Suppose $M_{100}(\mathbb{R})$ denote the space of $100\times 100$ with real entries matrices with a natural norm.
I am trying to show that there exists some $\delta>0$ such that if $M\in M_{100}(\mathbb{R})$ and $||M||<\delta$, then there exists some $N\in M_{100}(\mathbb{R})$ such that $N^2+N=M$ and $\|N+I\|<10^{-5}$.
I don't immediately see why this is or should be true, and I am completely lost on coming up with a proof. Any hint/help will be greatly appreciated. Also, does the above result extend to any $n\times n$ matrices?
This is basically an exercise in the holomorphic functional calculus. Solve the equation $x^2 + x = y$ for $x$ near $x = -1$, and write the solution as a Taylor series about $y = 0$. Then your $N$ will be that same power series with $M$ substituted for $y$. It works for any submultiplicative matrix norm on $n \times n$ matrices, or more generally in any Banach algebra, and with $10^{-5}$ replaced by any positive number.