= Quadratic lower bound for a nonsymmetric matrix
{title2=$x^TBx\geq c|x|^2\Longrightarrow |Bx|\geq c|x|,\quad c>0$}
Let $B$ be a real square <matrix> and suppose $x^TBx\geq c|x|^2$ for every real <vector> $x$, with $c>0$. The <Cauchy-Schwarz inequality> gives $x^TBx\leq|x|\,|Bx|$, hence $|Bx|\geq c|x|$ for all $x$. In particular the <kernel> is zero, so $B$ is an <invertible matrix>. Applying the bound to $x=B^{-1}y$ gives the <operator norm> estimate $\|B^{-1}\|\leq1/c$ in the <Euclidean norm>.
Symmetry of $B$ is unnecessary: $x^TBx=x^THx$ for its <symmetric part of a matrix> $H=(B+B^T)/2$. Thus the hypothesis is exactly a positive lower bound on that <symmetric matrix>, even when the antisymmetric part is nonzero.
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