CLASSIFICATION OF EXTREMAL 2‐PLANES OF MATRICES: rigidity and continuous families

For a linear subspace S of Mm×n(C), we define L(S) = Span{AB† : A, B in S} and R(S) = Span{A†B : A, B in S}, and study the extremal equality dim L(S) = dim R(S) = dim S. The central result is a rank-compression principle: if S is a singular 2-plane of maximal rank r, the extremal conditions force S to be unitarily equivalent to a subspace contained in M_r(C) ⊕ 0ₙ₋ᵣ. This compression reduces the classification to the regular cases of lower dimension.As an application, we obtain the complete orbit classification of extremal 2-planes for every n. The regular branch is classified by its invariants (p, [t], [u]) modulo block exchange, and the singular branch by the same invariants applied to the maximal-rank block r after compression. In particular, in the singular branch there is rigidity at maximal rank 2 and a continuous family of orbits from maximal rank 3 onwards, and the singular branch of M_4(C) is the first where a continuous family of non-TRO orbits appears.We extend the classification to the case of extremal 3-planes in M_3(C): every solution is regular and unitarily equivalent to a subspace of diagonal matrices, and the singular branch of M_3(C) is empty. Commutativity, on the other hand, is not preserved under bilateral unitary equivalence, as shown by the cyclic 3-plane Span{E_12, E_23, E_31}.Diagonalizability of the regular branch is universal for d ≤ 3, but the TRO property is not guaranteed by the extremal equality. For d ≥ 4, the existence of a non-diagonalizable extremal 4-plane shows that diagonalizability ceases to be universal. These results delimit precisely the boundary between rigidity and bifurcation in spaces of matrices, for every dimension n.