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# Transpose

A transpose of a doubly indexed object is the object obtained by replacing all elements with . For a second-tensor rank tensor , the tensor transpose is simply . The matrix transpose, most commonly written , is the matrix obtained by exchanging 's rows and columns, and satisfies the identity

 (1)

Unfortunately, several other notations are commonly used, as summarized in the following table. The notation is used in this work.

 notation references This work; 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)

The transpose of a matrix or tensor is implemented in the Wolfram Language as Transpose[A].

The product of two transposes satisfies

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

where Einstein summation has been used to implicitly sum over repeated indices. Therefore,

 (7)

Antisymmetric Matrix, Congruent Matrices, Conjugate Matrix, Conjugate Transpose, Symmetric Matrix

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## References

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.

Transpose

## Cite this as:

Weisstein, Eric W. "Transpose." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Transpose.html