Constructing the identity element of a matrix algebra generated by a single matrix

Question

Setup: Let us define the matrix algebra $\mathcal{A}$ generated by a single self-adjoint matrix $M$ to be the span of all natural powers of $M$ (the matrix and the span are over complex numbers) $$\mathcal A :=\left<M\right>=span\{M,M^2,M^3...\}.$$

Question: Is there an element $I_\mathcal{A}\in\mathcal{A}$ that acts as the multiplicative identity on the elements of $\mathcal{A}$? Is there a constructive way to show it?

Remarks:

1. I know that when we have the identity matrix as part of the algebra $\left<I,M\right>$ the algebra is just the span of spectral projections (including the kernel projection) of $M$.

2. I expect the same to be true for $\left<M\right>$ (excluding the kernel projection) and the identity $I_\mathcal{A}$ to be just the sum of all the spectral projections.

3. The problems start when I try to construct the spectral projections in $\left<M\right>$ without using the identity or construct the identity $I_\mathcal{A}$ without using the spectral projections (if I have one it is easy to construct the other).

Thank you.

Answer 1

Yes, the identity element always exists. Since $M$ is diagonalisable, its minimal polynomial is in the form of $xf(x)$ or $f(x)$ where $f$ is some (possibly constant) polynomial with $f(0)\ne0$. Let $g$ be the polynomial given by $\frac{f(x)-f(0)}{-f(0)}$. Then $g(0)=0$. Hence $g(M)\in\mathcal A$. Also, as $Mf(M)=0$, we have $Mg(M)=M$, i.e. $g(M)$ is the identity element in $\mathcal A$.