Cones, graphs and optimal scalings of matrices |
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Authors: | B David Saunders Hans Schneider |
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Institution: |
a Mathematics Department, University of Wisconsin, Wisconsin, Madison |
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Abstract: | Characterizations are given of the optimal scalings of a complex square matrix within its diagonal similarity class and its restricted diagonal equivalence class with respect to the maximum element norm. The characterizations are in terms of a finite number of products, principally circuit and diagonal products. The proofs proceed by reducing the optimal scaling problems from the multiplicative matrix level in succession to an additive matrix level, a graph theoretic level, and a geometric level involving duality theorems for cones. At the geometric level, the diagonal similarity and the restricted diagonal equivalence problems are unified. |
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