| Izvestiya: Mathematics |
|
|
|
|
|
A further sufficient condition for the determinantal conjecture Yaroslav N. Shitov
Bibliographic databases:
   § 1. IntroductionThe determinantal conjecture of Marcus [17] and de Oliveira [10] is a well-known problem in theory of matrices. This conjecture states that, for any pair $(A,\,B)$ of $n\times n$ normal matrices over $\mathbb C$ and with the spectra $\sigma(A)=(a_1,\dots,a_n)$ and $\sigma(B)=(b_1,\dots,b_n)$,
$$
\begin{equation}
\det(A+B)\ \text{ lies in the convex hull of } \ \bigcup_{\psi\in\mathcal{S}_n} \biggl\{\prod_{i=1}^n(a_i+b_{\psi_i})\biggr\},
\end{equation}
\tag{1.1}
$$
where $\mathcal{S}_n$ is the family of all permutations $\psi\colon i\to\psi_i$ of the set $\{1,\dots, n\}$. There has been a line of publications [1]–[9], [11], [14]–[16], [18], [19], [22] showing different conditions on $\sigma(A)$, $\sigma(B)$ to be sufficient for the determinantal conjecture. According to the survey [13], the best known results towards this direction include the validity of the conjecture in the following cases:
The latest of these results dates back to 2007, so the progress in this direction seems to have stuck for almost fifteen years [13], [21]. In this paper, we break this barrier and significantly improve one of the above mentioned conditions. Theorem 1. Let $A$ and $B$ be $n\times n$ normal matrices with eigenvalues $(a_1,\dots,a_n)$ and $(b_1,\dots,b_n)$, respectively. If $a_1,\dots,a_n\in\mathbb{R}$ and, additionally, $b_j\in\mathbb{R}$ for at least $n-3$ indices $j\in\{1,\dots,n\}$, then condition (1.1) is satisfied. This improves the result of Kovačec [15] of 1999, who confirmed the conjecture in case when, instead of three eigenvalues of $B$ as in the current paper, just one such eigenvalue is not required to be real, and an older result of Fiedler [11] of 1970, who proved (1.1) for normal matrices $A$, $B$ with all real eigenvalues. Our approach is algebraic, which is in contrast to many papers that use non-trivial tools in analysis [2], [3], [8], [9]. A particular idea that we believe is novel for the study of the determinantal conjecture is the use of Cauchy’s interlace theorem for minors of a Hermitian matrix, see [20] for a detailed discussion and history and [12] for a recent short proof. We are ready to proceed with the proof, which appears as a combination of Cauchy’s interlace theorem with another, already quite standard, fact that the Möbius transformations preserve the property of matrices to satisfy or not satisfy the assertion of the conjecture [2], [9], [13], [16]. In the forthcoming § 2, we specify our notation and give formulations of several previously known statements. In § 3, we present the core of our proof. § 2. PreliminariesWe begin with some standard notation. Definition 2. If $A$ is an $n\times n$ matrix with complex entries, we define $\sigma(A)$ as the multiset $(a_1,\dots,a_n)$ consistsing of the eigenvalues of $A$. Of course, we declare that an eigenvalue of multiplicity $k$ appears $k$ times in $\sigma(A)$. Definition 3. For two multisets $a=(a_1,\dots, a_n)$ and $b=(b_1,\dots,b_n)$ with complex elements, we define $\Delta(a,b)$ as the convex hull of where $\mathcal{S}_n$ is the family of all permutations $\psi\colon i\to\psi_i$ of the set $\{1,\dots, n\}$. Remark 4. Since the definition of $\Delta(a,b)$ corresponds to the convex hull in (1.1), the determinantal conjecture can be stated simply as for all normal matrices $A$, $B$. As explained in the introduction, our proof covers the case when $A$ is Hermitian, and $B$ has at most three non-real eigenvalues. Since we can find a unitary matrix $U$ such that $UBU^*$ is diagonal, and since the transformation $X\to UXU^*$ preserves the determinant, we can assume without loss of generality that $B$ is diagonal. Definition 5. For all integers $m, k$ satisfying $m\geqslant k\geqslant 3$, we define $\mathcal{A}(m,k)$ as the set of all pairs $(A,B)$ consisting of $m\times m$ matrices in which $A$ is Hermitian, $B$ is diagonal, and $B$ has real numbers at the positions $(1,1),\dots,(m-k,m-k)$ and arbitrary complex numbers at all other diagonal positions. It is easy to see that property (2.1) is preserved by taking limits, so one can restrict to generic matrices when proving the determinantal conjecture. Therefore, we need to adopt some language to deal with generic matrices. Definition 6. Let $\mathcal{F}$ be a subfield of $\mathbb{R}$, and let $u$ be a family of real numbers. We write $\mathcal{F}(u)$ to denote the smallest subfield of $\mathbb{R}$ containing both $\mathcal{F}$ and $u$. Definition 7. If $M_1,\dots,M_k$ are complex matrices, we define $\mathcal{F}(M_1,\dots,M_k):=\mathcal{F}(\mu)$, where $\mu$ is the set consisting of all the real and imaginary parts of any entry of any matrix in $(M_1,\dots,M_k)$. We write to denote the transcendence degree of $\mathcal{F}(M_1,\dots,M_k)$ over $\mathcal{F}$. If $\mathcal{F}=\mathbb{Q}$, then we simply write $\operatorname{trdeg}(M_1,\dots,M_k)$ instead of (2.2). Remark 9. On the right-hand side of (2.3), the matrix $A$ gives the $m^2$ summand, and $B$ gives the remaining $m+k$ degrees, of which $m-k$ correspond to the real diagonal entries, and the remaining $2k$ come from the other $k$ diagonal entries. We proceed with several well-known results. Theorem 11 (Cauchy’s interlace theorem [12], [20]). Let $A$ be an $n\times n$ Hermitian matrix, and let $A'$ be the matrix obtained from $A$ by removing its first row and first column. If $\sigma(A)=(a_1,\dots,a_n)$ and $\sigma(A')=(\alpha_1,\dots,\alpha_{n-1})$ with then $\alpha_i\in[a_i,a_{i+1}]$ for all $i\in\{1,\dots,n-1\}$. Theorem 12 (see [2], [13], [16]). Let $A$, $B$ be $n\times n$ normal matrices, and let $c\in\mathbb{C}$ be eigenvalue of neither $A$ nor $-B$. Then the matrices satisfy $\det(A+B)\in\Delta(\sigma(A),\sigma(B))$ if and only if $\det(A'+B')\in\Delta(\sigma(A'),\sigma(B'))$. § 3. The proofWe begin the core of our argument with three simple observations. Observation 13 (see [8], [13]). If $(A_t)$, $(B_t)$ are two sequences of $n\times n$ complex matrices that satisfy $A_t\to A$, $B_t\to B$ as $t\to+\infty$, and the condition holds for all $t$, then $\det(A+B)\in\Delta(\sigma(A),\sigma(B))$. This result is a consequence of the continuity of $\det$ and $\Delta$. Observation 14. If $X$ is a non-singular matrix with complex entries, then for any subfield $\mathcal{F}\subset\mathbb{R}$. Proof. We have $ \mathcal{F}(X^{-1})\subseteq\mathcal{F}(X)$, as the real and imaginary parts in $X^{-1}$ belong to $\mathcal{F}(X)$, and the opposite inclusion follows because $(X^{-1})^{-1}=X$ Lemma 15. Let $(A, B)\in\mathcal{A}(m,k)$ be generic. If $\det(A+B)\in\Delta(\sigma(A),\sigma(B))$, then $\det(A+B)$ is an interior point of $\Delta(\sigma(A),\sigma(B))$. Proof. Assume that $a_1,\dots,a_m$ and $b_1,\dots,b_m$ are the eigenvalues of $A$ and $B$, respectively. If the lemma is false, then $\det(A+B)$ lies on an edge of the polygon $\Delta(\sigma(A),\sigma(B))$, and, for some permutation $\pi\colon \{1,\dots,m\}\to\{1,\dots,m\}$, the condition
$$
\begin{equation}
\frac{\det(U\operatorname{diag}(a_1,\dots, a_m)U^*+B)-(a_1+b_{\pi_1})\cdots(a_m+b_{\pi _ m})}{(a_1+b_1)\cdots(a_m+b_m)-(a_1+b_{\pi_1})\cdots(a_m+b_{\pi _m})}\in\mathbb{R}
\end{equation}
\tag{3.2}
$$
holds for generic unitary matrix $U$, which implies (3.2) for all unitary matrices in an open neighbourhood of $U$ and leads to a contradiction. The following is some notation specific for our argument. Definition 16. Let $a=(a_1,\dots,a_n)$, $\beta=(\beta_1,\dots,\beta_{n-1})$ be two multisets of $n$ and $n-1$ complex numbers, respectively. We define $\overline{\Delta}(a,\beta)$ as the convex hull of where $\psi\colon i\to\psi_i$ runs over all injective mappings $\{1,\dots,n-1\}\to\{1,\dots,n\}$. Lemma 17. For $n\geqslant 2$, consider three multisets where $\beta$ consists of complex numbers, and $a$, $\alpha$ are real and satisfy Then $\Delta(\alpha,\beta)\subseteq\overline{\Delta}(a,\beta)$. Proof. If $n=2$, then $\overline{\Delta}(a,\beta)$ is the segment between $a_1+\beta_1$ and $a_2+\beta_1$, and $\Delta(\alpha,\beta)$ is the single point $\alpha_1+\beta_1$, and the desired statement that
$$
\begin{equation*}
\alpha_1+\beta_1\quad\text{belongs to the convex hull of } \ \{a_1+\beta_1,a_2+\beta_1\}
\end{equation*}
\notag
$$
follows by the interlace inequalities. This covers the case $n=2$, and so we can assume $n>2$ and proceed by induction. In order to prove the desired statement, we need to check that all vertices of $\Delta(\alpha,\beta)$ belong to $\overline{\Delta}(a,\beta)$, and hence, after an appropriate relabeling of $\beta$, it is sufficient to check that
$$
\begin{equation}
(\alpha_1+\beta_1)\cdots(\alpha_{n-2}+\beta_{n-2})\cdot(\alpha_{n-1}+\beta_{n-1})\in\overline{\Delta}(a,\beta).
\end{equation}
\tag{3.3}
$$
In view of the inductive assumption, the point $ (\alpha_1+\beta_1)\cdots(\alpha_{n-2}+\beta_{n-2}) $ belongs to both $\overline{\Delta}(a', \beta')$ and $\overline{\Delta}(a'', \beta')$, where $a'$ and $a''$ are obtained from $a$ by removing the $a_{n-1}$ and $a_n$ terms, respectively, and, similarly, $\beta'=(\beta_1,\dots,\beta_{n-2})$. Therefore, the points $ (\alpha_1+\beta_1)\cdots(\alpha_{n-2}+\beta_{n-2})\cdot(a_{n-1}+\beta_{n-1}) $ and $ (\alpha_1+\beta_1)\cdots(\alpha_{n-2}+\beta_{n-2})\cdot(a_n+\beta_{n-1}) $ belong to $\overline{\Delta}(a,\beta)$, and hence the inequalities $a_{n-1}\leqslant\alpha_{n-1}\leqslant a_n$ imply (3.3). The next result is immediate from Theorem 11 and Lemma 17. Corollary 18. Let $A$ be a Hermitian $n\times n$ matrix, let $A'$ be its principal $(n-1)\times(n-1)$ submatrix, and let $\beta$ be a multiset of $n-1$ complex numbers. Then We are now ready to proceed with the main technical contribution of this paper. Theorem 19. Let $m$ and $k$ be integers such that $m\geqslant k\geqslant 3$. If condition (2.1) holds for every pair $(A,B)\in\mathcal{A}(m,k)$, then condition (2.1) holds for every pair $(A,B)\in\mathcal{A}(n,k)$ as well, for all integers $n\geqslant m$. Proof. We work by induction on $n$ starting from $n=m$, which corresponds to the assumption of the theorem. In the rest of this proof, we assume that $n>m$, and our aim is to complete the induction step. According to Observation 13, we can assume that $(A,B)$ is a generic pair in $\mathcal{A}(n,k)$. We recall that $A$ is a Hermitian matrix, and $B$ is a diagonal matrix with real numbers $b_1,\dots,b_{n-k}$ at the corresponding positions $(1,1), \dots, (n-k,n-k)$, and with complex numbers $b_{n-k+1},\dots,b_n$ at the remaining diagonal entries.
Now we use the Möbius transformations and define
$$
\begin{equation*}
\overline{A_\varepsilon}=(A+(b_1-\varepsilon)I)^{-1}\quad \text{and} \quad \overline{B_\varepsilon}=(B-(b_1-\varepsilon)I)^{-1}
\end{equation*}
\notag
$$
for all appropriate $\varepsilon$. By Theorem 12, the desired assertion follows if we prove
$$
\begin{equation}
\det(\overline{A_\varepsilon}+\overline{B_\varepsilon})\in\Delta \bigl(\sigma(\overline{A_\varepsilon}),\sigma(\overline{B_\varepsilon})\bigr)
\end{equation}
\tag{3.4}
$$
with a sufficiently small real $\varepsilon>0$. Towards this end, we take which gives
$$
\begin{equation}
\overline{A_\varepsilon}=\overline{A}+\delta'(\varepsilon),\qquad \overline{B_\varepsilon}=\operatorname{diag}\biggl(\frac1{\varepsilon},\frac1{b_2-b_1}, \dots, \frac1{b_n-b_1} \biggr) +\delta''(\varepsilon),
\end{equation}
\tag{3.5}
$$
where $\delta'/\varepsilon$ and $\delta''/\varepsilon$ remain bounded as $\varepsilon\to 0$. We take $\overline{a}:=\sigma(\overline{A})$ and
$$
\begin{equation*}
\beta_j:=\frac{1}{b_j-b_1}\quad\text{for all } \ j\in\{2,\dots,n\},
\end{equation*}
\notag
$$
and also
$$
\begin{equation}
\overline{b}=\biggl(\frac1{\varepsilon},\beta_2,\dots,\beta_n\biggr),\qquad \beta=(\beta_2,\dots,\beta_n).
\end{equation}
\tag{3.6}
$$
In view of formula (3.5), the vertices of
$$
\begin{equation}
\Delta\bigl(\sigma(\overline{A_\varepsilon}),\sigma(\overline{B_\varepsilon})\bigr)
\end{equation}
\tag{3.7}
$$
differ from the corresponding points in $\Delta(\overline{a},\overline{b})$ by quantities that remain bounded as $\varepsilon\to0$, and hence, since the coordinates in $\overline{a}$ and $\beta$ in (3.6) do not depend on $\varepsilon$, every vertex of $\overline{\Delta}(\overline{a},\beta)/\varepsilon$ lies at distance at most $C$ of a vertex of polygon (3.7), where $C$ is some positive real number independent of $\varepsilon$. Therefore,
$$
\begin{equation}
\text{the $C$-contraction of }\frac{1}{\varepsilon}\cdot\overline{\Delta}(\overline{a},\beta)\text{ is a subset of } \Delta\bigl(\sigma(\overline{A_\varepsilon}),\sigma(\overline{B_\varepsilon})\bigr)
\end{equation}
\tag{3.8}
$$
for all sufficiently small $\varepsilon>0$, where the $C$-contraction of a polygon $S\subset\mathbb{C}$ is the set of all points in $S$ that lie at least at the distance $C$ from the boundary of $S$.
As we can see, condition (3.8) describes the right-hand side of the desired formula (3.4). Concerning the left-hand side, equation (3.5) implies
$$
\begin{equation}
\biggl|\det(\overline{A_\varepsilon}+\overline{B_\varepsilon}) -\frac{1}{\varepsilon}\cdot\det(\widetilde{A}+\widetilde{B})\biggr|<C'
\end{equation}
\tag{3.9}
$$
for all $\varepsilon>0$, where $\widetilde{A}$, $\widetilde{B}$ are the matrices obtained from $\overline{A}$, $\overline{B}$ by removing their corresponding first rows and first columns, and $C'$ is some positive real number independent of the choice of $\varepsilon$. Using the inductive assumption, we get that
$$
\begin{equation}
\det(\widetilde{A}+\widetilde{B})\in\Delta\bigl(\sigma(\widetilde{A}),\sigma(\widetilde{B})\bigr).
\end{equation}
\tag{3.10}
$$
Since the choice of $(A,B)$ is generic, the field $\mathbb{Q}(A+b_1I)$ has full transcendence degree, equal to $n^2$, over $\mathbb{Q}(b_2,\dots,b_n)$. This implies, according to Observation 14, that $\mathbb{Q}(\overline{A})$ has full transcendence degree over $\mathbb{Q}(b_2,\dots,b_n)$ as well, and hence the pair $(\widetilde{A}, \widetilde{B})$ is generic. This allows us to apply Lemma 15 on condition (3.10), and, finally, we conclude that $\det(\widetilde{A}+\widetilde{B})$ is an interior point of $\Delta(\sigma(\widetilde{A}),\sigma(\widetilde{B}))$.
Further, we use Corollary 18 to see that $\Delta(\sigma(\widetilde{A}),\beta)\subseteq\overline{\Delta}(\overline{a},\beta)$, and hence $\det(\widetilde{A}+ \widetilde{B})$ is an interior point of $\overline{\Delta}(\overline{a},\beta)$ as well. Since neither $\det(\widetilde{A}+\widetilde{B})$ nor $\overline{\Delta}(\overline{a},\beta)$ depend on $\varepsilon$, there exists some positive real number $d>0$ independent of $\varepsilon$ such that $\det(\widetilde{A}+\widetilde{B})$ belongs to the $d$-contraction of $\overline{\Delta}(\overline{a},\beta)$, and hence
$$
\begin{equation}
\frac{1}{\varepsilon}\cdot\det(\widetilde{A}+\widetilde{B})\ \text{ belongs to the } \frac{d}{\varepsilon}\text{-contraction of } \ \frac{1}{\varepsilon}\cdot\overline{\Delta}(\overline{a},\beta).
\end{equation}
\tag{3.11}
$$
A comparison of (3.9) and (3.11) allows us to choose $\varepsilon$ so that
$$
\begin{equation}
\det(\overline{A_\varepsilon}+\overline{B_\varepsilon})\ \text{ belongs to the $C$-contraction of } \ \frac{1}{\varepsilon}\cdot\overline{\Delta}(\overline{a},\beta)
\end{equation}
\tag{3.12}
$$
for arbitrary $C$ fixed in advance, and now conditions (3.8) and (3.12) show (3.4). Theorem 19 is proved. We know from Theorem 10 that condition (2.1) holds in $\mathcal{A}(3,3)$. Therefore, as a corollary of Theorem 19, we get the result announced in the abstract. Proof of Theorem 1. It is well known that $B$ can be taken diagonal without loss of generality. In fact, we can find a unitary matrix $U$ such that $UBU^*$ is diagonal, and then we can restrict to this case because the transformation $X\to UXU^*$ preserves the determinant. Therefore, it suffices to prove (2.1) for all pairs $(A,B)\in\mathcal{A}(n,3)$, which follows by an immediate application of Theorems 10 and 19.
Citation in format AMSBIB
\Bibitem{Shi25} |
|
|
Contact us: |
Terms of Use
|
Registration to the website |
Logotypes |
|