矩阵半张量积理论与应用研究中心.pdf

Ý ŒÜþÈ‘Ž¹ƒ˜: lÝ Ý ŒÜþÈ §“Ð ¥I‰Æ êƆXÚ‰ÆïÄ ¢ŒÆÝ ŒÜþÈn؆A^ïÄ¥% ¢ŒÆêÆÆ 2022c9 29F ìÀ! ¢ ̇SN 1 Ý ŒÜþÈ d5 2 Ý ŒÜþÈ 5Ÿ 3 3ëYÄ 4 {¤†yG 5 XÚ¥ A^ ( 2 / 37 I. Ý + Ý Ý ŒÜþÈ d5 gŽå u¥I nØ´ ú@å u¥I ˜‡êi©|. 3{IxÔ'æA«ŒÆ ÇKatz Ͷêi¤˜Ö[1] ¥•Ñ:“Ý gŽ{¤aÈ, § ¦^– ŒJˆ ÇŠ, ¥IÆö^§5)‚5•§|.” 3=IÆöCrilly Ö[2] ¥•J : Ý å u“ú c200c, ¥IêÆ[¦^ êi ”. Ý nØ´ùü Ö¥•˜J ŒV(¢´=„ . ©g¥I êÆ©|§ [1] V.J. Katz, A History of mathematics, Brief Version, Addison-Wesley, New York, 2004. [2] T. Crilly, 50 Mathenatical Ideas You Really Need to Know, È, “\،ؕ 50‡êÆ•£”§<¬e >ч , ®, 2012. 3 / 37 + Ý nØ ) žmµ›Ê-V¶M©<µ Cauchy({§1789-1857): 1 ª(determinant); ¦È(multiplication); Š‘(adjoint); Sylvester(=§1841-1897): Ý (matrix); A Š(eigenvalue ¦È •(rank of product); Cayley(=§1821-1895): A õ‘ª(characteristic polynomial); Caley-Hamiltonian ½n£Caley-Hamiltonian Theorem); Frobenies( , 1849-1917): ƒq(similar)¶ÜÓ(congruence •(rank)¶‚5Ã'(linearly independence)¶ ··· 4 / 37 + ép ê|Ý •{ &¢ Ù•, ˜‡‚5¼êf (x1 , · · · , xn ) éN´^•þÈL«µ f (x1 , · · · , xn ) = n X ci xi = cx, (1) i=1 ùp, c = [c1 , · · · , cn ], x = [x1 , · · · , xn ]T . ˜‡V‚5¼ê•éN´^Ý /ªL«: g(x1 , · · · , xm y1 , · · · , yn ) = m X n X ai,j xi yj = xT Ay, (2) i=1 j=1 ùp,  a11  a21  A =  ..  . a12 · · · a22 · · · am1 am2 · · ·  a1n a2n   .  amn 5 / 37 3Nõ‰Æ¯K¥¬Ñyn‚5½•p f (x1 , · · · , xp y1 , · · · , yn , z1 , · · · , zm ) = ‚5¼ê"= p n m X X X dk,i,j xk yi zj . k=1 i=1 j=1 (3) n‚5¼êUØU^Ý L«Q? l ›-Vl›c“m©, IS ˜ Æöm©&?^áN ÈL«n ê| $Ž, Xã1 ¤«. 6 / 37 laye kth dk11 dk12 · · · dk1n dk21 dk22 · · · dk2n ··· ··· ··· dkm1 dkm2 · · · dkmn r d111 d112 · · · d11n d121 d122 · · · d12n ··· ··· ··· d1m1 d1m2 · · · d1mn dp11 dp12 · · · dp1n dp21 dp22 · · · dp2n ··· ··· ··· dpm1 dpm2 · · · dpmn ã 1: ˜‡áN 7 / 37 + lõ‚5$Ž Ý ŒÜþÈ áN •3¯Kµ $Ž5KE,, " ˜„5, éJí2 n ±þ •p ê|þ " •Ä(3) ¥ n‚5¼êf : Rp × Rm × Rn → R. @o, ©þxk ∈ R, Kéy ∈ Rm , z ∈ Rn , ·‚k f (xk , y, z) = m X n X dk,i,j yi zj xk , ½x (4) i=1 j=1 §´˜‡ g.ª, Ù( Ý •áN ã1 ¥ 1k Ïd§f (x, y, z) Œ±wŠ'uy, z g., Ù( Ý " • 8 / 37 A(x) = p X Ak xk , (5) k=1 ùpAk ´áN 1k " Ïd, 'ux, y, z n‚5¼ê(4) Œ±üg•'uy, z V‚5¼ê: f (x, y, z) = yT A(x)z. ùp, ( Ý /r§ ¤ (5) UÄ^Ý (6) ¦ÈL«Q? ·‚Ø”/ª A(x) = [A1 , A2 , · · · , Ap ] ∗ x, (7) ùp “¦È” Ò´Ý ŒÜþȧ^n L«ù«¦{" ÏLù«“ꔆ“¬” ¦È, Ý ŒÜþȈ üg 8 "|^ù‡ n, ?Ûpgê|¦{ÑŒ±ü ˜g ê|†˜gê| ¦È" 9 / 37 Q y3b½f : ni=1 Vi → R§•n ‚5¼ê, ùp, Vi •ri ‘• þ˜m, Ä.•{δr1i , δr1i , · · · , δrrii }.
f δrk11 , δrk22 , · · · , δrknn = dk1 ,k2 ,··· ,kn , ki = 1, · · · , ri ; i = 1, 2, · · · , n. ò¤k~ê•i1Sü ˜1: D = [d1,1,··· ,1 , d1,1,··· ,1 , · · · , d1,1,··· ,rn ; · · · ; dr1 ,r2 ,··· ,1 , dr1 ,r2 ,··· ,2 , · · · , dr1 ,r2 ,··· ,rn ]. @o, aqngœ¹, ·‚k f (x1 , · · · , xn ) = D n x1 n x2 · · · n xn . (8) 10 / 37 + ˜„z Ý ŒÜþÈ Definition 1.1 A •m × n Ý , B •p × q Ý , n †p •t = lcm(n, p), KA †B ŒÜþȽ•

A n B = A ⊗ It/n B ⊗ It/p . ùp⊗ ´kronecker È:  a11 B a12 B · · · a21 B a22 B · · ·  A ⊗ B =  ..  . an1 B an2 B · · · • ú ê (9)  a1n B a2n B    ann B 11 / 37 Example 1.2     1 (i) X = 1 2 3 −1 §Y = . @o§ 2       X n Y = 1 2 · 1 + 3 −1 · 2 = 7 0 . (ii)  T   X = −1 2 1 −1 2 3 §Y = 1 2 −2 . @ o§         −3 −1 1 2 XnY = ·1+ ·2+ · (−2) = . 2 −1 3 −6 (iii)   1 2 1 1 A = 2 3 1 2 , 3 2 1 0   1 −2 . B= 2 −1 12 / 37 Example 1.2(Cont’d) @o§        1   −2 1 2 1 1   1 2 1 1 2    −1     1   −2   2 3 1 2 2 3 1 2 AnB =    2 −1          1   −2  3 2 1 0 3 2 1 0 2 −1    3 4 −3 −5 = 4 7 −5 −8 . 5 2 −7 −4 Example 1.3( È$Ž) U = Ux i + Uy j + Uz k ∈ R3 , V = Vx i + Vy j + Vz k ∈ R3 §K ~ V = (Uy Vz − Uz Vy )i − Ux Vz − Uz Vx )j + (Ux Vy − Uy Vx )k. U× 13 / 37 Example 1.3(Cont’d) ( È$Ž) XJ^Ý ŒÜþÈ, K ~ V = Muv, U× (10) ùpu = (Ux , Uy , Uz )T , v = (Vx , Vy , Vz )T ,   0 0 0 0 0 1 0 −1 0 M = 0 0 −1 0 0 0 1 0 0 . 0 1 0 −1 0 0 0 0 0 È ù«L«•ªÃØ3êŠOŽ½nØí 5NõB|"~Xéõ- È þѬ‘ ~ u2 × ~ ···× ~ un = M n−1 nni=1 ui . u1 × 14 / 37 II. Ý ŒÜþÈ 5Ÿ + U«gÊÏÝ ¦{ Proposition 2.1 A ∈ Mm×n , B ∈ Mp×q . XJn = p, K A n B = AB. (11) Ïd§Ý ŒÜþÈ´Ý ÊÏÈ ˜‡í2œ ¿…§§ U« A ¤kÊÏÈ Ì‡5Ÿœ Proposition 2.2 (i) (© Æ) A n (αB + βC) = αA n B + βA n C; (αB + βC) n A = αB n A + βC n A, α, β ∈ R. (12) 15 / 37 Proposition 2.2(cont’d) (2) ((ÜÆ) A n (B n C) = (A n B) n C. (13) (A n B)T = BT n AT . (14) Proposition 2.3 (i) (ii) A †B þŒ_§K (A n B)−1 = B−1 n A−1 . (15) 16 / 37 + «_uÊÏÝ ¦{ 5Ÿ ÊÏÝ ¦{†ê¦ƒ' (i) ‘ê •›¶ (ii) ØŒ m. 8I: ŽÑ“ØŒ †5”! üŒf:: Proposition 2.4(Pseudo-Commutativity) A ∈ Mm×n . (i) -Z ∈ Rt •˜1•þ§K A n Z = Z n (It ⊗ A); (16) (ii) -Z ∈ Rt •˜ •þ§K Z n A = (It ⊗ A) n Z. (17) 17 / 37 + m Ý Definition 2.5 m Ý W[m,n] ∈ Mmn×mn ½ÂXeµ   W[m,n] = In ⊗ δm1 , In ⊗ δm2 , · · · , In ⊗ δmm . (18) Proposition 2.6 (i) X ∈ Rm §Y ∈ Rn •ü •þ§K W[m,n] n X n Y = Y n X. (ii) (19) u ∈ Rm §v ∈ Rn •ü1•þ§K u n v n W[m,n] = v n u. (20) 18 / 37 Proposition 2.7 −1 T = W[m,n] = W[n,m] . W[m,n] (21) + ÊÏȣפ é' ŒÜŒÜþÈn CP × STP n Property Similar Similar Applicability linear, bilinear multi-linear Commutativity No Pseudo-Commutative 19 / 37 + @ÏÝ ŒÜþÈ Ö 20 / 37 III 3 ë Y Ä XÚ¥ A^ + õ‚5N : (i) ÜþOŽ¶ (Üþ| ¿) (ii) 6/M þ o“êOŽµ [·, ·] : V(M) × V(M) → V(M). (iii) k•+OŽµ G × G → G. (iv) õ õ‘ª. 21 / 37 + õ õ‘ª Px = (x1 , x2 , · · · , xn )T •˜g •þ§@o, ˜‡k ‘ªŒL«• P(x) = Mp xk . n õ (22) Example 3.1 ‰½P(x) = x12 + 4x1 x2 − x22 , §ŒL«• P(x) = Mp x2 , ùpx2 = (x12 , x1 x2 , x2 x1 , x22 )T , MP = (1, 4, 0, −1). MP Ø´•˜ , ~XMP = (1, 2, 2, −1) •é. 22 / 37 Example 3.1(cont’d) y3 P(x) •˜‡m gõ‘ª, KP(x) ŒL«• P(x) = m X Mk x k , (23) k=0 ùpMk •˜nk ‘1•þ. 23 / 37 + õ õ‘ª‡©úª : Theorem 3.2 x = (x1 , · · · , xn )T ∈ Rn , K D(xk+1 ) = Φnk xk , (24) ùp Φnk = k X Ins ⊗ W[nk−s ,n] . (25) s=0 24 / 37 + õ )Û¼ê NÐmª Proposition 3.3 • f (x), x ∈ Rn f (x) = f (x0 ) + Df (x0 )(x − x0 ) + 2!1 D2 f (x0 )(x − x0 )2 + 3!1 D3 f (x0 )(x − x0 )3 + · · · . (26) |^(26), •þ| NЪ˜ Œ± "2|^‡©ú ª(24)-(25), ¼ê, •þ|, éó•þ| 鉽•þ| o ꕘ˜Œ±OŽÑ5. l ¦š‚5››XÚAÛn Ø “êz!§Sz¢y¤•ŒU. 25 / 37 + ŒÜþÈ•{3>åXÚ A^ 26 / 37 IV. { ¤† y G + Ý ŒÜþÈŒ¯P : (1) I[g,‰ÆÄ7: ”š‚5°•-½››“êzAÛ •{9ó§A^”(1999-2002). (2) låÚ 1˜ŸØ©1998-2001 §“Ð, Semi-tensor product of matrices, part 1, Swap matrix and left semi-tensor product; part 2, Properties; part 3, Tensor form of polynomials; part 4, Some applications; part 5, Tensor algebra, ¥‰ XÚ¤, v‡? ÒµE0005-E0009§2000 c1 7 F. Cheng D. Semi-tensor product of matrices and its application to Morgan.s problem. Science in China, Series F, 2001, 44(3): 195-212. 27 / 37 (3) 3ëYÄ XÚ¥ A^2001-2007 •k¿Â óŠŒU´‡©XÚ-½• OŽ[1]. Ÿ¦(Øä. (4) 1˜ 'uÝ ŒÜþÈ ;Í[2] 2007. 1˜g – ¢ŒÆ2008. (5) 2008 cЧ1ng¥aV>››¬Æ. ˜uŒÆëZ A Ç w. m©'uÙ ä ïÄ. [1] Cheng D., Ma J., Lun Q., Mei S., Quadratic form of stable sub-manifold for power systems, Int. J. Robust Nonlin. Contr., 14: 773-788, 2004. [2] §“Ð, àö‘. 5Ý ŒÜþÈ- n؆A^6, ®: ‰Æч , 2007 (1 ‡, 2011). 28 / 37 (6) 'uÙ ä ;Í, 2011, •¹k•Æ‰ .. 29 / 37 (7) Automatica 2008-2010 /•ZnØ/•{ØØ©ø” [1], 2011. (8) 1˜Ÿ'uƉ،ÜþÈ•{Ø©[2], 2013. [1] Cheng D, Qi H. Controllability and observability of Boolean networks, Automatica, 45: 1659-1667, 2009. [2] Guo P., Wang Y., Li. H., Algebraic formulation and strategy optimization for a class of evolutionary networked games via semi-tensor product method, Automatica, 49(11), 3384-3389, 2013. 30 / 37 (9) ‰Æ #щE¤Òø, (u ¬: kvk# “ê½ AÛ( ?) m©&?Ý ŒÜþȆ“ê9AÛ ' X. 2014+ ;Í 31 / 37 (10) ¢ŒÆÝ 2018. Œ Ü þ È n Ø 9 A ^ ï Ä ¥ % ¤ á, 32 / 37 + yG 8c, Ý ŒÜþÈnØ® A^uõƉ •) Ü6XÚ ©Û†››¶ k•Æ‰†Æ‰››Ø¶ “ê†A˜AÛ(•‘XÚ) ››¶ k•gÄŶ ã؆è.››¶ “ê?è¶ " nØïÄ¥§ 33 / 37 ŒÜþÈ•{• ^u˜ >åXÚ››¶ ·ÜÄå•!E ó§ O¥"~X `z››¶ " 8c, Ý ŒÜþÈ Ø©Šö•)£Ø ÚO)µ ¥I!¿Œ|!±Ú !F !{I!=I! Ûd!a ;!Hš!#\·! I!eŒ|æ!\<Œ!<Ý!: ß|! I!žK!âAC.Ë, " 34 / 37 ISü •)£Ø ÚO)µ ®ŒÆ!˜uŒÆ!M Tó’ŒÆ!M Tó§Œ Æ!À ŒÆ!“ ° ŒÆ! ®nóŒÆ! ®e> ŒÆ!HmŒÆ!ìÀŒÆ!ìÀ“‰ŒÆ! ¢ŒÆ! þ° ÏŒÆ!ÓLŒÆ!ÀHŒÆ!H®“‰ŒÆ!¤ Ñ>f‰EŒÆ!¥HŒ®!uHnóŒÆ!úô“‰Œ Æ!uÀ‰EŒÆ!ìÀ‰EŒÆ!à ó’ŒÆ!¥‰ XÚ¤!¥‰ &ES ¤, " 35 / 37 V. ( “êzAÛ•{k– zµ ¥I “µAÛ“êz{ Ç©d: êÆÅ z ‡©AÛ “êz &EÔnXÚ, <óœU Ý ŒÜþÈ•{ ù´˜‡4ä M5 #••, Nõ#¯K, #nØ, # A^k–mu. ù´˜‡´uÆÃ, ´uë† #Ɖ. ¢ŒÆ?uƉc÷. •"\ \\! (ÅØŒ”!) 36 / 37 ! Q&A