Consider a square and symmetric real matrix. The positive definiteness, defined as, $$ \sum_{i,j=1}^nA_{ij}x_ix_j>0\qquad\forall \mathbf{x}=(x_1,\ldots,x_n)\in\mathbb{R}^n,\ \mathbf{x}\neq\mathbf{0} $$ could be characterised by various means, for example
- by the positiveness of its eigenvalues,
- by the alternating sign of the coefficients of its characteristic polynomial,
- by the positiveness of the leading principal minors
In $\mathbb{R}^2$, considering the matrix $$ \begin{pmatrix} a & b \\ b & c \end{pmatrix} $$ corresponding to the quadratic form $ax^2+2bxy+cy^2$, the positive definiteness conditions could be expressed as $$ a>0,\ c>0,\ -\sqrt{ac}<b<\sqrt{ac} $$ while for non negative values of $x,y$ (not both null) one obtains $$ a>0,\ c>0,\ -\sqrt{ac}<b\qquad\quad $$
There exist some general results about $x_i\geq0$ (not all null) in $\mathbb{R}^n$, in particular for $n=3$? $$ \sum_{i,j=1}^nA_{ij}x_ix_j>0\qquad\forall \mathbf{x}=(x_1,\ldots,x_n)\in[0,\infty)^n,\ \mathbf{x}\neq\mathbf{0} $$
