# Zero-Diagonal Matrix and Positive Definitness?

Can a $n \times n$ symmetric matrix $A$ with diagonal entries that are all equal to zero, be positive definite (or negative definite)?

• What is $x^TAx$ when $x$ is a basis vector? – user147263 Nov 25 '15 at 21:58
Of course. For example this is positive defined $$\sigma_1=\begin{pmatrix} 0 & 1 \\ 1 & 0 \\ \end{pmatrix}$$ While this is negative defined: $$\sigma_1'=\begin{pmatrix} 0 & -1 \\ -1 & 0 \\ \end{pmatrix}$$ You need to be careful that zero on the diagonal doesn't imply much about the eigenvalues. With a change of basis you can change the coefficient on the diagonal. For example let us suppose that $$T=\begin{pmatrix} 1 & 1 \\ 1 & 0 \\ \end{pmatrix}$$ Then $$T^{-1}\sigma_1T=\begin{pmatrix} 1 & 1 \\ 0 & -1 \\ \end{pmatrix}$$
• Actually your example shows that this is not the case when $n=2$. I appreciate your help. – Mim Nov 26 '15 at 23:25