Transpose of a matrix

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, and 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 lesson.

notationreferences
A^(T)This 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 Mathematica as Transpose[A].

The product of two transposes satisfies

(B^(T)A^(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^(T)A^(T).
(7)