- [[Finite Euclidean Inner Product]] - [[Norm, for Beginners]] --- ### **Intro** Positive Definiteness, or semi-definiteness are applied to Linear operator, and their behaviors inside of an innr product. **Definite Matrix**: > A matrix $A$ is said to be positive semi-definite, when $\langle x, Ax\rangle \ge 0 \;\forall x \in \mathbb{C}^n$. If it's not specifically specified, we always assume that definition of definiteness is defined on the field of $\mathbb{C}^n$. Denoted as $A \succeq 0$. **Semi-Definite Matrix** > A matrix $A$ is semi-definite has to be Positive Definite first if the only value $x$ such that $\langle x, Ax\rangle$ is $x = 0$. Using the same token, one can definite Negative-definite matrices, and negative semi-definite matrices. --- ### **Properties** **Energy Norm**: A positive definite matrix/operator introduces a norm over the vector space, the norm is defined as: $ \Vert x\Vert_A = \sqrt{\langle x, Ax\rangle} $ Proof: Skipped --- ### **Theorems**