## Sylvester's theorem

### Prof. Dr. Christian Bär

University of Potsdam, Germany

Sylvester's theorem says that every symmetric bilinear form on a finite dimensional real vector space can be represented by a diagonal matrix with $r_+$ many $1$'s, $r_-$ many $-1$'s, and $r_0$ many $0$'s on the diagonal.
Determine the quantities $r_+$, $r_-$ und $r_0$ for the bilinear form
$$\beta_A: \mathbb{R}^n \times \mathbb{R}^n\to \mathbb{R}, \quad\quad\beta_A(x,y) =x^\top \cdot A \cdot y,$$
where $A$ is a random symmetric $n\times n$-matrix.

After clicking on "update" the result will be shown so that you can compare it with what you got.
At the same time a new matrix is generated.
You can choose the size $n$ of the matrix from the dropdown menu.