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A multilinear singular value decomposition
 SIAM J. Matrix Anal. Appl
, 2000
"... Abstract. We discuss a multilinear generalization of the singular value decomposition. There is a strong analogy between several properties of the matrix and the higherorder tensor decomposition; uniqueness, link with the matrix eigenvalue decomposition, firstorder perturbation effects, etc., are ..."
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Abstract. We discuss a multilinear generalization of the singular value decomposition. There is a strong analogy between several properties of the matrix and the higherorder tensor decomposition; uniqueness, link with the matrix eigenvalue decomposition, firstorder perturbation effects, etc., are analyzed. We investigate how tensor symmetries affect the decomposition and propose a multilinear generalization of the symmetric eigenvalue decomposition for pairwise symmetric tensors.
Existence And Uniqueness Of Optimal Matrix Scalings
, 1994
"... . The problem of finding a diagonal similarity scaling to minimize the scaled singular value of a matrix arises frequently in robustness analysis of control systems. We show that the set of optimal diagonal scalings is nonempty and bounded if and only if the matrix that is being scaled is irreducibl ..."
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Cited by 12 (0 self)
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. The problem of finding a diagonal similarity scaling to minimize the scaled singular value of a matrix arises frequently in robustness analysis of control systems. We show that the set of optimal diagonal scalings is nonempty and bounded if and only if the matrix that is being scaled is irreducible. For an irreducible matrix, we derive a sufficient condition for the uniqueness of the optimal scaling. Key words. Diagonal similarity scalings, scaled singular value minimization, irreducible matrices. AMS(MOS) subject classifications. 65F35,15A60,15A12, 47A55 Notation. R (C) denotes the set of real (complex) numbers. R+ stands for the set of positive real numbers. For z 2 C, Re z is the real part of z. The set of m \Theta n matrices with real (complex) entries is denoted R m\Thetan (C m\Thetan ). I stands for the identity matrix, with size determined from context. For a matrix P 2 C m\Thetan , P T stands for the transpose and P stands for the complex conjugate of P T . kPk ...
EXISTENCE AND UNIQUENESS OF OPTIMAL MATRIX SCALINGS*
"... Abstract. The problem of finding a diagonal similarity scaling to minimize the scaled singular value of a matrix arises frequently in robustness analysis of control systems. It is shown here that the set of optimal diagonal scalings is nonempty and bounded if and only if the matrix that is being sca ..."
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Abstract. The problem of finding a diagonal similarity scaling to minimize the scaled singular value of a matrix arises frequently in robustness analysis of control systems. It is shown here that the set of optimal diagonal scalings is nonempty and bounded if and only if the matrix that is being scaled is irreducible. For an irreducible matrix, a sufficient condition is derived for the uniqueness of the optimal scaling. Key words, diagonal similarity scalings, scaled singular value minimization, irreducible matrices AMS subject classifications. 65F35, 15A60, 15A12, 47A55 Notation. R (C) denotes the set of real (complex) numbers. R+ stands for the set of positive real numbers. For z E C, Re z is the real part of z. The set of rn n matrices with real (complex) entries is denoted Rmn (cmn). I stands for the identity matrix with size determined from context. For a matrix P Cmn pT stands for the transpose and P * stands for the complex conjugate of pT. iipi is the spectral norm (maximum singular value) of P given by the square root of the maximum eigenvalue of P*P. (For a vector v Cn, Ilvll is just the Euclidean norm.) For P Cnn, Tr P stands for the trace, that is, the sum of the diagonal entries of P. 1. Introduction. Given