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Bi-monotone maps on the set of all variance-covariance matrices with respect to minus partial order

Gregor Dolinar, Dijana Ilišević, Bojan Kuzma, Janko Marovt

Abstract

Let $H_{n}^{+}(\mathbb{R})$ be the cone of all positive semidefinite $n\times n$ real matrices. We describe the form of all surjective maps on $H_{n}^{+}(\mathbb{R}) $, $n\geq 3$, that preserve the minus partial order in both directions.

Bi-monotone maps on the set of all variance-covariance matrices with respect to minus partial order

Abstract

Let be the cone of all positive semidefinite real matrices. We describe the form of all surjective maps on , , that preserve the minus partial order in both directions.
Paper Structure (4 sections, 10 theorems, 78 equations, 3 figures)

This paper contains 4 sections, 10 theorems, 78 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Incidence relation among concentric ellipses
  4. Preservers of the minus partial order

Key Result

Proposition 1.1

Let $L_{1}=(y,X_{1}\theta ,\sigma ^{2}D)$ and $L_{2}=(y,X_{2}\beta ,\sigma ^{2}D)$ be any two linear models. Then $X_{1}\leq ^{-}X_{2}$ if and only if there exists a matrix $A$ with $\func{Im}A^{T}\subseteq \func{Im}X_{2}^{T}$ and $L_{1}$ is the model $L_{2}$ constrained by $A\beta =0$.

Figures (3)

  • Figure 1: $E_1,E_2$ touch the unit circle and intersect at a concentric circle $K_\rho$. The image of $K_\rho$ is an ellipse passing through the symmetrically placed points of intersection of $\Phi(E_1)$ and $\Phi(E_2)$. These points belong to the interior of the shaded rectangle $[-1,1]\times[-1,1]$.
  • Figure 2: Four ellipses with axes symmetrically at an angle $\phi$ relative to $x$ and $y$, respectively, touching the same inner circle as well as the unit circle, and intersecting vertically/horizontally at the same concentric circle drawn in thick gray. Their images on the right picture must intersect at the image of the gray thick circle, i.e., at the gray thick ellipse and must touch the image of the inner circle, i.e., an inner ellipse.
  • Figure 3: Towards the proof of Step 11. The inner-most circle can be arbitrarily small, so arbitrarily narrow ellipse touches a unit circle and has axis on a ray passing through a given point. On the right are their $\Phi$-images.

Theorems & Definitions (19)

  • Proposition 1.1
  • Theorem 1
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 9 more