X Parameterization

Parameterization of a $d\times d$ (complex) hermitian matrix $A$ into a real vector $(x_i)$ of $d^2$ elements. The parameterization is linear, and preserves inner products: $\mathrm{tr}(A\,A') = \sum_i x_i x_i'$.

The parameterization is defined as follows: the first $d$ entries of $(x_i)$ are the diagonal entries of $A$. The following $d(d-1)/2$ entries are the real parts of the off-diagonal entries, and the yet followoing $d(d-1)/2$ entries are the imaginary parts of the off-diagonal entries. All off-diagonal entries are normalized by a factor $1/\sqrt{2}$ to preserve inner products. The off-diagonals are listed in the lower triangular part and row-wise. More precisely, we have (define as shorthand $d'= d(d-1)/2$):

$$\begin{align*} A = \begin{pmatrix} x_{1} & \ast & \ast & \ldots & \ast \\ (x_{d+1} + i x_{d+d'+1})/\sqrt2 & x_{2} & \ast & \ldots & \ast \\ (x_{d+2} + i x_{d+d'+2})/\sqrt2 & (x_{d+3} + i x_{d+d'+3})/\sqrt2 & x_{3} & & \ast \\ \vdots & & & \ddots & \ast \\ (x_{d'} + i x_{2d'})/\sqrt2 & \ldots & & (x_{d+d'} + i x_{2d'+d})/\sqrt2 & x_{d} \end{pmatrix}. \end{align*}$$

The upper triangular off-diagonals are set of course such that $A$ is hermitian.

See Tomographer::DenseDM::ParamX.