I have a set of functions expanded in a finite number of spherical harmonics (up to degree $L$), $$ \eta_k^n(\theta,\phi) = \sum_{l=0}^L \sum_{m=-l}^l d_{kl}^{nm} Y_l^m(\theta,\phi) $$ Similar to the spherical harmonics, my $\eta_k^n$ functions satisfy the condition $$ \eta_k^{-n} = (-1)^n (\eta_k^n)^{*} $$ Written in vector form, we have $$ \boldsymbol{\eta} = D \mathbf{Y} $$ where $\boldsymbol{\eta}$ and $\mathbf{Y}$ are vectors of length $(L+1)^2$ and $D$ is a square matrix of size $(L+1)^2 \times (L+1)^2$ with elements $d_{kl}^{nm}$. My functions $\eta_k^n$ do not satisfy an orthogonality condition, and so I would like to define a new set of functions $$ u_k^n = \sum_{lm} b_{kl}^{nm} \eta_l^m $$ or $$ \mathbf{u} = B \boldsymbol{\eta} $$ such that the $u_k^n$ are orthonormal with respect to the usual inner product over the unit sphere, $$ <f,g> = \int d\Omega f g^{*} $$ The coupling matrix for the $u_k^n$ is given by \begin{align} C_{\mathbf{u}} &= \int d\Omega \mathbf{u} \mathbf{u}^{\dagger} \\ &= \int d\Omega B \boldsymbol{\eta} \boldsymbol{\eta}^{\dagger} B^{\dagger} \\ &= B C_{\boldsymbol{\eta}} B^{\dagger} \end{align} where $C_{\boldsymbol{\eta}} = D D^{\dagger}$ is the coupling matrix of the $\eta_k^n$ functions. In order to make the $u_k^n$ functions orthonormal, I require that $$ C_{\mathbf{u}} = B C_{\boldsymbol{\eta}} B^{\dagger} = I $$ So my problem becomes: given $C_{\boldsymbol{\eta}} = D D^{\dagger}$, find a matrix $B$ which satisfies $B C_{\boldsymbol{\eta}} B^{\dagger} = I$.
This problem is not unique, as there are multiple (possibly infinite?) matrices $B$ which will satisfy that equation for a given matrix $C_{\boldsymbol{\eta}}$. However, I would like to impose the additional constraint that the orthonormal functions $u_k^n$ also satisfy the symmetry condition, $$ u_k^{-n} = (-1)^n (u_k^n)^{*} $$ This leads to the following condition on the elements of $B$: $$ b_{kl}^{-n,m} = (-1)^{n+m} (b_{kl}^{n,-m})^{*} $$ So at last I arrive at my question:
Given a positive-definite matrix $C_{\boldsymbol{\eta}}$, find a matrix $B$ which satisfies $B C_{\boldsymbol{\eta}} B^{\dagger} = I$ and also satisfies the constraint $$ b_{kl}^{-n,m} = (-1)^{n+m} (b_{kl}^{n,-m})^{*} $$ If anyone has any ideas on how to approach a problem like this, I would greatly appreciate it!
EDIT: If we remove the constraint, then one possible way to solve the (unconstrained) problem is to perform a Cholesky decomposition of the matrix $C_{\boldsymbol{\eta}} = L L^{\dagger}$. Then we have $$ B C_{\boldsymbol{\eta}} B^{\dagger} = (B L) (B L)^{\dagger} = I $$ and so $B L \in O((L+1)^2)$ where $O(n)$ denotes the orthogonal group of $n \times n$ matrices. One possible choice is simply $B = L^{-1}$.
This simple solution however does not satisfy my desired constraint, so a more sophisticated approach is needed.
