Singular value decomposition for the Takagi factorization of symmetric matricesстатья
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Аннотация:We describe a simple implementation of the Takagi factorization of symmetric matrices A = U \Lambda U^T with unitary U and diagonal \Lambda\ge 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 A_\epsilon = A + \epsilon R = U_\epsilon \Lambda_\epsilon U^T_\epsilon,
\epsilon \ge 0 with distinct singular values, the unitary matrices U_\epsilon are discontinuous at the point e = 0, if the singular values of A are multiple, but the composition U_\epsilon \Lambda_\epsilon U^T_\epsilon
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.