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Let $S(A)$ denote the orbit of a complex or real matrix $A$ under a certainequivalence relation such as unitary similarity, unitary equivalence, unitarycongruences etc. Efficient gradient-flow algorithms are constructed todetermine the best approximation of a given matrix $A_0$ by the sum of matricesin $S(A_1), ..., S(A_N)$ in the sense of finding the Euclidean least-squaresdistance $$\min \{\|X_1+ ... + X_N - A_0\|: X_j \in S(A_j), j = 1, >..., N\}.$$Connections of the results to different pure and applied areas are discussed.

Least-Squares Approximation by Elements from Matrix Orbits Achieved by Gradient Flows on Compact Lie Groups