Matrix Representation
Any quadratic form can be expressed as
where are the coordinates of with respect to some chosen basis, and is a certain symmetric matrix with entries in, that depends on and on the basis.
This formula can also be written as where is the standard inner product of, and is the vector of defined by
The quadratic form is trivial if and only if all the entries are 0. If is the real numbers, there is always a basis such that is a diagonal matrix. In this case, the signs of the diagonal elements determine whether the quadric is degenerate or not.
Read more about this topic: Quadric (projective Geometry)
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