Angle decomposition of matrices Original Research Article

An algorithm (ADUM) is developed to decompose an arbitrary N × N unitary matrix M into ½N(N — 1) simple factor matrices. Each factor matrix has the form of an N × N unit matrix, except for a 2 × 2 complex rotation submatrix located at an appropriate position on the diagonal. The factor matrices each contain a rotation angle and between 0 and 3 phase angles, adding up to a total of N2 independent real angles. This can be summarized into an N × N real angle matrix Γ, containing the same information as the unitary matrix M. The factorisation can be extended to Hermitian or even generally complex matrices by applying an eigenvalue expansion or, alternatively, a singular value decomposition. Several applications to physical problems are discussed, and it is shown that ADUM is a powerful tool in the interpolation of matrices which depend on external parameters because it efficiently represents the degrees of freedom of a matrix while guaranteeing that matrix properties are maintained.