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# Perspectives of matrix convex functions

Edited by Robion C. Kirby, University of California, Berkeley, CA, and approved March 10, 2011 (received for review February 14, 2011)

## Abstract

In this paper, we generalize the main results of [Effros EG, (2009) *Proc Natl Acad. Sci USA* 106:1006–1008]. Namely, we provide the necessary and sufficient conditions for jointly convexity of perspective functions and generalized perspective functions.

We denote by and , the self-adjoint *n* × *n* matrices with spectra in the closed interval and the strictly positive *n* × *n* matrices with spectra in , respectively. Throughout this paper, it is assumed that *f* is a real-valued continuous function on . We say that *f* is matrix (operator) convex if for all and all *c*∈[0,1]. Also, *f* is matrix concave if -*f* is matrix convex. Given , the value *f*(*L*) is defined by the spectral theorem, i.e., if is the spectral decomposition of *L*, then . We define the perspective function *g* (associated to *f*) on by Let *g* be a perspective function. We say that *g* is jointly convex if for all , , and all *c*∈[0,1]. Also, *g* is jointly concave if -*g* is jointly convex.

Let *h* > 0 be a continuous function. We define the generalized perspective function (associated to *f* and *h*) on with Recall that if for any continuous function *f*, *f*(*L*) commutes with any operator commuting with *L* (including *L* itself), then we obtain the perspective function which is considered in ref. 1 by Edward G. Effros.

In this note, we provide the necessary and sufficient conditions between matrix convexity and jointly convexity of perspective and generalized perspective functions.

## Main Results

Recently, Edward G. Effros (1) proved the following theorem.

Suppose that *f* is operator convex. When restricted to positive commuting matrices *L*,*R*, the perspective function is jointly convex in the sense that if *L* = *cL*_{1} + (1 - *c*)*L*_{2} and *R* = *cR*_{1} + (1 - *c*)*R*_{2} where [*L*_{j},*R*_{j}] = 0 (*j* = 1,2), and 0 ≤ *c* ≤ 1, then

In the following, we remove the conditions [*L*,*R*] = 0, and [*L*_{j},*R*_{j}] = 0 (*j* = 1,2) in the above theorem. Moreover, we prove that the jointly convexity of *g* implies the matrix convexity of *f*. We use Theorem 2.1 of ref. 2 to prove the main theorem of our paper.

The function *f* is matrix convex if and only if the perspective function *g* is jointly convex.

Let *f* be matrix convex and let , , and 0 ≤ *c* ≤ 1. Put *L*≕*cL*_{1} + (1 - *c*)*L*_{2} and *R*≕*cR*_{1} + (1 - *c*)*R*_{2}. The matrices *T*_{1}≕ = (*cR*_{1})^{1/2}*R*^{-1/2} and *T*_{2}≕((1 - *c*)*R*_{2})^{1/2}*R*^{-1/2} satisfy . From matrix convexity of *f* and Theorem 2.1 of ref. 2, we have This inequality means that *g* is jointly convex. Conversely, let the perspective function *g* be jointly convex. It is clear that *f*(*L*) = *g*(*L*,1). We have where *L*_{1},*L*_{2}∈*H*_{n}(*I*) and 0 ≤ *c* ≤ 1. Hence, *f* is matrix convex.

The function *f* is matrix concave if and only if the perspective function *g* is jointly concave.

In ref. 1 for given continuous functions *f* and *h*, and commuting positive matrices *L* and *R* Edward G. Effros defined and then proved the following theorem.

Suppose that *f* is matrix convex, *f*(0) ≤ 0 and that *h* is matrix concave with *h* > 0. Then (*L*,*R*)⟼(*f*△*h*)(*L*,*R*) is jointly convex on positive commuting matrices *L*,*R* in the sense of Theorem 2.1.

In the following, we prove that the jointly convexity of *f*△*h* implies the matrix convexity of *f* and the matrix concavity of *h*.

Suppose that *f* and *h* are continuous functions with *f*(0) < 0 and *h* > 0. Then *f* is matrix convex and *h* is matrix concave if and only if the generalized perspective function *f*△*h* is jointly convex.

Let *f* be matrix convex and *h* be matrix concave. Let , , and 0 ≤ *c* ≤ 1. Put *L*≕*cL*_{1} + (1 - *c*)*L*_{2} and *R*≕*cR*_{1} + (1 - *c*)*R*_{2}. Define *T*_{1}≕(*ch*(*R*_{1}))^{1/2}*h*(*R*)^{-1/2} and *T*_{2}≕((1 - *c*)*h*(*R*_{2}))^{1/2}*h*(*R*)^{-1/2}. The concavity of *h* implies that . From matrix convexity of *f* and Theorem 2.1 of ref. 3, we have

Conversely, let the generalized perspective function *f*△*h* be jointly convex. It is clear that and . Let and 0 ≤ *c* ≤ 1. Then, This inequality means that *f* is matrix convex. Let and 0 ≤ *c* ≤ 1. Then, Because *f*(0) < 0, Hence, *h* is matrix concave.

Under the same hypotheses of Theorem 2.5, we have the following assertions:

if

*f*,*h*are matrix concave, then the generalized perspective function*f*△*h*is jointly concave.if the generalized perspective function

*f*△*h*is jointly concave, then*f*is matrix concave and*h*is matrix convex.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. E-mail: madjid.eshaghi{at}gmail.com.

Author contributions: M.E.G. designed research; A.E. and I.N. performed research; and M.E.G., A.E., and I.N. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

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