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, Cindy 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 <cindy42...@gmail.com <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 > > > > -- > > You received this message because you are subscribed to the Google > Groups > > "sage-support" group. > > To post to this group, send email to > > sage-s...@googlegroups.com<javascript:>. > > > To unsubscribe from this group, send email to > > sage-support...@googlegroups.com <javascript:>. > > Visit this group at http://groups.google.com/group/sage-support?hl=en. > > > > > -- You received this message because you are subscribed to the Google Groups "sage-support" group. To post to this group, send email to sage-support@googlegroups.com. To unsubscribe from this group, send email to sage-support+unsubscr...@googlegroups.com. Visit this group at http://groups.google.com/group/sage-support?hl=en.