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- W1986541128 abstract "For a real matrix A, the separation of $A^T $ and A is sep $(A^T , - A) = min | A^T X + XA | / | X |$, where $| cdot |$ represents the Frobenius matrix norm. We discuss the conjecture that the minimizer X is symmetric. This conjecture is related to the numerical stability of methods for solving the matrix Lyapunov equation. The quotient is minimized by either a symmetric matrix or a skew-symmetric matrix and is maximized by a symmetric matrix. The conjecture is true if A is 2-by-2, if A is normal, if the minimum is zero, or if the real parts of the eigenvalues of A are of one sign. In general the conjecture is false, but counterexamples suggest that symmetric matrices are nearly optimal." @default.
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- W1986541128 date "1987-01-01" @default.
- W1986541128 modified "2023-09-27" @default.
- W1986541128 title "On the Singular “Vectors” of the Lyapunov Operator" @default.
- W1986541128 doi "https://doi.org/10.1137/0608003" @default.
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