Quantum Physics Division

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On a subspace metric based on matrix rank

O. M. Baksalary^{1}
,
G. Trenkler^{2}

^{1} Faculty of Physics, Adam Mickiewicz University, ul. Umultowska 85, PL 61-614 Poznań, Poland

^{2} Department of Statistics, Dortmund University of Technology, Vogelpothsweg 87, D-44221 Dortmund, Germany

**Linear Algebra and Its Applications, 432, p. 1475-1491 (2010)**

**DOI:** 10.1016/j.laa.2009.11.007

Abstract:

The metric between subspaces M, N ⊆ Cn, 1, defined by δ (M, N) = rk (PM - PN), where rk (·) denotes rank of a matrix argument and PM and PN are the orthogonal projectors onto the subspaces M and N, respectively, is investigated. Such a metric takes integer values only and is not induced by any vector norm. By exploiting partitioned representations of the projectors, several features of the metric δ (M, N) are identified. It turns out that the metric enjoys several properties possessed also by other measures used to characterize subspaces, such as distance (also called gap), Frobenius distance, direct distance, angle, or minimal angle.

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