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MathGroup Archive 2008

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Linear algebra with generic dimensions

  • To: mathgroup at smc.vnet.net
  • Subject: [mg93610] Linear algebra with generic dimensions
  • From: dudesinmexico <dudesinmexico at gmail.com>
  • Date: Tue, 18 Nov 2008 07:19:44 -0500 (EST)

I am looking for a way to do linear algebra computations where the
dimensions of matrices and vectors are
symbolic. Let me give an example to make this more clear. Say that you
have a matrix whose
generic element is defined as T_ij=rho^(j-i). If I want the square of
the Frobenius norm of T, I can write

Sum[Rho^(2 (j - i)), {i, 0, N - 1}, {j, 0, N - 1}], Element[{i, j},
Integers]

and Mathematica gives as an answer a function of N and Rho:

(Rho^(2 - 2 N) (-1 + Rho^(2 N))^2)/(-1 + Rho^2)^2

and this is what I want, a function of matrix size N and, in this
case, a matrix parameter.

However, If I use the built-in Norm[,"Frobenius"], I cannot specify an
array with a generic dimension,
and this is true of all the linear algebra functions. I think that
what I need is a new "matrix type" holding the
expression for a generic matrix element as a function of its indices
and the names of the variables
holding the dimensions. Then I could overload built-in functions like
Dot[], Norm[], Tranpose[], etc..
with new functions.

Has this ever been done before? Is there any package or example
showing how do implement these ideas?

Thanks
-Arrigo


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