Monotone maps on matrices of index one

New matrix partial order based on spectrally orthogonal matrix decomposition

“…In Section 4, sets of matrices which are majorized by an idempotent are considered. Finally, we establish that the 2,3 ≤ -order on M n (C) is not G-based for any n > 1.…”

“…The notion of spectrally orthogonal matrix decompositions was introduced in [2]. These decompositions are used in the present paper, so we recall below some basic definitions and properties.…”

“…However, for general matrices (that is, diagonalizable or not), this tool (see [1,p.603]) has not been so much exploited in relation to partial orders. In this paper, we apply matrix canonical forms defined in [2], namely spectrally orthogonal matrix decompositions, which are useful for general matrices. This gives us the possibility to introduce and investigate some new partial orders.…”

“…In Section 3, we establish that 1 ≤ is equivalent to the cn-order, which gives a characterization for the latter via a spectrally orthogonal decomposition. Also this section contains various examples, among them, there is an example showing that 2,3 ≤ indeed provides a new relation on M n (C), in particular, it is different from the cn-order and other known order relations. In Section 4, sets of matrices which are majorized by an idempotent are considered.…”

Graphs Defined by Orthogonality

“…The aim of this paper is to introduce the notion of the orthogonality graph of a ring and investigate it comprehensively in the cases of the matrix algebra and some of its subsets. If, in Definition 2.15, we take M n (F) as R, then we obtain the orthogonality graph O (M n (F)) of the n×n matrix ring over a field F. The relation of 700 orthogonality appears in [25,31,34], where some partial orders on the matrix algebra and matrix transformations monotone with respect to these orders are studied. Matrix orders are widely used in various fields of algebra and have applications in mathematical statistics and many other areas of mathematics [15].…”