Unitarization: Constructing Special Unitary Matrices by Generalized Cross Products and Unitary Left Multiplication

In analogy with orthogonalization, we define unitarization as the process of obtaining a special unitary matrix, of determinant 1, from a general complex matrix. The complex Gram–Schmidt process leads to SO(n, ℂ) with the bilinear inner product and to U(n) with the Hermitian inner product. In both cases, taking the last column to be the generalized cross product of the preceding columns (its complex conjugate in the Hermitian case) makes the determinant exactly 1. In three dimensions the quaternion product also replaces the projection for the second column, and in four dimensions the triple cross product determines the last column. We then define unitary left and right multiplication, SU(4) matrices determined by their first column alone, and show that the space of their generators is exactly 𝔰𝔭(2) ≅ 𝔰𝔬(5). This yields a QR decomposition in SU(4) without the Gram–Schmidt process, and shows that, unlike SO(4), SU(4) has no isoclinic decomposition by left and right multiplication.