Zero-Diagonal Matrix and Positive Definitness?

Question

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

Thanks in advance!

Answer 1

The answer is negative and actually even more is true: the matrix cannot be positive definite if there is at least one diagonal element which is equal to $0$.

By definition, an $n\times n$ symmetric real matrix $A$ is positive definite if $$ x^TAx>0 $$ for all non-zero $x\in\mathbb R$.

Suppose that $a_{ii}=0$ for some $i=1,\ldots,n$, where $a_{ii}$ denotes the $i$-th element on the diagonal of $A$. Suppose that all entries of $x\in\mathbb R^n$ are equal to $0$ except the $i$-th entry which is not equal to $0$. Such an $x$ is hence a non-zero vector since there is one entry which is not equal to $0$. We have that $$ x^TAx=a_{ii}x_i^2=0 $$ for all non-zero $x_i\in\mathbb R$ since $a_{ii}=0$. If follows that the matrix $A$ is not positive definite.

Of course a matrix with a diagonal entry equal to $0$ can still be positive semi-definite.

I hope this helps.