Given matrix $M$, how to find some matrix $A$ such that $M=A^\dagger A$?

Question

Given the matrix

$$M= \begin{bmatrix} 1+\alpha + n_3 & n_1 - i n_2 \\ n_1 + i n_2 & 1+ \alpha - n_3\end{bmatrix}$$

with $\alpha, n_1, n_2, n_3$ all real, how can this be expressed in terms of some matrix $A$ such that $M=A^\dagger A$?

This condition is needed in quantum measurement theory, see for example the first page, section II of this article https://arxiv.org/pdf/2001.04749.pdf.

Answer 1

Since it is such a small system, I simply made an ansatz for $A$ and solved for $M=A^\dagger A$. E.g.

$Assumptions = {{A11,A11i,A12,A12i,A21,A21i,A22,A22i,a,n1,n2,n3} \[Element] Reals}
A = {{A11 + I*A11i, A12 + I*A12i}, {A21 + I*A21i, A22 + I*A22i}}
AtA = ConjugateTranspose[A] * A // Simplify
M = {{1+a+n3,n1-I*n2},{n1+I*n2,1+a-n3}}
Solve[M==A, {A11,A11i,A12,A12i,A21,A21i,A22,A22i}] // Simplify

Since $M$ isn't symmetric, so Cholesky decomposition can't be used.