Obtaining auxiliary matrices

In the case when the density-kernel $K$ is expanded in terms of an auxiliary matrix $T$ e.g. in order to construct a positive semi-definite density-matrix, it is necessary to be able to calculate the auxiliary matrix $T$ which corresponds to a given density-kernel $K$ by

$$K = T T^{\dag } .$$ (8.14)

This can be achieved by minimising the function ${\cal I}(T)$ given by

$${\cal I}(T) = {\rm Tr} \left[ (K - T T^{\dag })^2 \right]$$ (8.15)

whose derivative with respect to $T$ is

$$\frac{\partial {\cal I}(T)}{\partial T^{\alpha \beta}} = -4 \left[ T^{\dag } (K - T T^{\dag }) \right]_{\beta \alpha} .$$ (8.16)

This derivative vanishes at the minimum, and so we find that the matrix $T$ which minimises ${\cal I}(T)$ is the desired auxiliary matrix (the solution $T=0$ corresponds to a local maximum). We therefore choose to minimise ${\cal I}(T)$ by the conjugate gradients method to obtain the auxiliary matrix.