# Permuted Graph Matrices and Their Applications

## 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 Im 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.

## Notes

### Acknowledgements

The author is grateful to C. Mehl, B. Meini and N. Strabič for their useful comments on an early version of this chapter.

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