In some cases, we need to construct a correlation matrix with a predefined set of eigenvalues, which is not trivial since arbitrary symmetric matrices with a given set of eigenvalues may not satisfy correlation constraints (e.g., unit diagonal elements).
A practical method to generate such matrices is based on the Method of Alternating Projections (MAP), as introduced by Waller (2018). This approach iteratively adjusts a matrix between two sets until convergence. It goes like this:
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