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.
The transpose of a matrix or tensor is implemented in the Wolfram Language as Transpose[A].
The product of two transposes satisfies
where Einstein summation has been used to implicitly sum over repeated indices. Therefore,
 |
(7) |
See also
Antisymmetric Matrix, Congruent Matrices, Conjugate Matrix, Conjugate Transpose, Symmetric Matrix
Explore with Wolfram|Alpha
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.
Referenced on Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Transpose." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Transpose.html