Invariance of Matrix Entries Under Conjugation by Subsets of $GL_n$

Question

Let $A\in\mathbb{R}^{n\times n}$ and define conjugation by $GL_n$ on $\mathbb{R}^{n\times n}$ in the usual way (e.g. for all $A\in\mathbb{R}^{n\times n}$ and $T\in GL_n$, $A\mapsto T^{-1}AT$). Are there any ways to derive conditions on $T$ so that a given subset of the entries of $A$ will be invariant under conjugation?

For example, let $$ A = \begin{bmatrix}a & b\\ c & d \end{bmatrix} $$ and suppose we want $b$ to be invariant under conjugation. Then we are looking for the subset $U \subset GL_n$ such that, for all $T \in U$, $$ T^{-1}AT = \begin{bmatrix}a' & b \\ c' & d' \end{bmatrix}. $$ Due to the application from which this question arises, it is also enough to be able say whether or not $U$ consists of just the identity matrix.

I would also appreciate any pointers to relevant literature.

Answer 1

Let $T=\begin{pmatrix} t_1 & t_2\cr t_3 & t_4\end{pmatrix}$ with $\det(T)=1$. Then $T^{-1}AT$ has upper right corner equal to $b$ if and only if $$ at_2t_4 + bt_4^2 - ct_2^2 - dt_2t_4=b. $$ This follows by a direct computation. Suppose that $c\neq 0$. Then we can solve the quadratic equation for $t_2$ over the complex numbers. This will give, in general, two non-zero solutions, hence there are more than just the multiples of the identity fixing $b$ under conjugation.

Answer 2

We assume that $T=[t_{i,j}]\in SL_n(\mathbb{C})$ ($\det(T)=1$). Note that the underlying field is $\mathbb{C}$ and not $\mathbb{R}$.

We choose (for example) $U=\{(i,j);i\leq j\}$ (the indices of the entries of the upper right part of a matrix) and we require that $(Adj(P)AP-A)_{i,j}=0$ for every $(i,j)\in U$.

Then we obtain a system of $\dfrac{n(n+1)}{2}+1$ equations of degree $n$ in the $n^2$ unknowns $t_{i,j}$. The question is: are these equations algebraically independent? (note that (*): the equality of index $(n,n)$ is a consequence of the equalities of indices $(i,i),i=1,\cdots,n-1$).

Experiments for $n=3$ (using Grobner basis) "show" that if $A$ is generic (randomly chosen), then the answer is yes (modulo (*)). That is, the solutions in $P\in SL_3(\mathbb{C})$ depend on $n(n-1)/2=3$ parameters.