RANK(TITLE)-HUNTING: DIFFICULTIES OF RANK-DETERMINATION FOR
MATRICES AND OTHER COMPLICATED DATA ARRAYS

Róbert Rajkó

Department of Unit Operations and Environmental Engineering
College Faculty of Food Engineering University of Szeged
H-6701 Szeged, POB 433, E-mail: rajko@sol.cc.u-szeged.hu

One of the most crucial points of using the multivariate(dimensional) chemometric methods [1,2] is the determination of the rank of the investigated data array [3,4,5]. For the rank of a matrix (this rank can be considered as the "singularity measure" of the matrix) the following definitions can be given:

  1. The rank of a matrix is equal to the maximum order of its nonzero minors.
  2. The rank of a matrix is the number of dyads, which are derived from the minimal dyadic decomposition of the matrix.
  3. The rank of a matrix is the maximum number of its linearly independent columns or rows.
The above definitions are equivalent of course. The rank of a scalar or a vector is always 1, except for the zero scalar and vector which have rank of zero. If the data array has more than two dimensions (3-arrays, 4-arrays etc.) the rank definition 2 is seemed the only solution which can be generalized, introducing triads, …, multiads. However, the difference between the rank properties of 3-arrays and 2-arrays (matrices) is already tremendous [7]: there are straight-forward algorithms to calculate the rank of a matrix (e.g., Gauss-elimination), but there is no known algorithm for 3-arrays; it is well-known that rmax() = min{N,M}, but rmax() is unknown and difficult to determine, though the weak inequalities max{N,M,K} Łrmax() Ł min{NM,NK,MK} are known; the dyadic decomposition of a matrix is never unique, in contrast, the triadic decomposition of a 3-array is frequently unique.

The presentation is about the determination of rank belonging to stochastic matrices, and studying how small values can be considered as zero.
 

[1] Massart, D.L., Vandeginste, B.G.M., Buydens, L.M.C., de Jong, S., Lewi, P.J. and Smeyers-Verbeke, J.: Handbook of Chemometrics and Qualimetrics: Part A and Part B. Elsevier, Amsterdam, 1997 and 1998

[2] Horvai György (Szerk.): Sokváltozós adatelemzés (kemometria), Nemzeti Tankönyvkiadó, Budapest, 2001.

[3] Malinowski, E.R.: Factor analysis in chemistry. Wiley, New York, 1991.

[4] Rajkó Róbert: Kalibráció a kémiai méréseknél. Az analitikai kémiai információ minősége Magyar Kémiai Folyóirat, 107, 45-59, 2001.

[5] Rajkó Róbert: Analitikai mérések teljesítményjellemzőinek kritikai vizsgálata többváltozós kalibráció esetén Magyar Kémikusok Lapja, submitted 2001.

[6] Rózsa Pál: Lineáris algebra és alkalmazásai. Tankönyvkiadó, Budapest, 1991.

[7] Coppi, R and Bolasco S.: Multiway data analysis. Elsevier, Amsterdam, 1989