What are the positive/negative intertial exponent of the quadratic form $q(A)=tr(A^2)$ for $A\in \Bbb R^{n\times n}$.

Question

What are the positive/negative intertial exponent of the quadratic form $q(A)=tr(A^2)$ for $A\in \Bbb R^{n\times n}$.

Choosing the canonical base $E_{ij}$, then ...

Answer 1

Let $\mathcal H$ and $\mathcal K$ be respectively the subspace of all symmetric matrices and the subspace of all skew symmetric matrices. Then $q$ is positive on $\mathcal H\setminus0$ and negative on $\mathcal K\setminus0$. Also, $\mathbb R^{n\times n}=\mathcal H\oplus\mathcal K$. Therefore $n_+(q)=\dim\mathcal H$ and $n_-(q)=\dim\mathcal K$.