 Quadratic lower bound for a nonsymmetric matrix

Let $B$ be a real square matrix and suppose $x^{T}Bx\ge c|x{|}^2$ for every real vector $x$, with $c>0$. The Cauchy-Schwarz inequality gives $x^{T}Bx\le |x||Bx|$, hence $|Bx|\ge c|x|$ 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 $\Vert B^{-1}\Vert \le 1/c$ in the Euclidean norm.

Symmetry of $B$ is unnecessary: $x^{T}Bx=x^{T}Hx$ 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.