# Ordering positive definite matrices

## Abstract

We introduce new partial orders on the set \(S^+_n\) of positive definite matrices of dimension *n* derived from the affine-invariant geometry of \(S^+_n\). The orders are induced by affine-invariant cone fields, which arise naturally from a local analysis of the orders that are compatible with the homogeneous geometry of \(S^+_n\) defined by the natural transitive action of the general linear group *GL*(*n*). We then take a geometric approach to the study of monotone functions on \(S^+_n\) and establish a number of relevant results, including an extension of the well-known Löwner-Heinz theorem derived using differential positivity with respect to affine-invariant cone fields.

## Keywords

Positive definite matrices Partial orders Monotone functions Monotone flows Differential positivity Matrix means## Mathematics Subject Classification

15B48 34C12 37C65 47H05## 1 Introduction

Well-defined notions of ordering of elements of a space are of fundamental importance to many areas of applied mathematics, including the theory of monotone functions and matrix means in which orders play a defining role [2, 11, 14, 17]. Partial orders play a key part in a wide variety of applications across information geometry where one is interested in performing statistical analysis on sets of matrices. In such applications, the choice of order relation is often taken for granted. This choice, however, is of crucial significance since a function that is not monotone with respect to one order, may be monotone with respect to another.

We outline a geometric approach to systematically generate orders on homogeneous spaces. A homogeneous space is a manifold that admits a transitive action by a Lie group, in the sense that any two points on the manifold can be mapped onto each other by elements of a group of transformations that act on the space. The observation that cone fields induce conal orders on continuous spaces, combined with the geometry of homogeneous spaces forms the basis of the approach taken in this paper. The aim is to generate cone fields that are invariant with respect to the homogeneous geometry, thereby defining partial orders built upon the underlying symmetries of the space. A smooth cone field on a manifold is often also referred to as a causal structure. The geometry of invariant cone fields and causal structures on homogeneous spaces has been the subject of extensive studies from a Lie theoretic perspective; see [12, 13, 18], for instance. Causal structures induced by quadratic cone fields on manifolds also play a fundamental role in mathematical physics, in particular within the theory of general relativity [22].

The focus of this paper is on ordering the elements of the set of symmetric positive definite matrices \(S^+_n\) of dimension *n*. Positive definite matrices arise in numerous applications, including as covariance matrices in statistics and computer vision, as variables in convex and semidefinite programming, as unknowns in fundamental problems in systems and control theory, as kernels in machine learning, and as diffusion tensors in medical imaging. The space \(S^+_n\) forms a smooth manifold that can be viewed as a homogeneous space admitting a transitive action by the general linear group *GL*(*n*), which endows the space with an affine-invariant geometry as reviewed in Sect. 2. In Sect. 3, this geometry is used to construct affine-invariant cone fields and new partial orders on \(S^+_n\). In Sect. 4, we discuss how differential positivity [9] can be used to study and characterize monotonicity on \(S^+_n\) with respect to the invariant orders introduced in this paper. We also state and prove a generalized version of the celebrated Löwner-Heinz theorem [11, 17] of operator monotonicity theory derived using this approach. In Sect. 5, we consider preorder relations induced by affine-invariant and translation-invariant half-spaces on \(S^+_n\), and provide examples of functions and flows that preserve such structures. Finally, in Sect. 6, we review the notion of matrix means and establish a connection between the geometric mean and affine-invariant cone fields on \(S^+_n\).

## 2 Homogeneous geometry of \(S^+_n\)

*n*has the structure of a homogeneous space with a transitive

*GL*(

*n*)-action. The transitive action of

*GL*(

*n*) on \(S^+_n\) is given by congruence transformations of the form

*almost effective*in the sense that \(\pm I\) are the only elements of

*GL*(

*n*) that fix every \(\varSigma \in S^+_n\). The isotropy group of this action at \(\varSigma = I\) is precisely the orthogonal group

*O*(

*n*), since \(\tau _Q: I \mapsto QIQ^T=I\) if and only if \(Q\in O(n)\). Thus, we can identify any \(\varSigma \in S^+_n\) with an element of the quotient space

*GL*(

*n*) /

*O*(

*n*). That is

*C*yields a unique decomposition \(C=PQ\) of

*C*into an orthogonal matrix \(Q\in O(n)\) and a symmetric positive definite matrix \(P\in S^n_+\). Now note that if \(\varSigma \) has Cholesky decomposition \(\varSigma = CC^T\) and

*C*has a Cauchy polar decomposition \(C=PQ\), then \(\varSigma =PQQ^TP=P^2\). That is, \(\varSigma \) is invariant with respect to the orthogonal part

*Q*of the polar decomposition. Therefore, we can identify any \(\varSigma \in S^+_n\) with the equivalence class \([\varSigma ^{1/2}]=\varSigma ^{1/2}\cdot O(n)\) in the quotient space

*GL*(

*n*) /

*O*(

*n*).

*GL*(

*n*) consists of the set \(\mathbb {R}^{n\times n}\) of all real \(n\times n\) matrices equipped with the Lie bracket \([X,Y]=XY-YX\), while the Lie algebra of

*O*(

*n*) is \(\mathfrak {o}(n)=\{X\in \mathbb {R}^{n\times n}: X^T=-X\}\). Since any matrix \(X\in \mathbb {R}^{n\times n}\) has a unique decomposition \( X=\frac{1}{2}(X-X^T) + \frac{1}{2}(X+X^T)\), as a sum of an antisymmetric part and a symmetric part, we have \(\mathfrak {gl}(n)=\mathfrak {o}(n)\oplus \mathfrak {m}\), where \(\mathfrak {m}=\{X\in \mathbb {R}^{n\times n}: X^T=X\}\). Furthermore, since \({\text {Ad}}_Q(S)=QSQ^{-1}=QSQ^T\) is a symmetric matrix for each \(S\in \mathfrak {m}\), we have

## 3 Affine-invariant orders

### 3.1 Affine-invariant cone fields

*I*that is \({\text {Ad}}_{O(n)}\)-invariant:

*X*that are characterized by its spectrum. This observation leads to the following result.

### Proposition 1

*n*real eigenvalues of the symmetric matrix

*X*.

For instance, \({\text {tr}}(X)\) and \({\text {tr}}(X^2)\) are both functions of *X* that are spectrally characterized and indeed \({\text {Ad}}_{O(n)}\)-invariant. Quadratic \({\text {Ad}}_{O(n)}\)-invariant cones are defined by inequalities on suitable linear combinations of \(({\text {tr}}(X))^2\) and \({\text {tr}}(X^2)\).

### Proposition 2

### Proof

*X*are zero; i.e., if and only if X = 0. \(\square \)

The dual cone \(C^*\) of a subset *C* of a vector space is a very important notion in convex analysis. For a vector space \(\mathcal {V}\) endowed with an inner product \(\langle \cdot ,\cdot \rangle \), the dual cone can be defined as \( C^*=\{y\in \mathcal {V}:\langle y,x\rangle \ge 0, \ \forall x\in C\}\). A cone is said to be self-dual if it coincides with its dual cone. It is well-known that the cone of positive semidefinite matrices is self-dual. The following lemma will be used to characterize the form of the dual cone \((\mathcal {K}_{\Lambda }^{\mu })^*\) for each \(\mu \in (0,n)\) with respect to the standard inner product on \(\mathbb {R}^n\).

### Lemma 1

*I*is given by the ray \(\{X\in T_{I}S^+_n:X=\lambda I, \lambda \ge 0\}\). By affine-invariance, (19) reduces to \(\{X\in T_{\varSigma }S^+_n:X=\lambda \varSigma , \lambda \ge 0\}\) for \(\mu =n\), which describes an affine-invariant field of rays in \(S^+_n\).

It should be noted that of course not all \({\text {Ad}}_{O(n)}\)-invariant cones at *I* are quadratic. Indeed, it is possible to construct polyhedral \({\text {Ad}}_{O(n)}\)-invariant cones that arise as the intersections of a collection of spectrally defined half-spaces in \(T_{I}S^+_n\). The clearest example of such a construction is the cone of positive semidefinite matrices in \(T_{I}S^+_n\), which of course itself has a spectral characterization \(\mathcal {K}(I) = \{X\in T_I S^+_n: \lambda _i(X)\ge 0, \; i = 1, \ldots , n\}\).

### 3.2 Affine-invariant pseudo-Riemannian structures on \(S^+_n\)

At this point it is instructive to note the following systematic analysis of all affine-invariant pseudo-Riemannian structures on \(S^+_n\) before continuing with our treatment of affine-invariant cone fields. This elegant characterization presents the affine-invariant Riemannian metrics of (6) and the quadratic affine-invariant cone fields of (19) within a unified and rigorous mathematical framework. Recall that a pseudo-Riemannian metric is a generalization of a Riemannian metric in which the metric tensor need not be positive definite, but need only be a non-degenerate, smooth, symmetric bilinear form. The signature of such a metric tensor is defined as the ordered pair consisting of the number of positive and negative eigenvalues of the real and symmetric matrix of the metric tensor with respect to a basis. Note that the signature of a metric tensor is independent of the choice of basis by Sylvester’s law of inertia. A metric tensor on a smooth manifold \(\mathcal {M}\) is called Lorentzian if its signature is \((1,\dim \mathcal {M}-1)\).

### 3.3 Affine-invariant partial orders on \(S^+_n\)

A smooth cone field \(\mathcal {K}\) on a manifold \(\mathcal {M}\) gives rise to a *conal order* \(\prec _{\mathcal {K}}\) on \(\mathcal {M}\), defined by \(x\prec _{\mathcal {K}} y\) if there exists a (piecewise) smooth curve \(\gamma :[0,1]\rightarrow \mathcal {M}\) with \(\gamma (0)=x\), \(\gamma (1)=y\) and \(\gamma '(t)\in \mathcal {K}(\gamma (t))\) whenever the derivative exists. The closure \(\le _{\mathcal {K}}\) of this order is again an order and satisfies \(x\le _{\mathcal {K}} y\) if and only if \(y\in \overline{\{z:x\prec _{\mathcal {K}} z\}}\). We say that \(\mathcal {M}\) is *globally orderable* if \(\le _{\mathcal {K}}\) is a partial order. Here we will prove that the conal orders induced by affine-invariant cone fields on \(S^+_n\) define partial orders. That is, we will show that the conal orders satisfy the antisymmetry property that \(\varSigma _1\le _{\mathcal {K}} \varSigma _2\) and \(\varSigma _2\le _{\mathcal {K}} \varSigma _1\) together imply \(\varSigma _1=\varSigma _2\), for any affine-invariant cone field \(\mathcal {K}\) on \(S^+_n\). In other words, we will prove that there do not exist any non-trivial closed conal curves in \(S^+_n\). In the following, we will make use of the preimage theorem [3] given below. Recall that given a smooth map \(F:\mathcal {M}\rightarrow \mathcal {N}\) between manifolds, we say that a point \(y\in \mathcal {N}\) is a *regular value* of *F* if for all \(x\in F^{-1}(y)\) the map \(dF\vert _x:T_x\mathcal {M}\rightarrow T_y\mathcal {N}\) is surjective.

### Theorem 1

(The preimage theorem) Let \(F:\mathcal {M}\rightarrow \mathcal {N}\) be a smooth map of manifolds, with \(\dim \mathcal M=m\) and \(\dim \mathcal {N}=n\). If \(x\in \mathcal {N}\) is a regular value of *F*, then \(F^{-1}(c)\) is a submanifold of \(\mathcal {M}\) of dimension \(m-n\). Moreover, the tangent space of \(F^{-1}(c)\) at *x* is equal to \(\ker (dF\vert _x)\).

*F*. Hence, \(F^{-1}(c)\) is a submanifold of codimension 1 for any choice of \(c>0\). Furthermore, as \({\text {im}}(F) = \mathbb {R}^+=\{c\in \mathbb {R}:c>0\}\), the collection of submanifolds \(\{F^{-1}(c)\}_{c>0}\) forms a foliation of \(S^+_n\). Since \(\det \varSigma >0\) for any \(\varSigma \in S^+_n\), (24) implies that \({\text {ker}}(dF\vert _{\varSigma })=\{X\in T_{\varSigma }S^+_n:{\text {tr}}(\varSigma ^{-1}X)=0\}\). Thus, the tangent spaces to the submanifolds \(\{F^{-1}(c)\}_{c>0}\) are described by the affine-invariant distribution \(\mathcal {D}_{\varSigma }\) of rank \(\dim S^+_n-1 = n(n+1)/2-1\) on \(S^+_n\) defined by \(\mathcal {D}_{\varSigma }:=\{X\in T_{\varSigma }S^+_n:{\text {tr}}(\varSigma ^{-1}X)=0\}\).

### Proposition 3

### Proof

Proposition 3 clearly implies that \(S^+_n\) equipped with any of the cone fields described by (19) does not admit any non-trivial closed conal curves. Indeed, this result holds for all affine-invariant cone fields, not just quadratic ones. To see this, note that the permutation symmetry (10) of Proposition 1, implies that \({\text {tr}}(\varSigma ^{-1}X)\ne 0\) whenever \(X\in \mathcal {K}(\varSigma ){\setminus }\{0\}\). It thus follows by (26) that \(\det \circ \gamma :[0,1]\rightarrow \mathbb {R}^+\) is a strictly monotone function for any non-trivial conal curve \(\gamma \), which rules out the existence of closed conal curves. We thus arrive at the following theorem.

### Theorem 2

All affine-invariant conal orders on \(S^+_n\) are partial orders.

At this point it is worth noting a few interesting features of the collection of submanifolds \(\{F^{-1}(c)\}_{c>0}\) of \(S^+_n\). First note that if \(\gamma \) is an *inextensible* conal curve, then by (26) it must intersect each of the submanifolds \(F^{-1}(c)\) exactly once. That is, for each \(c>0\), \(F^{-1}(c)\) defines a *Cauchy surface* for the causal structure induced by any affine-invariant cone field. We also note the following results which connect these submanifolds to geodesics on \(S^+_n\) with respect to the standard affine-invariant Riemannian metric \(ds^2={\text {tr}}[(\varSigma ^{-1}d\varSigma )^2]\) on \(S^+_n\).

### Proposition 4

- (i)
If \(\varSigma _1,\varSigma _2\in S^+_n\) satisfy \(\det \varSigma _1=\det \varSigma _2=c\), then the geodesic from \(\varSigma _1\) to \(\varSigma _2\) lies in \(F^{-1}(c)\).

- (ii)
If \(X\in T_{\varSigma }S^+_n\) satisfies \({\text {tr}}(\varSigma ^{-1}X)=0\), then the geodesic through \(\varSigma \) in the direction of

*X*stays on the submanifold \(F^{-1}(\det \varSigma )\).

### Proof

*i*) Let \(\varSigma _1,\varSigma _2\in S^+_n\) satisfy \(\det \varSigma _1=\det \varSigma _2\). The geodesic \(\gamma \) from \(\varSigma _1\) to \(\varSigma _2\) is given by

*ii*) The geodesic \(\gamma \) from \(\varSigma \) in the direction of \(X\in T_{\varSigma }S^+_n\) takes the form \(\gamma (t)=\varSigma ^{1/2}\exp (t\varSigma ^{-1/2}X\varSigma ^{-1/2})\varSigma ^{1/2}\). If \({\text {tr}}(\varSigma ^{-1}X)=0\), then

### 3.4 Causal semigroups

*wedge*to be a closed and convex subset of a vector space that is also invariant with respect to scaling by positive numbers. Notice in particular that a wedge need not be pointed. Let \(\mathcal {M}=G/H\) be a homogeneous space,

*G*a Lie group with group identity element

*e*and Lie algebra \(\mathfrak {g}\),

*H*a closed subgroup with Lie algebra \(\mathfrak {h}\), and \(\pi :G\rightarrow \mathcal {M}\) the associated projection map. Assume that the Lie algebra \(\mathfrak {g}\) contains a wedge

*W*such that (

*i*) \(W\cap -W=\mathfrak {h}\) and (

*ii*) \({\text {Ad}}(h)W=W\) for all \(h\in H\). A wedge

*W*is said to be a

*Lie wedge*if \(e^{{\text {ad}}h}W=W\) for all \(h\in W\cap -W\). Denoting the left action of

*G*on \(\mathcal {M}\) by \(\tau _g:\mathcal {M}\rightarrow \mathcal {M}\), we have \(\pi \circ \lambda _g=\tau _g\circ \pi \), where \(\lambda _g\) is the left multiplication with

*g*on

*G*. Conditions (

*i*) and (

*ii*) ensure that \(d\pi \vert _g\circ d\lambda _g\vert _eW\) only depends on \(\pi (g)\), so that

*G*on \(\mathcal {M}\): \(d\tau _g\vert _x\mathcal {K}(x)=\mathcal {K}(\tau _g(x))\). These results can be found in [13]. The set \(S=\{g\in G: o \le _{\mathcal {K}} \tau _g(o)\}\), where \(o=\pi (e)\), is a closed semigroup of

*G*referred to as the causal semigroup of \((\mathcal {M},G,\mathcal {K})\). The following theorem is derived from [18].

### Theorem 3

The affine-invariant cone fields on \(S^+_n=GL(n)/O(n)\) can be viewed as projections of invariant wedge fields on the Lie group *GL*(*n*) in the sense of the above results. Since we have the reductive decomposition \(\mathfrak {gl}(n)=\mathfrak {o}(n)\oplus \mathfrak {m}\), it is easy to construct the corresponding wedge field *W* that satisfies conditions (*i*) and (*ii*) for a given affine-invariant cone field \(\mathcal {K}\). We will now use this structure and Theorem 3 to prove the following important result.

### Theorem 4

Let \(S^+_n\) be equipped with an affine-invariant cone field \(\mathcal {K}\) and the standard affine-invariant Riemannian metric \(ds^2={\text {tr}}[(\varSigma ^{-1}d\varSigma )^2]\). For any pair of matrices \(\varSigma _1,\varSigma _2\in S^+_n\), we have \(\varSigma _1\le _{\mathcal {K}}\varSigma _2\) if and only if the geodesic from \(\varSigma _1\) to \(\varSigma _2\) is a conal curve.

### Proof

*W*in \(\mathfrak {gl}(n)\) by

*W*satisfies the properties required of it in Theorem 3. If \(I\le _{\mathcal {K}}\varSigma \), it follows from Theorem 3 that there exists \(A\in W\) such that

### Remark 1

*n*real and positive eigenvalues of \(\varSigma _2\varSigma _1^{-1}\). We have thus used invariance to reduce the question of whether a pair of positive definite matrices \(\varSigma _1\) and \(\varSigma _2\) are ordered with respect to any of the quadratic affine-invariant cone fields to a pair of inequalities involving the spectrum of \(\varSigma _2\varSigma _1^{-1}\).

### 3.5 Visualization of affine-invariant cone fields on \(S^+_2\)

*n*forms a cone in the space of symmetric \(n\times n\) matrices. Moreover, \(S^+_n\) forms the interior of this cone. A concrete visualization of this identification can be made in the \(n=2\) case, as shown in Fig. 1a. The set \(S^+_2\) can be identified with the interior of the set \(K=\{(x,y,z)\in \mathbb {R}^3: z^2-x^2-y^2 \ge 0, \ z \ge 0\}\), through the bijection \(\phi :S^+_2\rightarrow {\text {int}} K\) given by

### 3.6 The Löwner order

*n*. First note that the cone at \(T_IS^+_n\) can be expressed as

## 4 Monotone functions on \(S^+_n\)

### 4.1 Differential positivity

*f*be a map of \(S^+_n\) into itself. We say that

*f*is

*monotone*with respect to a partial order \(\ge \) on \(S^+_n\) if \(f(\varSigma _1)\ge f(\varSigma _2)\) whenever \(\varSigma _1\ge \varSigma _2\). Such functions were introduced by Löwner in his seminal paper [17] on operator monotone functions. Since then operator monotone functions have been studied extensively and found applications to many fields including electrical engineering [1], network theory, and quantum information theory [6, 19]. Monotonicity of mappings and dynamical systems with respect to partial orders induced by cone fields have a local geometric characterization in the form of differential positivity [9]. A smooth map \(f:S^+_n\rightarrow S^+_n\) is said to be differentially positive with respect to a cone field \(\mathcal {K}\) on \(S^+_n\) if \(df\vert _{\varSigma }(\delta \varSigma )\in \mathcal {K}(f(\varSigma ))\) whenever \(\delta \varSigma \in \mathcal {K}(\varSigma )\), where \(df\vert _{\varSigma }:T_{\varSigma }S^+_n \rightarrow T_{f(\varSigma )}S^+_n\) denotes the differential of

*f*at \(\varSigma \). Assuming that \(\ge _{\mathcal {K}}\) is a partial order induced by \(\mathcal {K}\), then

*f*is monotone with respect to \(\ge _{\mathcal {K}}\) if and only if it is differentially positive with respect to \(\mathcal {K}\). To see this, recall that \(\varSigma _2\ge _{\mathcal {K}} \varSigma _1\) means that there exists some conal curve \(\gamma :[0,1]\rightarrow S^+_n\) such that \(\gamma (0)=\varSigma _ 1\), \(\gamma (1)=\varSigma _2\) and \(\gamma '(t)\in \mathcal {K}(\gamma (t))\) for all \(t\in (0,1)\). Now \(f\circ \gamma : [0,1]\rightarrow S^+_n\) is a curve in \(S^+_n\) with \((f\circ \gamma ) (0)=f(\varSigma _1)\), \((f\circ \gamma )(1)=f(\varSigma _2)\), and

### 4.2 The generalized Löwner-Heinz theorem

One of the most fundamental results in operator theory is the Löwner-Heinz theorem [11, 17] stated below.

### Theorem 5

Furthermore, if \(n\ge 2\) and \(r>1\), then \(\varSigma _1\ge _L \varSigma _2 \not \Rightarrow \varSigma _1^r\ge _L \varSigma _2^r\).

There are several different proofs of the Löwner-Heinz theorem. See [5, 11, 17, 20], for instance. Most of these proofs are based on analytic methods, such as integral representations from complex analysis. Instead we employ a geometric approach to study monotonicity based on a differential analysis of the system. One of the advantages of such an approach is that it is immediately applicable to all of the conal orders considered in this paper, while providing geometric insight into the behavior of the map under consideration. By using invariant differential positivity with respect to the family of affine-invariant cone fields in (19), we arrive at the following extension to the Löwner-Heinz theorem.

### Theorem 6

(Generalized Löwner-Heinz) For any of the affine-invariant partial orders induced by the quadratic cone fields (19) parametrized by \(\mu \), the map \(f_r(\varSigma )=\varSigma ^r\) is monotone on \(S^+_n\) for any \(r\in [0,1]\).

This result suggests that the monotonicity of the map \(f_r: \varSigma \mapsto \varSigma ^r\) for \(r\in (0,1)\) is intimately connected to the affine-invariant geometry of \(S^+_n\) and not its translational geometry. The structure of the proof of Theorem 6 is as follows. We first prove that the map \(f_{1/p}:\varSigma \mapsto \varSigma ^{1/p}\) is monotone for any \(p\in \mathbb {N}\). We then extend this result to maps \(f_{q/p}:\varSigma \mapsto \varSigma ^{q/p}\) for rational numbers \(q/p\in \mathbb {Q}\cap (0,1)\), before arriving at the full result via a density argument. We prove monotonicty by establishing differential positivity in each case. To prove the monotonicity of \(f_{1/p}:\varSigma \mapsto \varSigma ^{1/p}\), \(p\in \mathbb {N}\), we only need the following lemma [24].

### Lemma 2

*A*and

*B*are Hermitian \(n\times n\) matrices, then

The proof of the theorem for rational exponents is based on a simple observation whose proof nonetheless requires a few technical steps that are based on Proposition 5, which itself relies on Lemma 3 established in [7, 10].

### Lemma 3

*F*,

*G*be real-valued functions on some domain \(D\subseteq \mathbb {R}\) and \(\varSigma \),

*X*be Hermitian matrices, such that the spectrum of \(\varSigma \) is contained in

*D*. If (

*F*,

*G*) is an antimonotone pair so that \((F(a)-F(b))(G(a)-G(b))\le 0\) for all \(a,b\in D\), then

### Proposition 5

*X*is a Hermitian matrix, then

### Proof

*X*be a Hermitian matrix. Then, we have

### Proof of Theorem 6:

*p*-th root matrix function \(f_{1/p}\) is the

*p*-th power function \(f_p:\varSigma \mapsto \varSigma ^p\) and \(f_{1/p}\) contracts the invariant cone field \(\mathcal {K}\), \(f_p\) must expand \(\mathcal {K}\). Second, this expansion is greater for larger

*p*. That is, for positive integers \(p_1\le p_2\),

Finally, we extend the result to all real exponents \(r\in [0,1]\). Assume for a contradiction that there exists some \(r\in (0,1)\) and \(\varSigma _1,\varSigma _2\in S^+_n\) such that \(\varSigma _1\ge \varSigma _2\) and \(\varSigma _1^r < \varSigma _2^r\). Define \(E=\{x\in (0,1): \varSigma _1^x < \varSigma _2^x\}\) and note that \(E\ne \emptyset \) since \(r\in E\). As *E* is an open set in \(\mathbb {R}\), there exists some \(s\in \mathbb {Q}\cap E\) so that \(\varSigma _1^s < \varSigma _2^s\), which is a contradiction. Therefore, \(f_{r}\) is monotone for all \(r\in [0,1]\) with respect to any of the affine-invariant orders parametrized by \(\mu \). \(\square \)

### Remark 2

*X*, the inequality (55) of Proposition 5 with \(k=0\) becomes strict as

### 4.3 Matrix inversion

*f*is given by

### 4.4 Scaling and congruence transformations

### 4.5 Translations

It is important to note that translations do not generally preserve an affine-invariant order unless the associated affine-invariant cone field happens to also be translation invariant.

### Proposition 6

Let \(\le _{\mathcal {K}}\) denote the partial order induced by an affine-invariant cone field \(\mathcal {K}\) on \(S^+_n\). If \(\mathcal {K}\) is not translation invariant, then there exists a translation \(T_C:S^+_n\rightarrow S^+_n\), \(T_C(\varSigma )=\varSigma +C\) that does not preserve \(\le _{\mathcal {K}}\).

### Proof

## 5 Invariant half-spaces

### 5.1 An affine-invariant half-space preorder

To illustrate this we return to a puzzling aspect concerning the monotonicity of the function \(f_r(x)=x^r\) on the real line for \(r>0\) and its analogue result for positive semidefinite matrices. Namely, that the map \(f_r\) is monotone on \(S^+_n\) with respect to an affine-invariant partial order if \(r\in [0,1]\) but is not monotone on \(S^+_n\) for \(r>1\). We will show that the monotonicity on the real line for \(r>0\) is inherited in the matrix function setting in the form of a one-dimensional monotonicity expressed as the preservation of the affine-invariant half-space preorder for any \(r>0\).

### Proposition 7

The function \(f_r:\varSigma \mapsto \varSigma ^r\) is monotone on \(S^+_n\) with respect to the affine-invariant half-space preorder \(\preceq _{\mathcal {H}}\) for any \(r>0\).

### Proof

This result further highlights the natural connection between affine-invariance of causal structures on \(S^+_n\) and monotonicity of the matrix power functions \(f_r(\varSigma )=\varSigma ^r\). In particular, \(f_r\) is generally not monotone with respect to a preorder induced by a half-space field that is translation-invariant.

*K*in \(\mathbb {R}^3\) given by \(z^2-x^2-y^2\ge 0\), \(z\ge 0\) via a bijection \(\phi :\varSigma \mapsto (x,y,z)\). At \(\varSigma = \phi ^{-1}(x,y,z)\in S^+_2\), the inequality \({\text {tr}}(\varSigma ^{-1}X)\ge 0\) takes the form \(\ z\delta z - x \delta x - y \delta y \ge 0\), where \((\delta x, \delta y, \delta z)\in T_{(x,y,z)}K\) as shown in (45). The distribution \(\partial \mathcal {H}\) that consists of the hyperplanes which form the boundary of the half-space field \(\mathcal {H}_{\varSigma }\) are given by \(\ z\delta z - x \delta x - y \delta y = 0\). This distribution is clearly integrable with integral submanifolds of the form \( z^2-x^2 - y^2 = C\), where \(C\ge 0\) is a constant for each of the integral submanifolds, which form hyperboloids of revolution as shown in Fig. 4. As expected, these surfaces coincide with the submanifolds of constant determinant predicted in Sect. 3.3.

### 5.2 The Toda and QR flows

*Toda*flow is a well-know Hamiltonian dynamical system on the space of real symmetric matrices of fixed dimension

*n*, which can be expressed in the Lax pair form

*isospectral*. That is, the eigenvalues of

*X*(

*t*) and \(\varSigma (t)\) are independent of

*t*. Isospectral flows clearly preserve all translation invariant orders that possess spectral characterizations.

In [15], the following theorem is established for the projected Toda and QR flows. The projected flows refer to projections of the flows to the \(r\times r\) upper left corner principal submatrices of *X*(*t*) and \(\varSigma (t)\), i.e., the flows of \(X_r(t)=E_r^TX(t)E_r\) and \(\varSigma _r(t)=E_r^T\varSigma (t)E_r\), where \(E_r^T= [I_r \; 0]\).

### Theorem 7

For \(1\le r \le n\) and any symmetric matrix *X*(0) and symmetric positive definite matrix \(\varSigma (0)\), the ordered eigenvalues of the projected Toda flow orbit \(X_r(t)=E_r^TX(t)E_r\) and the projected QR flow orbit \(\varSigma _r(t)=E_r^T\varSigma (t)E_r\) are nondecreasing functions of *t*.

### Corollary 1

Let *f*(*x*) be any nondecreasing real-valued function and \(\alpha >0\). Then \(F(t)={\text {tr}}(f(E_r^TX(t)E_r))\) and \(G(t)={\text {tr}}(f(E_r^T\varSigma (t)^{\alpha }E_r))\) are nondecreasing functions of *t* for \(t\in \mathbb {R}\).

## 6 Matrix means

- 1.
\(M(\varSigma _1,\varSigma _2)=M(\varSigma _2,\varSigma _1)\)

- 2.
\(\varSigma _1 \le \varSigma _2 \implies \varSigma _1 \le M(\varSigma _1,\varSigma _2) \le \varSigma _2\)

- 3.
\(M(A^T\varSigma _1 A, A^T\varSigma _2 A) = A^T M(\varSigma _1,\varSigma _2)A\), for all \(A\in GL(n).\)

- 4.
\(M(\varSigma _1,\varSigma _2)\) is monotone in \(\varSigma _1\) and \(\varSigma _2\).

### Theorem 8

The geometric mean \(\#\) (102) defines a matrix mean for any affine-invariant order \(\le \) on \(S^+_n\).

### Proof

The geometric mean \(\varSigma _1\#\varSigma _2\) of two points \(\varSigma _1,\varSigma _2\in S^+_n\) is the midpoint of the geodesic joining \(\varSigma _1\) and \(\varSigma _2\) in \(S^+_n\) endowed with the standard Riemannian metric \(ds^2={\text {tr}}[(\varSigma ^{-1}d\varSigma )^2]\) [5]. This geometric interpretation immediately implies \(\varSigma _1\#\varSigma _2=\varSigma _2\#\varSigma _1\). Furthermore, given any affine-invariant order \(\le _{\mathcal {K}}\) induced by an affine-invariant cone field \(\mathcal {K}\) and a pair of matrices satisfying \(\varSigma _1\le _{\mathcal {K}}\varSigma _2\), the geodesic \(\gamma :[0,1]\rightarrow S^+_n\) from \(\varSigma _1\) to \(\varSigma _2\) is a conal curve by Theorem 4. Hence, the midpoint \(\varSigma _1\#\varSigma _2\) of \(\gamma \) clearly satisfies \(\varSigma _1 \le _{\mathcal {K}} \varSigma _1\#\varSigma _2 \le _{\mathcal {K}} \varSigma _2\). Since congruence transformations are isometries, for any \(A\in GL(n)\) the geodesic connecting \(A^T\varSigma _1 A\) to \(A^T\varSigma _2 A\) is given by \(\tilde{\gamma }(t)=A^T\gamma (t) A\). Thus, \((A^T\varSigma _1A)\#(A^T\varSigma _2A)=A^T( \varSigma _1\#\varSigma _2)A\). Finally, for fixed \(\varSigma _1\in S^+_n\), the function \(F(\varSigma )=\varSigma _1\#\varSigma \) is monotone with respect to any affine-invariant order since congruence transformations preserve affine-invariant orders and the function \(\varSigma \mapsto \varSigma ^{1/2}\) is monotone for any affine-invariant order. By symmetry, \(\#\) is also monotone with respect to its first argument. That is, the four conditions that define a matrix mean are all satisfied by the geometric mean for any choice of affine-invariant order. \(\square \)

## 7 Conclusion

The choice of partial order is a key part of studying monotonicity of functions that is often taken for granted. Invariant cone fields provide a geometric approach to systematically construct ‘natural’ orders by connecting the geometry of the state space to the search for orders. Coupled with differential positivity, invariant cone fields provide an insightful and powerful method for studying monotonicity, as shown in the case of \(S^+_n\). Future work can focus on exploring the applications of the new partial orders presented in this paper to the study of dynamical systems and convergence analysis of algorithms defined on matrices. It may also be fruitful to explore the implications of this work in convexity theory. New notions of partial orders mean new notions of convexity. In this context it may be natural to consider the concept of geodesic convexity on \(S^+_n\) with respect to the Riemannian structure on \(S^+_n\), as well as the usual notion of convexity on sets of matrices that is based on translational geometry.

## Notes

### Acknowledgements

We should like to thank the anonymous referees whose reviews resulted in significant improvements to the quality of this paper. We are particularly grateful to the reviewer who suggested the elegant characterization of affine-invariant pseudo-Riemannian structures presented in Sect. 3.2 and important clarifications in Sect. 3.4.

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