Finding Well-Conditioned Similarities to Block-Diagonalize Nonsymmetric Matrices Is NP-Hard |
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Institution: | Univ Calif Berkeley, Dept Math, Berkeley, CA 94720 and Univ Calif Berkeley, Lawrence Berkeley Lab, Berkeley, CA 94720. |
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Abstract: | Given an upper triangular matrix A ∈ Rn×n and a tolerance τ, we show that the problem of finding a similarity transformation G such that G−1AG is block diagonal with the condition number of G being at most τ is NP-hard. Let ƒ(n) be a polynomial in n. We also show that the problem of finding a similarity transformation G such that G−1AG is block-diagonal with the condition number of G being at most ƒ(n) times larger than the smallest possible is NP-hard. |
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