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1 SGC - 77
2 Bruhat
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4 Gram-Schmidt
5 Wigner Bruhat Bruhat Bruhat KGB iv
6 Z, Q, R, C A det A trace A rank A [14], [24] x x x y y y A = B A B P Q P Q
7 1 K 1.1 K K C R Q 3 K Q 1 [21] 1.1 V K K V V, (α, v) αv 1 K 0
8 2 K = C 1 M n (C), M m,n (C) M n, M m,n GL n (C) GL n 2.1 n V f : V V 2.1 λ C f : V V v V f(v) =λv v λ V {v 1,...,v n } f : V V A M n n A Au = λu (u C n,u 0) 1
9 3 3.1 f : V V m>0 f m =0 X M n (K) m>0 X m =0 ( ) a b X = c d a + d =0,ad bc =0 3.1 f End K V V {v 1,...,v n } f(v i )=v i+1 (1 i n 1) f(v n )=0 n v 1 f v 2 f... f v n f f End K V V {v 1,...,v m } f(v i )=v i+1 (1 i m 1) v m 0,f(v m )=0 {v 1,...,v m } U := v 1,...,v m f U
10 4 4.1 K K R C 1 V dim V = n 4.1 V B : V V K B(αu + βu,v)=αb(u, v)+βb(u,v) (α, β K; u, u,v V ). B(u, γv + δv )=γb(u, v)+δb(u, v ) (γ,δ K; u, v, v V ) B(u, v) B(u, v) =B(v, u) B(u, v) Q : V K Q(v) =B(v, v) (v V ) (4.1) V V {v 1,...,v n } ψ : V K n ψ(v i )=e i (1 i n) {e 1,...,e n } K n 1 K 0 Q Q K C
11 5 Gram-Schmidt 5.1 Gram-Schmidt R n (u, v) n (u, v) = u i v i = t uv (u, v R n ) (5.1) i=1 u R n i u i (4.4) (1) (, ):R n R n R (2) (u, v) =(v, u) (u, v R n ) (3) u R n (u, u) 0 u =0 3 1 v R n (v, v) 0 v = (v, v) v 1
12 6 K = C 6.1 K = C 6.1 V h : V V C 1 (1) h(u, v) h(αu + βu,v)=αh(u, v)+βh(u,v) (α, β C; u, u,v V ) α (2) h(u, v) h(u, γv + δv )=γh(u, v)+δh(u, v ) (γ,δ C; u, v, v V ) 1 Charles Hermite ( ).
13 f End K V X M n (C) g GL n (C) g 1 Xg = J m1 (α 1 ) J ml (α l ) (7.1) 3.7 X J m (α) m J m (α) =α1 m + N (m), N (m) := m 1 i=1 E i,i+1 (7.2) 1 (3.5) (7.1) A 1 A 2 1 N (m) = J m (0)
14 8 Bruhat X L U X = LU LU LU LU Bruhat KGB 8.1 {1, 2,...,n} [n] [n] [n] S n n σ S n 1 i n σ(i) σ(1),σ(2),...,σ(n) 1, 2,...,n σ S n σ ( ) 1 2 n σ = σ(1) σ(2) σ(n) σ ( ) ( ) ( ) σ = = =
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16 137
17 [1] Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics, vol. 34, American Mathematical Society, Providence, RI, 2001, Corrected reprint of the 1978 original. [2] Takeshi Ikeda, Schubert classes in the equivariant cohomology of the Lagrangian Grassmannian, Adv. Math. 215 (2007), no. 1, [3] Takeshi Ikeda and Hiroshi Naruse, Excited Young diagrams and equivariant Schubert calculus, Trans. Amer. Math. Soc. 361 (2009), no. 10, [4] N. Jacobson, Schur s theorems on commutative matrices, Bull. Amer. Math. Soc. 50 (1944), [5] Anthony W. Knapp, Lie groups beyond an introduction, second ed., Progress in Mathematics, vol. 140, Birkhäuser Boston Inc., Boston, MA, [6] Dragomir Ž. Doković and Kaiming Zhao, Rational Jordan decomposition of bilinear forms, Commun. Contemp. Math. 7 (2005), no. 6, [7] Nolan R. Wallach, Real reductive groups. I, Pure and Applied Mathematics, vol. 132, Academic Press Inc., Boston, MA, [8] 2003 [9] 2009 [10] 2006 [11] / UP 1982 [12] / 1960 [13] 1965 [14] 1974 [15] 1982 [16] [ ] 2002 [17] / 2005 [18] [19] I II 2001 [20] / 2003 [21] 1987 [22] 1988 [23] /1998 [24] 1995
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20 , , 135 4, 6 1 Borel 116 Cartan 116 Cartan 97, 101 Cayley-Hamilton 20 Gauss 120 Gram-Schmidt 69, 91 Jordan 43 Jordan 44 Jordan 44, 104, 107 KGB 131 Kronecker δ 7 Lagrange 67 LDU 122 LPU 122 LU 120, 122 Richardson 131 Sylvester 64 (F 1 (t),...,f l (t)) 31 [n] 1 n 113 A 87 V 17 U 72, 92 u v 90 X 13 v A B 15 u 1,...,u k 3 1 n 7 Ad(g) 15 Alt n (K) 16 A n 82, 122 B n 116 Bn 116 Bn R 82, 122 B Q (u, v) 60 γ σ 114 deg F (t) 31 δ ij Kronecker δ 7 Δ k (A) 120 det X
21 diag(a 1,...,a n ) 25 e 1,...,e n 4 E i,j 37 End K V, End V 14 exp A 50 F d (V ) 125 F(V ) 125 GL n (K) 13, 65 GL(V ) 64 Grass k (V ) 125 Her n (C) 89 Her skew n (C) 89 Hom K (V,W), Hom(V,W) 8 Im A 6 J n 67 K[x] 2 K[x] n n 5 Ker A 6 L A 9 LGrass k (V ) 135 N n 82, 122 O n (K) 65 O n (S) 65 O p,q (R) 85 O X 108 O(V ; Q) 65 p A (t) 19 φ f (t) 30 R A 9 rank A 13 R g 9 SL n (K) 65 SL(V ) 64 S n 113 SO n (K) 66 SO n (S) 66 SO(V ; Q) 66 Sp(V ) 67 Sp(V ; Σ) 67 Sym n (C) 131 Sym n(k) 85 Sym n (K) 16 Sym + n (R) 76 T n 116 trace A 16 t X 10, 16 U n 116 Un 116 U n (C) 90 U(V ; h), U(V ) 90 V (f; λ) 24 V (f; λ) 24 Borel, Armand ( ) 116 Cartan, Élie ( ) 97, 116 Cayley, Arthur ( ) 20 Gauss, Carl Friedlich ( ) 79, 120 Gram, Jorgen ( ) 69 Grassmann, Hermann ( ) 125 Hamilton, William Rowan ( ) 20 Hermite, Charles ( ) 86 ( ) 81 Jordan, Camille ( ) 44 Kronecker, Leopord ( ) 7 Lagrange, Joseph-Louis ( ) 67 Richardson, Roger ( ) 131 Schmidt, Erhard ( ) 69 Schubert, Hermann ( ) 130 Sylvester, James Joseph ( ) 64 Wigner, Eugene ( )
22 著者略歴 西山 にしやま 享きょう 1981 年京都大学理学部卒業 1986 年京都大学大学院理学研究科博士課程後期課程修了東京電機大学理工学部助手, 京都大学総合人間学部助教授, 京都大学大学院理学研究科准教授, を経て, 現在青山学院大学理工学部教授理学博士専門分野表現論, 調和解析 ( フーリエ解析 ), 幾何学的な不変式論, 群作用と軌道の構造主要著書 基礎課程微分積分 I, II ( サイエンス社,1998), 多項式のラプソディー ( 日本評論社,1999), よくわかる幾何学 ( 丸善,2004). 射影幾何学の考え方 数学のかんどころ 19( 共立出版,2013). 太田琢也 西山享共著 代数群と軌道 数学の杜 3( 数学書房,2015). 臨時別冊 数理科学 SGC ライブラリ -77 重点解説ジョルダン標準形行列の標準形と分解をめぐって ( 電子版 ) 著者西山享 2018 年 3 月 25 日初版発行 ISBN この電子書籍は 2010 年 10 月 25 日初版発行の同タイトルを底本としています. 数理科学編集部発行人森平敏孝 TEL.(03) FAX.(03) ホームページ http : // ご意見 ご要望は sk@saiensu.co.jp まで. 発行所 株式会社サイエンス社 TEL.(03) ( 代表 ) 東京都渋谷区千駄ヶ谷 本誌の内容を無断で複写複製 転載することは, 著作者および出版者の権利を侵害することがありますので, その場合にはあらかじめサイエンス社著作権担当者あて許諾をお求めください. 組版ビーカム
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微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行時のものです.
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