[math-fun] Different values of the dot product
Hello, I already posted this problem at sci.math but didn't get a lot of response. Maybe someone on this list can offer some genuine insight. Suppose we have the m different vectors x_1, x_2, ..., x_m in the R^n. No two vectors are the same and no vector is the zero vector. Furthermore we calculate for every 1<=i<j<=m the normal dot product x_i.x_j How many different numbers are we at least going to get, regardless of the choice of x_1, x_2, ..., x_m? Certainly, if m<=n, we can set x_1 = (1,0,0...,0)^T, x_2 = (0,1,0...)^T, ... - each dot product will be zero, we therefore only get one number as a result. This, of course, also works for any orthogonal basis of R^n. But what if m>n? Is anything known about this problem? Any ideas, results, bounds? Stefan
Quoting Stefan Steinerberger <stefan.steinerberger@gmail.com>:
I already posted this problem at sci.math but didn't get a lot of response. Maybe someone on this list can offer some genuine insight.
Suppose we have the m different vectors x_1, x_2, ..., x_m in the R^n. No two vectors are the same and no vector is the zero vector. Furthermore we calculate for every 1<=i<j<=m the normal dot product x_i.x_j
How many different numbers are we at least going to get, regardless of the choice of x_1, x_2, ..., x_m?
Isn't this a Gram matrix and isn't it kind of like a determinant, from which you can deduce the volumo of the parallelogram spanned by its rows (or columns)? As a symmetric matrix (with real vectors, as stipulated) it would have m^2 elements, equal pairs across the diagonal. Others could be equal as well, but rarely. - hvm ------------------------------------------------- www.correo.unam.mx UNAMonos Comunicándonos
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Stefan Steinerberger