If we have the matrix equation $AX^{(i)} = X^{(i+1)}$ where $A$ is a constant matrix, this is what we'd call a recursive function; in matrix form.
Moreover, if $X^{(i+1)} = X^{(i)}$, i.e. $AX = X$ our matrix $X$ represents an equilibrium state for our system.
With the last equation, multiplying by the inverse of $X$, assuming $X$ is inversible, we get the following: $A = I$. This is what's bothering me.
- Say I had already defined $A\not=I$ , and had found the equilibrium state for $X$, whatever that is, doesn't this contradict the definition above?
And another question, which I have also wondered about some:
- If we define the matrix $A$ with the property $A^2 = A$, which determinant can $A$ have?
Here I also tried multiplying by the inverse of $A$, and got that $A = I$. Hence, $\det A = 1$. Is this proof enough? Does this prove that any matrix $B^2=B \implies B = I$?
I have a suspicion that my reasoning is off because the proof above only shows for when $\det A\not=0$, i.e. $A$ is invertible.