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.