  We provide a polynomial lower bound on the minimum singular value of an
$m\times m$ random matrix $M$ with jointly Gaussian entries, under a polynomial
bound on the matrix norm and a global small-ball probability bound
$$\inf_{x,y\in S^{m-1}}\mathbb{P}\left(\left|x^* M y\right|>m^{-O(1)}\right)\ge
\frac{1}{2}.$$ With the additional assumption that $M$ is self-adjoint, the
global small-ball probability bound can be replaced by a weaker version.
  We establish two matrix anti-concentration inequalities, which lower bound
the minimum singular values of the sum of independent positive semidefinite
self-adjoint matrices and the linear combination of independent random matrices
with independent Gaussian coefficients. Both are under a global small-ball
probability assumption. As a major application, we prove a better singular
value bound for the Krylov space matrix, which leads to a faster and simpler
algorithm for solving sparse linear systems. Our algorithm runs in
$\tilde{O}\left(n^{\frac{3\omega-4}{\omega-1}}\right)=O(n^{2.2716})$ time where
$\omega<2.37286$ is the matrix multiplication exponent, improving on the
previous fastest one in
$\tilde{O}\left(n^{\frac{5\omega-4}{\omega+1}}\right)=O(n^{2.33165})$ time by
Peng and Vempala.
