For Hermitian matrix $\mathbf{A}$, the following statements are equivalent, and any one can serve as the definition of the positive definiteness of $\mathbf{A}$ [Mey00, Sec. 7.6; Woe16, Sec. 7.4]:

• $\vec{x}^\ast\mathbf{A}\vec{x}>0$, for all $\vec{x}\neq\vec{0}$. Note $\vec{x}^\ast\mathbf{A}\vec{x}$ is called a quadratic form.

  In Dirac notation, $\langle x|\mathbf{A}|x\rangle\gt0$, for all $|x\rangle\neq\vec{0}$. Note $|0\rangle\color{red}{\neq}\vec{0}$.

• The eigenvalues of $\mathbf{A}$ are all positive.
• $\mathbf{A}$ can be put in the form $\mathbf{A}=\mathbf{B}^\ast\mathbf{B}$ or $\mathbf{A}=\mathbf{B}\mathbf{B}^\ast$, where $\mathbf{B}$ is a nonsingular matrix.

Similarly, the following statements are equivalent, and any one can serve as the definition of the positive semidefiniteness of $\mathbf{A}$:

• $\vec{x}^\ast\mathbf{A}\vec{x}\geq0$, for all $\vec{x}\neq\vec{0}$.

  In Dirac notation, $\langle x|\mathbf{A}|x\rangle\geq0$, for all $|x\rangle\neq\vec{0}$.

• The eigenvalues of $\mathbf{A}$ are all nonnegative.