I n A Az, z n Ax = b,, x 2 = x x x., x x Ax = b. G A{, 4}, Ax = b x = Gb + I GAz, z G A{}, x = Gb Ax = b Gb 2 x 2, x, Ax = b G A{, 4}., {, 4}-. y = Gb. x Ax = b , Ax = b x = Gb + I n A Az. x 2 2 = x x = A b + I n A Az A b + I n A Az = A b A b + I n A Az I n A Az +A b I n A Az + z I n A A A b. A b I n A Az = A Ax I n A Az = x A A I n A Az = x A AI n A Az = x A A A AA Az =. z I n A A A b = z I n A AA Ax = z A A A AA Ax =. x 2 2 = A b A b + I n A Az I n A Az A b A b. 94

19 y = A b Ax = b. A Moore-Penrose A A {}- A, Ax = b, y = A b , A C m n, G C n m, G A {, 4}- G XA = I n, GA I n F = min X C n m XA I n F Ax = b. x = Gb Ax = b G A{, 3}., Ax = b {, 3}-. x = Gb + I GAy, y 6.5.4, Ax = b. A AA b = A b. P AQ = A Ax = A b Ir P, Q m n. A Ir X = Q P, Y Z n m AA = P Ir Q Ir X Q Y Z m n AA = AA, P Ir X, m n n m P = P Ir X P = P Ir X P. P. m m Ir X P P = P P Ir X A Ay = A AA b = Q Ir = Q Ir Ir X n m = Q Ir y = A b Ax = b. P b = A b. n m. P P Ir X n m P P P b m m P b m m 95

20 6.5.2 A C m n, G C n m, G A {, 3}- G AX = I m, AG I m F = min X C n m AX I m F., AX = I m I m AvecX = veci m X F - vecx l 2, X AX = I m vecx , vecx vecx = I m A,3 veci m + I mn I m A,3 I m Ay, y vecx = I m A,3 veci m + I mn I m A,3 Ay, y X = A,3 + I n A,3 AY, Y C n m, X A {, 3} A C m n, Ax = b, Ax = b, y = A b. A 6.5.4, y = A b Ax = b. x Ax = b, x A Ax = A b, x = A b + z, z A Ax =. A Ax = Ax =, z Ax =, Az =. x x = A b + z A b + z = A b A b + z z + z A b + A b z, z A b = z A AA b = z A A A b = A Az A b =, A b z = A AA b z = A b A A z = A b A Az =, x x = A b A b + z z A b A b y = A b Ax = b., 6.5.6, x x = A b A b z =, x = A b,. AXB = C, Penrose AXB = C AA CB B = C; X = A CB + Y A AY BB. 96

21 , Penrose.,, N. origin node destination node OD. N n OD. k OD x k OD i, j, x k i j. X = x,, x n T N. N m, i y i. Y = y,, y m T N. {, j OD i, a ij =,. A = a ij m n N., : AX = Y N, X., OD n m, 6.5.7,. ˆX ˆX X, { min X T X AX = Y , ˆX = A Y ˆX = A,4 Y.. Ax = b? 2. Ax = b?? AXB = C? P, P 2, : P = P + P 2 P P 2 = P 2 P = ; 2 P = P P 2 P P 2 = P 2 P = P 2 ; 3 P, I P, T P T T. 3. R 3 L e =,, T. M α =,, T β =,, T, P L,M x = 2, 3, T M L ; 2 P L x = 2, 3, T L

22 . A «= A,. 7. A C m n, U C m m V C n n. UAV = V A U. 8. H Hermite, H = H. 9. A = A A 2 Hermite ra 2 = ra.. : A, A A = AA, A n = A n, n.. E ij Moore-Penrose {} F [x] 3 f, g = R fxgx d x, Moore-Penrose 4. rbc = rb. D B = BCD, CBC B {}-. 2 rbc = rc. D C = DBC, BC B C {}-. 5. A m n, B m r, AA B = B D B = AD; 2 A m n, B r m, BA A = B D B = DA A Moore-Penrose {}-, C A ; : A, A, ; 2 A, A.,, : AB = B A A ABB A = BB A BB A AB = A AB {, 2}- : A = 2 «; 2 A A; i i A. 27. {, 3}- : A 2 2 A; 2 A = C A. 98

23 {, 4}- 2 2 A; « {, 2}-? 2 {, 4}-? {, 2}- {, 3}- {, 4}-? 35. {, 2}- {, 3}- {, 4}-? 36. {, 2}- {, 3}- {, 4}-? 37. A λ. A = λ 2 A; 2 A {, 2}-, {, 3}- {, 4}- ; A L, M C n. : P L+M = P L + P M P L + P M = P L + P M P L + P M ; 2 P L M = 2P LP L + P M P M = 2P M P L + P M P L. 39. : A = A,4 AA,3. 4. A, A 2 8 2, b =,,, T, b 2 =,, 2 T. A x = b A 2 x = b 2. «« A =, b =. Ax = b. 42. A = «2, b = «. Ax = b. 44. AXB =, A = B A. b =,,, T, Ax = b? 2 b =,,, T, Ax = b?, ;,. 46. Ax = b AA b = b. A C m n, b C m. 47. AXB = C,, 99

24 A A; B = 2 «; C A; 2 A 2 2 A; B A; C 2 A. 48. Ax = a x = A a + I A Ay y A a + NA. :, A b I A Ay, y; 2 Ax = a Bx = b A a B b NA + NB; 3 Ax = a Bx = b A, B, C, D n, AX = B XC = D. : AD = BC; 2..:. 5. min{x T x}, Ax = b,. 5. min{trx T X 2trX}, XA = ˆX = I AA. 52. U W V, α, β V. α + U β + W. : α + U β + W = α + P U P U + P W β α + U W ; 2 α + U β + W = α + P U + P W P W β α + U W ; 3 α + U β + W = α + I P W P U P W β α + U W. : 75. 2

4 A C n n, AA = A A, A,,, Hermite, Hermite,, A, A A, A, A 4 (, 4,, A A, ( A C n n, A A n, 4 A = (a ij n n, λ, λ,, λ n A n n ( (Schur λ i n

4 A C n n, AA = A A, A,,, Hermite, Hermite,, A, A A, A, A 4 (, 4,, A A, ( A C n n, A A n, 4 A = (a ij n n, λ, λ,, λ n A n n ( (Schur λ i n ,?,,, A, A ( Gauss m n A B P Q ( Ir B = P AQ r(a = r, A Ax = b P Ax = P b, x = Qy, ( Ir y = P b (4 (4, A A = ( P Ir Q,,, Schur, Cholesky LU, ( QR,, Schur,, (,,, 4 A AA = A A Schur, U U AU = T AA = A A