= Positive-definite operator
{title2=$\langle Av,v\rangle>0\quad(v\ne0)$}
A <self-adjoint operator> $A$ is positive definite when $\langle Av,v\rangle>0$ for every nonzero vector $v$ in its domain. In finite dimension it is invertible and has positive <eigenvalues>. In infinite dimension, strict positivity of individual quadratic-form values need not give a bounded inverse; the stronger uniform bound $\langle Av,v\rangle\geq c\|v\|^2$ with $c>0$ supplies coercivity.
For maps $\beta:U\to V$ and $\gamma:V\to W$ between finite-dimensional <inner product spaces>, with $\operatorname{im}\beta=\ker\gamma$, the operator $\beta\beta^*+\gamma^*\gamma$ is positive definite. Its <quadratic form> is $\|\beta^*v\|^2+\|\gamma v\|^2$, whose zero set is $\operatorname{im}\beta\cap(\operatorname{im}\beta)^\perp=\{0\}$. This uses the <adjoint operator> relation $\ker\beta^*=(\operatorname{im}\beta)^\perp$.
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