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Conditions for the existence and uniqueness of optimal matrix scalings

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Open Problems in Mathematical Systems and Control Theory

Part of the book series: Communications and Control Engineering ((CCE))

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Abstract

Given M ∈ C n × n, consider the problem of finding

$$ {f_{\min }}\left( M \right) = \inf \left\{ {\left\| {DM{D^{ - 1}}} \right\|\left| {D \in \mathcal{D}} \right.} \right\}, $$
((6.1))

where ∥·∥ denotes the spectral norm, and Ɗ is a set of “scalings” defined by

$$ \mathcal{D} = \left\{ {D|_{{D_i} = D_i^* \in \;\;{C^{ki \times ki}},{d_i} \in \;R,\;Trace\left( D \right) = 1}^{D \in {C^{n \times n}},\;D = \;diag\left( {{D_1},...,{D_m},{d_1}{I_l}_{_1},...,dp{I_{lp}}} \right)}} \right\}. $$
((6.2))

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References

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Author information

Authors and Affiliations

  1. School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN, 47907-1285, USA

    Venkataramanan Balakrishnan

Authors
  1. Venkataramanan Balakrishnan
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Editor information

Editors and Affiliations

  1. Institute of Mathematics, University of Leige, Sart Tilman B37, B-4000, Liege, Belgium

    Vincent Blondel (PhD) (PhD)

  2. Department of Mathematics, Rutgers University, Hill Center 724, 08903, New Brunswick, USA

    Eduardo D. Sontag (PhD) (PhD)

  3. Centre for Arificial Intelligence and Robotics, Raj Bhavan Cirlce - High Grounds, 560001, Bangalore, India

    Mathukumalli Vidyasagar (PhD) (PhD)

  4. Institute of Mathematics, Rijksuniversiteit te Groningen, Postbus 800, 9700, Av Groningen, The Netherlands

    Jan C. Willems (PhD) (PhD)

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© 1999 Springer-Verlag London Limited

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Balakrishnan, V. (1999). Conditions for the existence and uniqueness of optimal matrix scalings. In: Blondel, V., Sontag, E.D., Vidyasagar, M., Willems, J.C. (eds) Open Problems in Mathematical Systems and Control Theory. Communications and Control Engineering. Springer, London. https://doi.org/10.1007/978-1-4471-0807-8_6

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  • DOI: https://doi.org/10.1007/978-1-4471-0807-8_6

  • Publisher Name: Springer, London

  • Print ISBN: 978-1-4471-1207-5

  • Online ISBN: 978-1-4471-0807-8

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