Revisiting the Marcus–de Oliveira Conjecture

Abstract

The Marcus–de Oliveira determinantal conjecture claims that the determinant of the sum of two normal matrices A and B with the prescribed spectra $\sigma \left(A\right)=\{a_1,\cdots ,a_n\}$ and $\sigma \left(B\right)=\{b_1,\cdots ,b_n\},$ respectively, is contained in the convex hull of the points $z_{\sigma }={\prod }_{i=1}^n(a_i+b_{\sigma \left(i\right)})$ for $\sigma \in S_n$, the symmetric group of degree n. The conjecture was independently proposed by Marvin Marcus in 1973 and de Oliveira in 1982, inspired by a result obtained by Miroslav Fiedler in 1971. We survey the main achievements relating to this open problem in matrix analysis. Some related results and questions that it has raised are also briefly reviewed. This overview aims to bring the attention of researchers to this problem and to stimulate the development of original approaches and techniques in the area. Ideally, this work may inspire further progress towards the solution of this long-standing conjecture.

1. Introduction

The Marcus–de Oliveira determinantal conjecture claims that the determinant of the sum of two normal matrices A and B with the prescribed spectra of, respectively,

$\sigma \left(A\right)=\{a_1,\cdots ,a_n\},\sigma \left(B\right)=\{b_1,\cdots ,b_n\},$

is contained in the convex hull of the points

$z_{\sigma }=\prod _{i=1}^n(a_i+b_{\sigma \left(i\right)})$

for $\sigma \in S_n$, which is the symmetric group acting on n objects.

This long-standing conjecture was independently proposed by Marcus in 1973 [1] and by de Oliveira as a research problem in 1982 [2], inspired by a result obtained by Fiedler in 1971 in the Hermitian case [3]. The mentioned result states the conjecture for the case that $A,B$ are Hermitian matrices, and it can be easily extended to the case that $\sigma \left(A\right)\cup \sigma \left(B\right)$ lies on a line through the origin.

Throughout, $M_n$, $N_n$, and $U_n$ will denote the sets of $n\times n$ complex matrices, normal matrices, and unitary matrices. As usual, $I_n\in N_n$ denotes the identity matrix.

For $A,B\in M_n,$ the determinantal range [4] is defined as

$\Delta (A,B):=\{det(A+UB{U}^{*}):U\in U_n\}.$

The Marcus–de Oliveira conjecture, which became known by the acronym MOC, claims that for $A,B\in N_n,$ with prescribed spectra $\sigma \left(A\right)$ and $\sigma \left(B\right),$ the following applies:

$\Delta (A,B)\subseteq conv\{z_{\sigma }:\sigma \in S_n\},$

where conv denotes the convex hull in the complex plane.

The $z_{\sigma }$’s are called the $\sigma$-points of $\Delta (A,B)$, and they are generated by permutation matrices.

When formulating the conjecture, de Oliveira observed the curious fact that “there are some true relations such that when replacing sums by products and products by sums, we obtain other true relations”. In this perspective, he considered the set

$W(A,B)=\{\mathrm{Tr}\left(AUB{U}^{*}\right):U\in U_n\},$

for $A,B\in M_n$. This set is the well-known B-numerical range of $A,$ a generalization of the classical numerical range $W\left(A\right)=W(A,I_n)$, introduced by Toeplitz and Hausdorff in the first quarter of the last century (see, e.g., [4] and references therein). Notice that in the literature, $W(A,B)$ is usually denoted by $W_B\left(A\right),$ and B is commonly replaced by C.

When proposing the conjecture, de Oliveira provided a short proof for what he calls the “dual” result of the MOC:

$W(A,B)\subseteq conv\{\sum _{i=1}^n{a}_i{b}_{\sigma \left(i\right)},\sigma \in S_n\}.$

Some observations are in order. Normal matrices are unitarily similar to the diagonal matrices of their eigenvalues; that is, $A=\mathrm{diag}(a_1,\cdots ,a_n),$ and $B=\mathrm{diag}(b_1,\cdots ,b_n)$, respectively. Since $\Delta (A,B)$ is invariant under unitary similarity transformations, without loss of generality in the MOC, we may assume A and B in diagonal form. For $U\in U_n,$ we easily find

$det(A+UB{U}^{*})={\displaystyle \frac{AU+UB}{detU}}={\displaystyle \frac{Z\circ U}{detU}},$

where $Z={[a_i+b_j]}_{i,j}$ and ∘ denote the Hadamard, or entrywise, product. Further, it can be easily seen that in the MOC, it suffices to consider U ranging over $S{U}_n,$ the special unitary group, which is formed by unitary matrices with the determinant 1.

The investigation of the MOC has attracted the interest of several researchers (see e.g., [5,6,7]), but despite many efforts to solve it affirmatively, or negatively, the conjecture remains open.

The remainder of this survey is organized as follows: In Section 2, we give an up-to-date overview of the major breakthroughs regarding the MOC, aiming at motivating new results and foster original ideas to attack the problem. In Section 3, the state of the art concerning related topics that it has motivated is described. It should be remarked that no progress on the MOC was obtained in the period from 2007 until 2021. In this year, a new class of normal dilation matrices affirming the conjecture were discovered [8]. In Section 4, determinantal inequalities for the sums of matrices are presented.

2. Classes of Matrices Stating the MOC

Following Rodtes in [8], we briefly write $(A,B)\in MOC$ if the pair of normal matrices $(A,B)$ affirms the Marcus–de Oliveira Conjecture.

The MOC has been proven for particular classes of matrices, enumerated in the sequel (see also [9]).

The MOC holds if A or B has at most one nonzero eigenvalue [2], 1982. The proof uses the well-known Binet–Cauchy formula.

As noticed in [10], 1985, since

$A+UB{U}^{*}=(A+\lambda I_n)+U(A-\lambda I_n)U^{*}$

for $\lambda$ a complex number and $U\in U_n,$ the MOC holds if all but possibly one of the $a_i$’s or $b_i$’s are equal.

In [11], 1985, the MOC was stated for $3\times 3$ matrices, using the Birkhoff theorem for doubly stochastic matrices.

The natural tentative to extend the proof for $n\ge 4$ fails.

Theorem 1

([12], 1986). The MOC holds if one of A or B is positive definite Hermitian and the other is skew-Hermitian.

Proof. 

Let $A=\mathrm{diag}(a_1,\cdots ,a_n)$, with $a_i>0$ $B=\mathrm{diag}(i{b}_1,\cdots ,i{b}_n),$ for $b_i\in \mathbb{R}$, $B_U=UB{U}^{*}$, and $z_U=det(A+i{B}_U)$, with $U\in U_n.$ For S Hermitian and any real $ϵ,$ the matrix ${\mathrm{e}}^{iϵS}$ is unitary. A complicated formula of the type

$det(A+i{\mathrm{e}}^{iϵS}{B}_U{\mathrm{e}}^{-iϵS})=det(A+i{B}_U)(1+\mathrm{Tr}\ast +\cdots )$

is derived, in which $A+i{B}_U$ and the commutator $[S,B_U]=S{B}_U-B_{U}S$ figure several times.

Let $z_U$ lie on a supporting line of $\Delta (A,B)$, perpendicular to the direction ${\mathrm{e}}^{i\varphi }$. Rotating $\Delta (A,iB)$ by an angle $-\varphi ,$ the supporting line becomes parallel to the imaginary axis, and so, the function

$z\to \Re ({\mathrm{e}}^{-i\varphi }det(A+{\mathrm{e}}^{iϵS}{B}_U))$

is stationary at $ϵ=0.$ From this fact and using the above formula, we obtain the commutation relation

${[A,(A+i{B}_U)]}^{-1}{\mathrm{e}}^{i\varphi }-{[A,(A-i{B}_U)]}^{-1}{\mathrm{e}}^{-i\varphi }=0.$

Technical computations held through six pages lead to the following crucial result:

If $z_U\ne 0$ lies on a supporting line of $\Delta (iB,A)$, then either $z_U$ is a $\sigma$-point or $z_U$ belongs to a regular curve that is contained locally in $\Delta (iB,A)$ and which has zero curvature at $z_U$.

To finish the theorem proof, let us fix a complex number w and consider a point $z_U$ so that $r= |z_U-w|$ is maximum. Thus, the disk $B(w,r)$ contains $\Delta (iB,A)$, and $z_U$ lies on a supporting line. Hence, it is either a $\sigma$-point or there exists a curve that is contained in $\Delta (iB,A)$ with zero curvature at $z_U$. This hypothesis is impossible, because the boundary of the disk that is centered at w and has a radius $|z_U-w|$ has nonzero curvature at $z_U$. Henceforth, $z_U$ is a $\sigma$-point.

We have therefore proven that $\Delta (iB,A)$ is the intersection of all the closed disks containing the $\sigma$-points, and this intersection is the convex hull of the $\sigma$-points. □

In [13], 1988, it was proven that if all of the $a_i$’s and $b_i$’s have the same modulus, then $\Delta (A,B)$ is contained in the line through the origin of argument $1/2{\sum }_{i=1}^n(arg{a}_i+arg{b}_i),$ and the MOC is true.

In [14], 1989, the MOC was shown to be true in the case in which all the $\sigma$-points lie on a straight line, not necessarily through the origin.

In [15], 1999, Drury showed the veracity of the conjecture when $n=4$ and U is a Householder reflection.

In [16], 1994, Theorem 1 was extended, showing that the MOC holds if the eigenvalues of A are positive and those of B lie on a line through the origin, or if the eigenvalues of A lie on a line ℓ and those of B lie on a line parallel to ℓ.

In [17], 2006, Theorem 1.2, Corollary 1.3, the MOC for the case of essentially Hermitian matrices was revisited, and in [18], 2007, Corollary 3, the veracity of the MOC was stated for scalar multiples of unitary matrices, using Moebius transformations.

To the best of our knowledge, there was no progress on the conjecture in the period from 2007 until 2021. In this year, a paper affirming the conjecture for a class of normal dilations was published [8].

For the sake of completeness, we sketch the proof of the aforementioned result. For this purpose, some considerations are in order. Let $a,b,c,d$ be real numbers and

(1)$X=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right],$

and define the Moebius transformation on $\mathbb{C}\setminus \{-d/c\}$ (or on $\mathbb{C}$ if $c=0$):

$\chi \left(z\right):=(az+b){(cz+d)}^{-1}.$

Then, $\chi \left(A\right)$ can be analogously defined for $A\in M_n$, when $det(cA+d{I}_n)\ne 0.$ Assume that $det(cA+d{I}_n)det(cB+d{I}_n)\ne 0.$ For $A,B$ normal matrices and $U\in U_n$, we find that

$det(\chi \left(A\right)-U\chi \left(B\right)U^{*})={\displaystyle \frac{{(detX)}^n}{det(cA+d{I}_n)}}det(cB+d{I}_n)det(A-UB{U}^{*}),$

and the following result holds:

Theorem 2.

Let $det\left(X\right)\ne 0$ and $det(cA+d{I}_n)det(cB+d{I}_n)$. The pair $(A,-B)\in$ MOC if and only if $\left(\chi \right(A),-\chi (B\left)\right)\in MOC$.

In [12], Theorem 2, it was shown that if the MOC holds for $(A,B)$, then it holds for $({(A-t{I}_n)}^{-1},{(B+t{I}_n)}^{-1})$ for any complex t not belonging to $\sigma (A\oplus -B).$ This result can be viewed as a special case of the previous theorem.

Let $X\in M_n$ and s be a complex number. It is clear that

(2)$N(X,s):=\left[\begin{array}{cc}X& {(X-s{I}_n)}^{*}\\ {(X-s{I}_n)}^{*}& X\end{array}\right],$

is a normal matrix of the order $2n.$ It can be seen that the eigenvalues of $N(X,s)$ lie on both real and imaginary axes, so the matrix is not essentially Hermitian or a scalar multiple of a unitary matrix, in which case the MOC would have already been known to hold. Notice that for any $U\in U_n,$ the matrix $UN(X,s)U^{*}$ is a normal dilation of X.

Theorem 3.

Let $X,Y\in M_n$ and $s,t\in \mathbb{C}.$ Then, $\left(N\right(X,s),N(Y,t\left)\right)\in MOC.$

For the sake of self-containment, we include the proof of this theorem. The proof uses the following lemmas, since it is the next one due to Drury [19].

Lemma 1.

Let $A,B\in M_n$ be Hermitian with eigenvalues of $a_1,\cdots ,a_n$ and $b_1,\cdots ,b_n$, respectively. Let $t_1,\cdots ,t_n$ be the eigenvalues of $A+B.$ Then,

$\prod _{i=1}^n(\lambda +t_j)\in conv\{\prod _{j=1}^n(\lambda +a_j+b_{\sigma \left(j\right)}):\sigma \in S_n\},$

where conv denotes the convex hull in the space of polynomials, and λ is an indeterminate.

Lemma 2.

Let $A,B\in M_n$ and $C,D\in M_m$ be normal. If $(A,B)\in MOC$ and $(C,D)\in MOC,$ then $(A\oplus C,B\oplus D)\in MOC.$

As a simple corollary of the above result, the next result follows:

Lemma 3.

Let $X,Y\in M_n$ and $\alpha ,\beta \in \mathbb{C}.$ Then, $(X-X^{*}+\alpha I_n,Y-Y^{*}+\beta I_n)\in MOC$, and $(X+X^{*}+\alpha I_n,Y+Y^{*}+\beta I_n)\in MOC$.

Proof of Theorem 3.

Let U be the following block unitary matrix:

(3)$U:=2^{1/2}\left[\begin{array}{cc}{I}_n& I_n\\ -I_n& I_n\end{array}\right]\in U_{2n}.$

An easy computation yields that

(4)$U\left[\begin{array}{cc}M& N\\ N& M\end{array}\right]U^{*}=(M-N)\oplus (M+N),$

for any $M,N\in M_n.$ Let

$A:=X+X^{*}+\overline{s}{I}_n,B:=Y-Y^{*}+\overline{t}{I}_n,C:=X+X^{*}-\overline{s}{I}_n,D:=Y+Y^{*}-\overline{t}{I}_n.$

By Lemma 1, the pairs $(A,B)$ and $(C,D)$ satisfy the MOC, and so, by Lemma 2, we have $(A\oplus C,B\oplus D)\in$ MOC. Hence,

$(N(X,s),N(Y,t))=(U(A\oplus C)U^{*},U(B\oplus D)U^{*})$

satisfies the MOC. □

3. Cornucopia of Results Concerning $\mathbf{\Delta }(\mathit{A},\mathit{B})$

The investigation of $\Delta (A,B)$, in particular its geometric properties, has motivated fruitful research. The complete characterization of this set is a rather difficult problem, only solved in very special cases.

The continuity of the map $U\to det(A+UB{U}^{*})$ implies that $\Delta (A,B)$ is a compact and connected set, since $U_n$ is compact and connected. It is also invariant under simultaneous unitary similarities of A and $B.$

One of the preliminary results concerning the geometry of $\Delta (A,B)$ was the characterization of certain nondifferentiable boundary points. A point P in the boundary $\partial \Delta$ of $\Delta$ is a corner if the intersection of a small enough P-centered disk with $\Delta$ is contained in a P-centered sector of angle less than $\pi .$

In [13], it was shown that if $0\ne z\in \partial \Delta (A,B)$ is a corner, then z is a $\sigma$-point. The proof uses the following formula:

$det(P+iQ)=detP(1+ϵ\mathrm{Tr}Q{P}^{-1})+\Theta \left({ϵ}^2\right)$

which is valid for $P,Q\in M_n$ and a nonsingular P, as well as the Fiedler techniques described in [3]. In [20], the case of the zero corner was covered, and the result was further extended to matrices $A,B\in M_n$ in Theorem 3.10 of [21,22].

For $n=2$ and the arbitrary matrices, $A,B\in M_n,$ $\Delta (A,B)$ is an elliptic disk. For the normal matrices A and B, the elliptic disk degenerates into a line segment, with the $\sigma$-points as endpoints [4].

The convexity, star-shapedness, or simple connectedness of $\Delta (A,B)$ are problems that naturally arise.

When $A,B\in N_3$, in [23], it was shown that $\Delta (A,B)$ is star-shaped, and the necessary and sufficient conditions for $\Delta (A,B)$ to be convex are also given.

In [24], the following example for $n=12$ shows that $\Delta (A,B)$ is not simply connected; that is, it can have a hole, and so, $\Delta (A,B)$ may not be convex or star-shaped.

Let

$A_1=\mathrm{diag}(1,{\displaystyle \frac{1+\sqrt{3}}{-1+\sqrt{3}}}),B_1=\mathrm{diag}(1,-1).$

Consider the $12\times 12$ diagonal block matrices

$A=A_1\oplus \cdots \oplus A_1,B=B_1\oplus \cdots \oplus B_1.$

When U ranges over $U_2,$$det(U{A}_1{U}^{*}+i{B}_1)$ describes the following line segment:

${\displaystyle \frac{2}{-1+\sqrt{3}}}\left[\sqrt{3}+i,\sqrt{3}-i\right].$

The set $det(VA{V}^{*}+iB)$, when V ranges over $U_{12},$ originates the points

$z\left(t\right)={({\displaystyle \frac{2\sqrt{3}}{-1+\sqrt{3}}}+{\displaystyle \frac{2it}{-1+\sqrt{3}}})}^6,t\in [-1,1],$

and these points generate a closed continuous curve surrounding the origin when t varies. On the other hand,

$|det(VA{V}^{*}+iB){|}^2={|detA|}^2\times |det(I_{12}+\left(A^{-1}{V}^{*}BV\right){|}^2=$

$={|detA|}^2\times |det(I_{12}+{\left(A^{-1}{V}^{*}BV\right)}^2)|.$

Since $A^{-1}$ is Hermitian and $UB{U}^{*}$ is positive definite, all the eigenvalues of $A^{-1}{V}^{*}BV$ are real, and so, $|det(VA{V}^{*}+iB){|}^2$ cannot be zero. Therefore, a hole exists in $\Delta (A,B)$.

In [25], the necessary and sufficient conditions for $\Delta (A,B)$ with $A,B\in M_n$ being a singleton or a line segment were obtained. Further results in this area are welcome.

4. Determinantal Inequalities

Motivated by the MOC, determinantal inequalities involving sums of matrices were intensively investigated by different researchers (see, e.g., [26]). One of the first results in this context is the following theorem concerning strictly dissipative matrices [4]. A complex square matrix written in the Cartesian decomposition form $M=H+iK$ is strictly dissipative if K is positive definite.

Theorem 4.

Assume $a_1\ge \cdots \ge a_n$ and $b_1\ge \cdots \ge b_n>0$ to be the eigenvalues of H and K, respectively, of a strictly dissipative matrix $M=H+iK$. Then, the following bounds hold:

$\prod _{j=1}^n|a_j+i{b}_j|\le |detM|\le \prod _{j=1}^n|a_j+i{b}_{n-j+1}|,$

where left and right inequalities turn into equalities if and only if M is a normal matrix with the eigenvalues $a_j+i{b}_j$ and $a_j+i{b}_{n-j+i}$, respectively.

In 1937, von Neumann, and in 1993, Miranda and R.C. Thompson obtained extremal results for the trace of the product $\mathrm{Tr}\left(AUBV\right)$, as $U,V$ both range over the unitary group or both over the special orthogonal group $S{O}_n$, formed by orthogonal matrices with the determinant 1.

Theorem 5.

Let $A,B$ be square matrices with singular values $a_1\ge \cdots \ge a_n\ge 0$ and $b_1\ge \cdots \ge b_n\ge 0.$ Then, the following hold:

• i. 

  For the complex matrices A and B, $max\left\{\right|\mathrm{Tr}\left(AUBV\right)|:U,V\in U_n\}={\sum }_{i=1}^n{a}_i{b}_i$ (von Neumann);

• ii. 

  For the real matrices A and B, $max\left\{\right|\mathrm{Tr}\left(AUBV\right)|:U,V\in S{O}_n\}={\sum }_{i=1}^{n-1}{a}_i{b}_i+signdet\left(AB\right)a_n{b}_n$ (Miranda and Thompson).

Parallel results have been sought to obtain analogous results, replacing the trace function by the determinant. The next theorem was independently obtained by different researchers (see, e.g., [13,27,28]).

Theorem 6.

Let $A,B$ be square complex matrices with singular values $a_1\ge \cdots \ge a_n\ge 0$ and $b_1\ge \cdots \ge b_n\ge 0.$ Then, the following holds:

$|det(A+B)|\le \prod _{i=1}^n(a_i+b_{n-i+1}).$

Further, $|det(A+B\left)\right|=0$ if and only if $a_1>b_n$ and $b_1\ge a_n$. Otherwise,

$|det(A+B)|\ge \prod _{i=1}^n(a_i-b_{n-i+1}).$

The inequalities obtained here were extended by Tam and Thompson to the context of real matrices and some other real simple Lie groups [29]. The same authors also generalized the Hermitian case that was obtained by Fiedler [3] to the Lie algebra of a compact connected subgroup of $U_n$ [30]. Also, the lower bound in the previous theorem was extended in [27], Theorem 5, to Schur concave functions.

The determinantal analogue to Miranda–Thompson trace inequality was obtained in [31], stating the following:

If $A,B$ are square real matrices with singular values $a_1\ge \cdots \ge a_n\ge 0$ and $b_1\ge \cdots \ge b_n\ge 0,$ then

$\underset{U,V\in S{O}_n}{max}det(A+UBV)\le \prod _{i=1}^n(a_i+b_{n-i+1}).$

The minimum ${min}_{U,V\in S{O}_n}det(A+UBV)$ was also determined, and it splits into four distinct cases that are ruled by the parity of n and inequalities that hold between $a_1,a_n,b_1,b_n.$

Moreover, the equality cases in the derived determinantal inequalities have been discussed, e.g., in [32]. In [33], a Fiedler-type theorem for the determinant of certain generalized Hermitian matrices was derived. The extention for new classes of matrices of properties of $\Delta (A,B)$, obtained for the case of normal matrices, is in order.

Author Contributions

Both authors have contributed substantially to the work reported. All authors have read and agreed to the published version of the manuscript.

Acknowledgments

The authors are grateful to the reviewers for careful reading of the manuscript and constructive comments.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

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