Quadratic Form

Equivalent quadratic forms

Two quadratic forms $Q_1$ and $Q_2$ are said to be the equivalent if they can be obtained from each other through a change of variable. That is, two quadratic forms with matrices $A$ and $B$ are equivalent if and only if there exists an invertible matrix $P$ such that

$$B=P^{T}AP$$

Thus, the two quadratic forms represent the same function from ${\mathbb{R}}^n$ to $\mathbb{R}$.