[Rd] transpose of complex matrices in R

[peter dalgaard]

On Jul 30, 2010, at 11:35 AM, Robin Hankin wrote:

> Hello everybody
> 
> When one is working with complex matrices, "transpose"  very nearly always means
> *Hermitian* transpose, that is, A[i,j] <- Conj(A[j,i]).
> One often writes A^* for the Hermitian transpose.
> 
> I have only once seen a  "real-life" case
> where transposition does not occur simultaneously with complex conjugation.
> And I'm not 100% sure that that wasn't a mistake.
> 
> Matlab and Octave sort of recognize this, as "A'" means the  Hermitian transpose of "A".
> 
> In R, this issue makes t(), crossprod(), and tcrossprod() pretty much useless to me.
> 
> OK, so what to do?  I have several options:
> 
> 1.  define functions myt(), and mycrossprod() to get round the problem:
> myt <- function(x){t(Conj(x))}
> 
> 2.  Try to redefine t.default():
> 
> t.default <- function(x){if(is.complex(x)){return(base::t(Conj(x)))} else {return(base::t(x))}}
> (This fails because of infinite recursion, but I don't quite understand why).
> 
> 3.  Try to define a t.complex() function:
> t.complex <- function(x){t(Conj(x))}
> (also fails because of recursion)
> 
> 4. Try a kludgy workaround:
>  t.complex <- function(x){t(Re(x)) - 1i*t(Im(x))}
> 
> 
> Solution 1 is not good because it's easy to forget to use myt() rather than t()
> and it does not seem to be  good OO practice.
> 
> As Martin Maechler points out, solution 2 (even if it worked as desired)
> would break the code of everyone who writes a myt() function.
> 
> Solution 3 fails and solution 4 is kludgy and inefficient.
> 
> Does anyone have any better ideas?

What's wrong with 

> t.complex <- function(x) t.default(Conj(x))
> M <- matrix(rnorm(4)+1i*rnorm(4),2)
> M
                     [,1]                  [,2]
[1,] 0.9907631-0.6927544i -0.0079213-1.1038222i
[2,] 1.2076160+0.4397778i  0.5926077-0.2140982i
> t(M)
                      [,1]                 [,2]
[1,]  0.9907631+0.6927544i 1.2076160-0.4397778i
[2,] -0.0079213+1.1038222i 0.5926077+0.2140982i

It's not going to help with the cross products though.

As a general matter, in my book, transpose is transpose and the other thing is called "adjoint". So another option is to use adj(A) for what you call myt(A), and then just remember to transcribe A^* to adj(A). 

I forget whether the cross products A^*A and AA^* have any special names in abstract linear algebra/functional analysis.

-- 
Peter Dalgaard
Center for Statistics, Copenhagen Business School
Solbjerg Plads 3, 2000 Frederiksberg, Denmark
Phone: (+45)38153501
Email: pd.mes at cbs.dk  Priv: PDalgd at gmail.com