Matrix equation that doesn't change its form under a change of basis. I would like to continue the thread with preserved properties under a change of basis (I'm developing my reasearch), however the problem is so distinct that I've decided to ask a new question (I hope the question will not be classified as duplicate)
My question  is now:
What are matrix equations that don't change own form under orthogonal change of basis ?
(i.e. equations are satisfied by an old and a new matrix after a change of basis) 
My candidates so far :
If $g_i(A)$ is any polynomial function without constant matrix (e.g. $g_i(A) =2A^2+3A$) then  following forms of equations are, I  assume ( please, Dear Reader, confirm or deny that it is a true), satisfying the condition mentioned above:


*

*$g_1(A)+g_2(A^T)+  g_3(AA^T)+g_4(A^TA) = 0$ ,

*$g_1(A)+g_2(A^T)+  g_3(AA^T)+g_4(A^TA) = I$ ,  

*$g_1(A)+g_2(A^T)+  g_3(AA^T)+g_4(A^TA) = -I$  (and others = any
$k*I$).


From this we have, for example, conditions for preserved symmetry, skew-symmetry, orthogonality, normality mentioned earlier by Om.  and much more...
Edit 1.
The same is probably true for equation 


*

*$g_1(A)g_2(A^T)g_3(AA^T)g_4(A^TA) = kI$


Edit 2. (5 days)
It's worth to notice that the pattern can be extended also for equations with many different matrices. From that we have preserved relations between matrices such as: 


*

*commutativity $AB-BA=0$  

*transpose  $A^T-B=0$  

*inverse $ AB=I $ .


(As we see proposition is rather important so it seems that it should have own name. But has it ? Maybe someone knows ?)

Are there any other such unchanged equations for matrices changed under new basis?
 A: In this post, let $A^\star$ denote the transpose conjugate of $A$. 
Assume that $A$ is normal, meaning that $A^\star A = A A^\star$. 
Then, if $p(z, \overline z)$ is a complex polynomial, the equation 
$$\tag{1}
p(A, A^\star)=0$$
(here $A^\star$ is the transpose conjugate) is unitarily invariant in the sense that if $A$ satisfies (1) then $A'=UAU^\star$ also satisfies (1). Note that the constant term of the polynomial must be evaluated to a scalar matrix, e.g. if $p(z, \overline z)=z\overline z +1$ then $p(A, A^\star)=AA^\star + I$.
If $A$ is not normal, the equation (1) must be replaced with 
$$
p_1(A)+p_2(A^\star) + p_3(AA^\star) + p_4(A^\star A)=0, $$ 
where $p_1, p_2, p_3, p_4$ are polynomials. This is exactly what you did above.

You can find more invariant equations, but this requires the introduction of the so-called "functional calculus":
https://en.wikipedia.org/wiki/Holomorphic_functional_calculus
The most important example is the exponential equation: 
$$
\tag{2} e^A =B.$$ 
If $A=UA'U^\star$ and $B=UB'U^\star$ then $(A, B)$ solve (2) if and only if $(A', B')$ solve (2). (Here $U$ is a unitary matrix).
