# Norm in a quadratic space

 1 Hi I'm working in a vector space where the inner product is not the usual one and I would need to access directly to the norm induced by inner product. I found how to create such a space but the 'norm' function gives me the usual norm, which is not the one I want. Here is an example : SP=matrix(QQ,[[2,0],[0,3]]) V=VectorSpace(QQ,2,inner_product_matrix=SP)  What I want to compute is e.g. e0=V.0 e0.inner_product(e0)  while e0.norm() gives me 1 (wrong answer for me) I assume this should be possible directly. Any idea how ? Thanks, asked Mar 01 '12 Bertrand 21 ● 2

 0 You could redefine the norm function in a subclass. Note that your e0 is an object of class Vector_rational_dense: sage: V=VectorSpace(QQ,2,inner_product_matrix=SP) sage: e0=V.0 sage: type(e0)  So you can create a subclass, with a new norm function: sage: from sage.modules.vector_rational_dense import Vector_rational_dense as Vrd sage: class MyVector(Vrd): ....: def norm(self): ....: return sqrt(self.inner_product(self)) ....:  and write a helper function to take objects from the old class and make ones in the new class: sage: def improve_norm(v): ....: return MyVector(v.parent(),v.list()) ....:  Here's how you could use it: sage: f0 = improve_norm(e0) sage: e0.norm() 1 sage: f0.norm() sqrt(2)  Since MyVector is a subclass of Vector_rational_dense, the two should behave entirely the same, except for the new methods of MyVector: sage: e0.parent() == f0.parent() True sage: e0 == f0 True  Be careful though; unless you do a bit more work, arithmetic with MyVector objects will produce Vector_rational_dense objects by default: sage: type(f0) sage: type(e0 + f0) sage: type(f0+f0) sage: type(3*f0)  this is because the addition and multiplication methods are still those of Vector_rational_dense, which return an object of that type. If you want to fix this, you could also redefine _rmul_, _add_, etc. posted Mar 01 '12 niles 3605 ● 7 ● 45 ● 101 http://nilesjohnson.net/
 0 Shouldn't the induced norm be sqrt(e0.inner_product(e0))? I would just write a function to do this: def mynorm(v): return sqrt(v.inner_product(v))  I've heard that Sage could do more to support working with non-standard inner product matrices. This might be an example of a place that could be improved. Alternatively, if you wanted to modify the code in devel/sage/sage/modules/free_module_element.py, you could submit a patch. In the norm function, I would check to see if the inner product matrix defines the dot product: if not self.parent().__inner_product_is_dot_product(): special-case code for inner product matrices  Also, you might change the __abs__ function. posted Mar 03 '12 Jason Grout 3305 ● 7 ● 28 ● 74 oops -- I forgot to add sqrt; fixing now . . . my understanding was that the OP needed to use the built-in method e0.norm() for some reason (presumably it's being called from some other function that also cannot be modified). If this is not the case, your solution is of course better :) niles (Mar 03 '12)

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Asked: Mar 01 '12

Seen: 82 times

Last updated: Mar 03 '12