Hi, David,
Thanks for your explanation about the minimize function in sage. I didn't
realize it's only for differentiable functions.
For the stuff regarding lattice, I think there may be some misunderstanding
here.
What I want is to find the minimum of a lattice.
A lattice L can be defined as
L={x=\lambda M| \lambda\in Z^m},
where M is the generator matrix of L and the gram matrix of L is equal to
MM^T.
The matrix
[1 2]
[3 4]
has determinant 4-2*3=-2, which is nonzero. Moreover, I use it as the generator
matrix for lattice, not the gram matrix. Thus I don't think it needs to be
symmetric or Hermitian.
As for the definition of minimum of a lattice, I assume it's defined for
all lattice, thus there should be no other restrictions on the generator
matrix (not gram matrix) except for it to be invertible.
According to the definition I found:
N(x)=x\cdot x=(x,x)=\sum x_i^2
for a vector x=(x_1,x_2,\dots,x_n) in a lattice and the minimum norm of
lattice L is
min{N(x): x\in L, x\neq 0}.
When I calculate Mv (M is the generator matrix and v is the vector (x,y)
with x,y both integers) I get a vector in L. Then I find the norm of Mv,
which is the norm of this vector in L.
What I need is the minimum of this value.
Did I get the wrong definition of the minimum of lattice?
Best Regards,
Xiaolu
On Thursday, September 6, 2012 6:50:00 PM UTC+8, David Loeffler wrote:
>
> Dear Cindy,
>
> Without wishing to cause offence, I think your problem isn't a Sage
> problem: it's that you don't understand the mathematical problem that
> you're trying to solve.
>
> Firstly, if V is an inner product space with basis v_1, ..., v_n and M
> is its Gram matrix (the matrix whose i,j entry is v_i paired with
> v_j), then the norm of the vector with coordinates x_1, .., x_n is not
> the usual norm of (M * [x_1; ...; x_n]); it's [x_1, ..., x_n] * M *
> [x_1; ...; x_n].
>
> Secondly, the matrix [1, 2; 3, 4] is not symmetric or Hermitian and
> its determinant is 0, so it is not the Gram matrix of a positive
> definite inner product space.
>
> Thirdly, the "minimize" function does what it says on the tin: it
> finds the minimum value of a function, and it does so by using
> calculus, assuming the function is differentiable. The minimum value
> of the norm of a vector in a positive definite inner product space is
> 0, the norm of the zero vector. You want the minimum value at a
> non-zero integer point and calculus is not going to help you with
> that.
>
> May I ask what motivates this long string of questions? Are you a
> student? If so, you should go back and read your undergraduate linear
> algebra notes a bit more carefully.
>
> Regards, David Loeffler
>
> On 6 September 2012 10:38, Cindy <[email protected] <javascript:>>
> wrote:
> > BTW, the generator matrix I used for the previous example is
> > [1 2]
> > [3 4]
> >
> > Thanks.
> >
> > Cindy
> >
> >
> > On Wednesday, September 5, 2012 7:31:48 PM UTC+8, David Loeffler wrote:
> >>
> >> > how can I get the minimum norm for the
> >> > ideal lattice (J,\alpha) using sage?
> >>
> >> What have you tried so far?
> >>
> >> David
> >
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