real quadratic forms


A quadratic form in an m -dimensional  real vectorspace V, is a bilineal map V\times V\to\mathbb{R} which can be determined by a m\times m-matrix A via

(v,w)\mapsto v^TAw

If we are allowed to write g(v,w)=v^TAw we can find that he (or she) possibly satisfy

  1. g(v,w)=g(w,v),  property dubbed symmetry
  2. g(v,v)\ge0, positive-definiteness
  3. g(v,v)=0 iff v=0, non-degeneracy

In case of affirmatively both three are satisfy g is called a metric on V and the pair

(V,g)

is called (real) Euclidean vectorspace…

MORE on \mathbb{R}^2

MORE on \mathbb{R}^3

2 Comments

Filed under math, multilinear algebra

2 responses to “real quadratic forms

  1. Amadeus

    Is it possible to have more than one metric?
    and if it is, what would that imply?

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