转置

双重索引对象的转置是通过将所有元素替换为得到的对象。对于二阶张量秩张量，张量转置就是。矩阵转置，最常用的写法是，是通过交换的行和列得到的矩阵，并满足恒等式

(1)

遗憾的是，常用的还有几种其他表示法，如下表所示。本文档中使用表示法。

表示法	参考文献
	本文档；Golub and Van Loan (1996), Strang (1988)
	Arfken (1985, p. 201), Griffiths (1987, p. 223)
	Ayres (1962, p. 11), Courant and Hilbert (1989, p. 9)

矩阵或张量的转置在 Wolfram 语言中实现为转置[A].

两个转置的乘积满足

			(2)
			(3)
			(4)
			(5)
			(6)

其中爱因斯坦求和已被用于对重复索引进行隐式求和。因此，

(7)

另请参阅

反对称矩阵, 合同矩阵, 共轭矩阵, 共轭转置, 对称矩阵

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参考文献

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 201, 1985.Ayres, F. Jr. Schaum's Outline of Theory and Problems of Matrices. New York: Schaum, pp. 11-12, 1962.Boothroyd, J. "Algorithm 302: Transpose Vector Stored Array." Comm. ACM 10, 292-293, May 1967.Brenner, N. "Algorithm 467: Matrix Transposition N Place [F1]." Comm. ACM 16, 692-694, Nov. 1973.Cate, E. G. and Twigg, D. W. "Algorithm 513: Analysis of In-Situ Transposition." ACM Trans. Math. Software 3, 104-110, March 1977.Courant, R. and Hilbert, D. Methods of Mathematical Physics, Vol. 1. New York: Wiley, 1989.Golub, G. H. and Van Loan, C. F. Matrix Computations, 3rd ed. Baltimore, MD: Johns Hopkins, 1989.Griffiths, D. J. Introduction to Elementary Particles. New York: Wiley, p. 220, 1987.Knuth, D. E. "Transposing a Rectangular Matrix." Ch. 1.3.3 Ex. 12. The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 3rd ed. Reading, MA: Addison-Wesley, pp. 182 and 523, 1997.Laflin, S. and Brebner, M. A. "Algorithm 380: In-Situ Transposition of a Rectangular Matrix. [F1]." Comm. ACM 13, 324-326, May 1970.Strang, G. Introduction to Linear Algebra. Wellesley, MA: Wellesley-Cambridge Press, 1993.Strang, G. Linear Algebra and its Applications, 3rd ed. Philadelphia, PA: Saunders, 1988.Windley, P. F. "Transposing Matrices in a Digital Computer." Computer J. 2, 47-48, Apr. 1959.

在 Wolfram|Alpha 中被引用

转置

请引用为

Weisstein, Eric W. "转置。" 来自 MathWorld--Wolfram Web 资源。 https://mathworld.net.cn/Transpose.html