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
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
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.
Referenced on Wolfram|Alpha:
Transpose
CITE THIS AS:
Weisstein, Eric W. "Transpose." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Transpose.html
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