I have a problem.
Let $A(t)$ be a $n \times n$ matrix for each $t \in [0,b]$ with the property for all vectors $x$ that $$x^TA(t)x \geq C|x|^2$$ where $C$ doesn't depend on $t$. Can I use this fact to tell me that $A(t)$ has entries all in $L^\infty(0,T)$?
All I know is that $A(t)^{-1}$ exists, and each of its entries are in $L^\infty(0,T)$. And $A(t)^{-1}$ is also positive-definite, BUT the constant is not necessarily independent of $t$.