Exact Dimensional Bound for Complex Matrix Subspaces Avoiding Nonzero Scalar Multiples of Unitary Matrices

Let M_d(ℂ) be the space of complex d × d matrices, and let C_d = {λU : λ ∈ ℂ, U ∈ U(d)} be the set of scalar multiples of unitary matrices. We determine the maximal complex dimension of a linear subspace S ⊆ M_d(ℂ) satisfying S ∩ C_d = {0}. We prove that m(d) = max {dimℂ S : S ⊆ M_d(ℂ), S ∩ C_d = {0}} = d(d − 1) for every d ≥ 2. The lower bound is attained by the subspace of matrices with zero last row. For d ≥ 4, the matching upper bound is proved by a variational argument: after obtaining an invertible element via Flanders' bounded-rank theorem, we maximize the squared modulus of the determinant on a normalized Frobenius sphere. A combinatorial defect inequality for the squared singular values yields a positive-definite Hessian block of dimension at least 2d − 1, which exceeds the real codimension of a subspace of dimension d(d − 1) + 1. This gives a contradiction to the maximality of the extremizer. The cases d = 2 and d = 3 are treated separately; the case d = 3 is closed using a result of L. Chen on maximally entangled states in seven-dimensional subspaces of ℂ³ ⊗ ℂ³.