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Transpose

A transpose of a doubly indexed object is the object obtained by replacing all elements a_(ij) with a_(ji). For a second-tensor rank tensor a_(ij), the tensor transpose is simply a_(ji). The matrix transpose, most commonly written A^T, is the matrix obtained by exchanging A's rows and columns. For an invertible square matrix, it satisfies the identity

(A^T)^(-1)=(A^(-1))^T.
(1)

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

notationreferences
A^TThis work; Golub and Van Loan (1996), Strang (1988)
A^~Arfken (1985, p. 201), Griffiths (1987, p. 223)
A^'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

(B^TA^T)_(ij)=(b^T)_(ik)(a^T)_(kj)
(2)
=b_(ki)a_(jk)
(3)
=a_(jk)b_(ki)
(4)
=(AB)_(ji)
(5)
=(AB)_(ij)^T,
(6)

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

(AB)^T=B^TA^T.
(7)

See also

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. Linear Algebra and its Applications, 3rd ed. Philadelphia, PA: Saunders, 1988.Strang, G. Introduction to Linear Algebra. Wellesley, MA: Wellesley-Cambridge Press, 1993.Windley, P. F. "Transposing Matrices in a Digital Computer." Computer J. 2, 47-48, Apr. 1959.

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Transpose

Cite this as:

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

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