Quadratic forms linearized by real matrices

Question

Could you please tell me what is meant by this: quadratic forms linearized by real matrices

I don't ask you to explain it, just simply tell me what is meant with it. Diagonalization or such? Thanks

Answer 1

If $(x, y)$ is a scalar product on a vector space, and $A$ a linear transformation on that space, then the mapping that takes $x$ to $(Ax, x)$ is called a quadratic form, and is said to be realized by $A$.