The task is to show that there's such environment $U \subset \mathbb{R}^{n,n}$ in the surroundings of unit matrix $I_n$ that for any $X\in U$ there can be found a matrix $Y\in \mathbb{R}^{n,n}$ that satisfies $X^{p}=Y^{q}$ ($q,p\in \mathbb{N}$ here)
Now we have to prove this one by using the implicit function theorem and I have no clue how to identify some "matrix building function" with a map $ \mathbb{R}^n×\mathbb{R}^m \mapsto \mathbb{R}^m$ which is required for the theorem to be applied. I tried but I cannot seem to get the dimensions to correspond with the theorem.
Id be grateful for any hint)
Hint: Let $f: \Bbb R^{n \times n} \times \Bbb R^{n \times n} \to \Bbb R^{n \times n}$ be the map defined by $$ f(X,Y) = X^p - Y^q. $$ We wish to prove that there is a function $g$ such that $Y = g(X)$ is a matrix for which $f(X,Y) = 0$ whenever $X$ in some neighborhood of $I$. Begin by noting that $F(I,I) = 0$.
Regarding the invertibility of the derivative requirement: recall that the derivative of $f$ with respect to $Y$ must be invertible. Computing and considering the derivative of a function with matrix inputs is tricky. One approach is as follows: expanding $f(I,I+H)$ yields $$ f(X,Y+H) = I^p - (I + H)^q =\\ I - I-\binom q1 H^1 - \binom q2 H^2 - \cdots = -qH + o(H). $$ Thus, the derivative $D_Y f(X,Y)$ at the point $X = I, Y = I$ is the linear map $$ D_Y f(X,Y)(H) = -qH. $$ This linear map is inverible.