行列
出典: フリー百科事典『ウィキペディア(Wikipedia)』 (2024/02/16 13:37 UTC 版)
脚注
参考文献
- Arnold, V. I.; Cooke, Roger (1992), Ordinary differential equations, Berlin, New York: Springer-Verlag, ISBN 978-3-540-54813-3
- Artin, Michael (1991), Algebra, Prentice Hall, ISBN 978-0-89871-510-1
- Association for Computing Machinery (1979), Computer Graphics, Tata McGraw–Hill, ISBN 978-0-07-059376-3
- Baker, Andrew J. (2003), Matrix Groups: An Introduction to Lie Group Theory, Berlin, New York: Springer-Verlag, ISBN 978-1-85233-470-3
- Bau III, David; Trefethen, Lloyd N. (1997), Numerical linear algebra, Philadelphia: Society for Industrial and Applied Mathematics, ISBN 978-0-89871-361-9
- Bretscher, Otto (2005), Linear Algebra with Applications (3rd ed.), Prentice Hall
- Bronson, Richard (1989), Schaum's outline of theory and problems of matrix operations, New York: McGraw–Hill, ISBN 978-0-07-007978-6
- Brown, William A. (1991), Matrices and vector spaces, New York: M. Dekker, ISBN 978-0-8247-8419-5
- Coburn, Nathaniel (1955), Vector and tensor analysis, New York: Macmillan, OCLC 1029828
- Conrey, J. B. (2007), Ranks of elliptic curves and random matrix theory, Cambridge University Press, ISBN 978-0-521-69964-8
- Fudenberg, D.; Tirole, Jean (1983), Game Theory, MIT Press
- Gilbarg, David; Trudinger, Neil S. (2001), Elliptic partial differential equations of second order (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-41160-4
- Godsil, Chris; Royle, Gordon (2004), Algebraic Graph Theory, Graduate Texts in Mathematics, 207, Berlin, New York: Springer-Verlag, ISBN 978-0-387-95220-8
- Golub, Gene H.; Van Loan, Charles F. (1996), Matrix Computations (3rd ed.), Johns Hopkins, ISBN 978-0-8018-5414-9
- Golub, Gene H.; Van Loan, Charles F. (2013), Matrix Computations (4th ed.), Johns Hopkins, ISBN 978-1-4214-0794-4
- Greub, Werner Hildbert (1975), Linear algebra, Graduate Texts in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90110-7
- Halmos, Paul Richard (1982), A Hilbert space problem book, Graduate Texts in Mathematics, 19 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-90685-0, MR675952
- Horn, Roger A.; Johnson, Charles R. (2013). Matrix analysis (Second ed.). Cambridge University Press. ISBN 978-0-521-54823-6. MR2978290
- Householder, Alston S. (1975), The theory of matrices in numerical analysis, New York: Dover Publications, MR0378371
- Krzanowski, W. J. (1988), Principles of multivariate analysis, Oxford Statistical Science Series, 3, The Clarendon Press Oxford University Press, ISBN 978-0-19-852211-9, MR969370
- Itõ, Kiyosi, ed. (1987), Encyclopedic dictionary of mathematics. Vol. I--IV (2nd ed.), MIT Press, ISBN 978-0-262-09026-1, MR901762
- Lang, Serge (1969), Analysis II, Addison-Wesley
- Lang, Serge (1987a), Calculus of several variables (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-96405-8
- Lang, Serge (1987b), Linear algebra, Berlin, New York: Springer-Verlag, ISBN 978-0-387-96412-6
- Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR1878556
- Latouche, G.; Ramaswami, V. (1999), Introduction to matrix analytic methods in stochastic modeling (1st ed.), Philadelphia: Society for Industrial and Applied Mathematics, ISBN 978-0-89871-425-8
- Manning, Christopher D.; Schütze, Hinrich (1999), Foundations of statistical natural language processing, MIT Press, ISBN 978-0-262-13360-9
- Mehata, K. M.; Srinivasan, S. K. (1978), Stochastic processes, New York: McGraw–Hill, ISBN 978-0-07-096612-3
- Mirsky, Leonid (1990), An Introduction to Linear Algebra, Courier Dover Publications, ISBN 978-0-486-66434-7
- Nocedal, Jorge; Wright, Stephen J. (2006), Numerical Optimization (2nd ed.), Berlin, New York: Springer-Verlag, p. 449, ISBN 978-0-387-30303-1
- Oualline, Steve (2003), Practical C++ programming, O'Reilly, ISBN 978-0-596-00419-4
- Press, William H.; Flannery, Brian P.; Teukolsky, Saul A.; Vetterling, William T. (1992), “LU Decomposition and Its Applications”, Numerical Recipes in FORTRAN: The Art of Scientific Computing (2nd ed.), Cambridge University Press, pp. 34–42
- Punnen, Abraham P.; Gutin, Gregory (2002), The traveling salesman problem and its variations, Boston: Kluwer Academic Publishers, ISBN 978-1-4020-0664-7
- Reichl, Linda E. (2004), The transition to chaos: conservative classical systems and quantum manifestations, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98788-0
- Rowen, Louis Halle (2008), Graduate Algebra: noncommutative view, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-4153-2
- Šolin, Pavel (2005), Partial Differential Equations and the Finite Element Method, Wiley-Interscience, ISBN 978-0-471-76409-0
- Stinson, Douglas R. (2005), Cryptography, Discrete Mathematics and Its Applications, Chapman & Hall/CRC, ISBN 978-1-58488-508-5
- Stoer, Josef; Bulirsch, Roland (2002), Introduction to Numerical Analysis (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-95452-3
- Ward, J. P. (1997), Quaternions and Cayley numbers, Mathematics and its Applications, 403, Dordrecht: Kluwer Academic Publishers Group, ISBN 978-0-7923-4513-8, MR1458894
- Wolfram, Stephen (2003), The Mathematica Book (5th ed.), Champaign, Ill: Wolfram Media, ISBN 978-1-57955-022-6
- Pappur Nagappa Shivakumar、K C Sivakumar、Yang Zhang: "Infinite Matrices and Their Recent Applications", Springer, ISBN 978-3319301792 (2016年5月25日)。
- 斎藤正彦『線形代数学』(第3版)東京図書、2017年4月10日。ISBN 978-4-489-02179-4。
物理学に関するもの
- Bohm, Arno (2001), Quantum Mechanics: Foundations and Applications, Springer, ISBN 0-387-95330-2
- Burgess, Cliff; Moore, Guy (2007), The Standard Model. A Primer, Cambridge University Press, ISBN 0-521-86036-9
- Guenther, Robert D. (1990), Modern Optics, John Wiley, ISBN 0-471-60538-7
- Itzykson, Claude; Zuber, Jean-Bernard (1980), Quantum Field Theory, McGraw–Hill, ISBN 0-07-032071-3
- Riley, K. F.; Hobson, M. P.; Bence, S. J. (1997), Mathematical methods for physics and engineering, Cambridge University Press, ISBN 0-521-55506-X
- Schiff, Leonard I. (1968), Quantum Mechanics (3rd ed.), McGraw–Hill
- Weinberg, Steven (1995), The Quantum Theory of Fields. Volume I: Foundations, Cambridge University Press, ISBN 0-521-55001-7
- Wherrett, Brian S. (1987), Group Theory for Atoms, Molecules and Solids, Prentice–Hall International, ISBN 0-13-365461-3
- Zabrodin, Anton; Brezin, Édouard; Kazakov, Vladimir; Serban, Didina; Wiegmann, Paul (2006), Applications of Random Matrices in Physics (NATO Science Series II: Mathematics, Physics and Chemistry), Berlin, New York: Springer-Verlag, ISBN 978-1-4020-4530-1
歴史に関するもの
- Bôcher, Maxime (2004), Introduction to higher algebra, New York: Dover Publications, ISBN 978-0-486-49570-5, reprint of the 1907 original edition
- Cayley, Arthur (1889), The collected mathematical papers of Arthur Cayley, I (1841–1853), Cambridge University Press, pp. 123–126
- Dieudonné, Jean, ed. (1978), Abrégé d'histoire des mathématiques 1700-1900, Paris: Hermann
- Hawkins, Thomas (1975), “Cauchy and the spectral theory of matrices”, Historia Mathematica 2: 1–29, doi:10.1016/0315-0860(75)90032-4, ISSN 0315-0860, MR0469635
- Knobloch, Eberhard (1994), “From Gauss to Weierstrass: determinant theory and its historical evaluations”, The intersection of history and mathematics, Sci. Networks Hist. Stud., 15, Basel, Boston, Berlin: Birkhäuser, pp. 51–66, MR1308079
- Kronecker, Leopold (1897), Hensel, Kurt, ed., Leopold Kronecker's Werke, Teubner
- Mehra, J.; Rechenberg, Helmut (1987), The Historical Development of Quantum Theory (1st ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-96284-9
- Shen, Kangshen; Crossley, John N.; Lun, Anthony Wah-Cheung (1999), Nine Chapters of the Mathematical Art, Companion and Commentary (2nd ed.), Oxford University Press, ISBN 978-0-19-853936-0
- Weierstrass, Karl (1915), Collected works, 3
関連項目
注釈
- ^ 下線や二重下線などを付けることもあるが、これはタイプライター原稿で用いられた太字書体を指示する書式の名残[2]
- ^ OEDによれば、数学用語としての "matrix" の最初の用例は J. J. Sylvester in London, Edinb. & Dublin Philos. Mag. 37 (1850), p. 369: "We ‥commence‥ with an oblong arrangement of terms consisting, suppose, of m lines and n columns. This will not in itself represent a determinant, but is, as it were, a Matrix out of which we may form various systems of determinants by fixing upon a number p, and selecting at will p lines and p columns, the squares corresponding to which may be termed determinants of the pth order.
- ^ これは与えられた行列の全ての成分が加法逆元を持つ限りにおいて、加法のみから定められることに注意。特にスカラー乗法が(任意のスカラーと任意の行列に対する演算として)定義されている必要はない。従って、同じサイズの任意の行列に対する減法を定めるならば、例えば係数域が加法についてアーベル群であれば十分であるが、通例として行列の係数域は何らかの可換環と仮定するから、それには環の加法群構造を用いればよい
- ^ 正方行列でない行列に対して行列式を考える理論も存在する。これは C. E. Cullis により導入された。[27]
- ^ 普通はさらに一般線型群の閉集合となることも要求する。
- ^ "Not much of matrix theory carries over to infinite-dimensional spaces, and what does is not so useful, but it sometimes helps." [42]
- ^ "Empty Matrix: A matrix is empty if either its row or column dimension is zero",[43] "A matrix having at least one dimension equal to zero is called an empty matrix", [44]
出典
- ^ a b c d e 斎藤2017、21頁。
- ^ https://raksul.com/dictionary/underline/
- ^ Shen, Crossley & Lun 1999 cited by Bretscher 2005, p. 1
- ^ Needham, Joseph; Wang Ling (1959). Science and Civilisation in China. III. Cambridge: Cambridge University Press. p. 117. ISBN 9780521058018
- ^ Cayley 1889, pp. 475–496, vol. II.
- ^ Dieudonné 1978, p. 96, Vol. 1, Ch. III.
- ^ Merriam–Webster dictionary, Merriam–Webster 2009年4月20日閲覧。
- ^ The Collected Mathematical Papers of James Joseph Sylvester: 1837–1853, Paper 37, p. 247
- ^ Knobloch 1994.
- ^ Hawkins 1975.
- ^ Kronecker 1897.
- ^ Weierstrass 1915, pp. 271–286.
- ^ Bôcher 2004.
- ^ Mehra & Rechenberg 1987.
- ^ a b c d 斎藤2017、23頁。
- ^ a b 斎藤2017、24頁。
- ^ a b 斎藤2017、25頁。
- ^ a b 斎藤2017、31頁。
- ^ 斎藤2017、89頁。
- ^ Brown 1991, Definition II.3.3.
- ^ Greub 1975, Section III.1.
- ^ Brown 1991, Theorem II.3.22.
- ^ a b 斎藤2017、34頁。
- ^ 斎藤2017、26頁。
- ^ http://www2.math.kyushu-u.ac.jp/~tnomura/EdAct/2010TKR.pdf
- ^ Stephen P. Boyd. “Crimes against Matrices” (pdf). 2013年3月2日閲覧。
- ^ 中神祥臣・柳井晴夫 著、『矩形行列の行列式』、丸善、2012年。ISBN 978-4-621-06508-2.
- ^ Greub 1975, Section III.2.
- ^ Coburn 1955, Ch. V.
- ^ Lang 2002, Chapter XIII.
- ^ Lang 2002, XVII.1, p. 643.
- ^ Lang 2002, Proposition XIII.4.16.
- ^ Reichl 2004, Section L.2.
- ^ Greub 1975, Section III.3.
- ^ Greub 1975, Section III.3.13.
- ^ Baker 2003, Def. 1.30.
- ^ Baker 2003, Theorem 1.2.
- ^ Artin 1991, Chapter 4.5.
- ^ Artin 1991, Theorem 4.5.13.
- ^ Rowen 2008, Example 19.2, p. 198.
- ^ Itõ 1987, "Matrix".
- ^ Halmos 1982, p. 23, Chapter 5.
- ^ Glossary, O-Matrix v6 User Guide.
- ^ MATLAB Data Structures
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