If a quadratic form is represented as a matrix, it may not be symmetric, but there must exist a representation that is symmetric (or Hermitian if complex). But is the reverse true? That is, do all symmetric and Hermitian matrices represent a quadratic form (except maybe in the case where matrices are all zeros)?
2026-03-26 10:57:56.1774522676
Do all symmetric matrices represent a quadratic form?
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Roughly speaking, for a field $F$ with $char F\ne 2$, there is a correspondence between symmetric bilinear forms and quadratic forms (Proposition 1.6 in Pfister's Book). So, the answer for your question is "yes" for real/complex symmetric matrices.
For a more detailed exposition on quadratic forms over fields (and a proof of the above cited Proposition) I recommend Introduction To Quadratic Forms Over Fields or Quadratic Forms with Applications to Algebraic Geometry and Topology.