Transpose of a matrix
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 lesson.
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 Mathematica 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)
|