Definite Matrix

Let \(M\subseteq \mathbb{R}^{n\times n}\) be a symmetric real-valued matrix. We call it

• positive definite if \(x^TMx>0\) for every \(x\in \mathbb{R}^n\setminus \{0\}\),
• positive semi-definite if \(x^TMx\ge 0\) for every \(x\in \mathbb{R}^n\),
• negative definite if \(x^TMx<0\) for every \(x\in \mathbb{R}^n\setminus \{0\}\),
• negative semi-definite if \(x^TMx\le 0\) for every \(x\in \mathbb{R}^n\) and
• indefinite if it is not one of those mentioned above.