numpy-discussion

[Numpy-discussion] Question about complex variables

[Karthikesh R. <ka...@ja...>]

Hi All,

i found something strange about complex matrices/arrays. Is it true that
transposing them does not yeald the mathematically correct transpose (the
signs of the imaginary part is reversed)?  This leads to wrong results
while calculating covariances etc ..

Actually i find that things go a bit wild with complex number, is it true?

Warm regards

karthik

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Karthikesh Raju,		    email: ka...@ja...
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Re: [Numpy-discussion] Question about complex variables

[Paulo J. S. S. <rs...@im...>]

Karthik,

You are making a common mistake. In Mathematics, the "transpose" of a
matrix is just the matrix with the axis transposed like numarray/numeric
does. What you called transpose is actually the "adjoint" or "conjugate
transpose" of a matrix. Actually, this last operation (the adjoint) is
the most interesting one in Linear Algebra/Functional Analysis. For more
details take a look at:

http://en.wikipedia.org/wiki/Conjugate_transpose
http://en.wikipedia.org/wiki/Transpose

In the reals both operations coincide, thus the confusion. This is the
reason why matlab can use the same symbol for both operations.

Now, enough Mathematics, let's go to numarray. To get the adjoint of a
matrix A in numarray, you can simply do:

conjugate(transpose(A))

If you want, you may define a function named adjoint to perform the
above operations. 

Note that this creates a new matrix, not only a new view of the original
matrix like "transpose" function in numarray. 

Hope it helps,

Paulo
-- 
Paulo José da Silva e Silva 
Professor Assistente do Dep. de Ciência da Computação
(Assistant Professor of the Computer Science Dept.)
Universidade de São Paulo - Brazil

e-mail: rs...@im...          Web: http://www.ime.usp.br/~rsilva

Teoria é o que não entendemos o    (Theory is something we don't)
suficiente para chamar de prática. (understand well enough to call) 
                                   (practice)