Singular value decomposition for the Takagi factorization of symmetric matrices

摘要

We describe a simple implementation of the Takagi factorization of symmetric matrices A=UΛUT with unitary U and diagonal Λ≥0 in terms of the square root of an auxiliary unitary matrix and the singular value decomposition of A. The method is based on an algebraically exact expression.For parameterized family Aε=A+εR=UεΛεUεT, ε≥0 with distinct singular values, the unitary matrices Uε are discontinuous at the point ε=0, if the singular values of A are multiple, but the composition UεΛεUεT remains numerically stable and converges to A.The factorization is represented as a fast and compact algorithm. Its demo version for Wolfram Mathematica and interactive numerical tests are available on Internet.