# Permuted Graph Matrices and Their Applications

Chapter

## Abstract

A permuted graph matrix is a matrix $$V \in \mathbb{C}^{(m+n)\times m}$$ such that every row of the m × m identity matrix I m appears at least once as a row of V. Permuted graph matrices can be used in some contexts in place of orthogonal matrices, for instance when giving a basis for a subspace $$\mathcal{U}\subseteq \mathbb{C}^{m+n}$$, or to normalize matrix pencils in a suitable sense. In these applications the permuted graph matrix can be chosen with bounded entries, which is useful for stability reasons; several algorithms can be formulated with numerical advantage with permuted graph matrices. We present the basic theory and review some applications from optimization or in control theory.

## Keywords

Invariant Subspace Negative Real Part Full Column Rank Algebraic Riccati Equation Grassmann Manifold
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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