transpose in covariant notation REVIEWED
DOCX · 18.9 KB
Open DOCX file
A brief working note by Phil, dated 3.6.16 with a 5.16.16 comment, noting that the covariant transpose (MT)ab = Mba defined with tilted indices differs from the ordinary matrix transpose that swaps rows and columns. It illustrates the conflict with an example matrix R and its (RT)12 element, and asks how the tensor doc handled it, citing eq. (7.9.3). Index positions and matrix entries were lost in extraction.
AI-written summary; may contain errors. This description is approximate.
Extracted text (machine-read; may contain errors)
Transpose in Covariant Notation PhL 3.6.16
5.16.16: An early and brief realization that there really are two different kinds of transposes. This stuff is not ensconced in both tensor doc and wedge doc.
On the one hand, I have been going with this general idea of a covariant transpose
(MT)ab = Mba (RT)ab = Rba
(MT)ab = Mba (RT)ab = Rba
(MT)ab = Mba (RT)ab = Rba
(MT)ab = Mba (RT)ab = Rba , (2.11.d.9)
But now I see that this is "at odds" with what one might mean by the transpose of a matrix. Consider
R =
The "matrix transpose" of this matrix is given by swapping the elements of the rows and columns, so
RT =
Then you would say
(RT)12 = first row and second column = R21
But the claim above is that
(RT)12 = R21
which is a completely different object. So we have two distinct ideas here.
"covariant transpose of an up-tilt matrix"
"matrix transpose of an up-tilt matrix"
How did I handle this in tensor doc?
In (7.9.3) I talk about RT as the covariant transpose above