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iccm2007 vol ii 1 4 open problems in matrix theory xingzhi zhan abstract we survey some open problems in matrix theory by briey describing their history and current state 2000 ...

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                  ICCM2007 · Vol. II · 1–4
                      Open problems in matrix theory
                                             Xingzhi Zhan∗
                                                Abstract
                        We survey some open problems in matrix theory by briefly describing
                        their history and current state.
                        2000 Mathematics Subject Classification: 15A15, 15A18, 15A60,
                        15A29, 15-02, 05B20, 05C07.
                        Keywords and Phrases: Matrix, open problem, survey
                       Sometimessolutions to challenging matrix problems can reveal con-
                  nections between different parts of mathematics. Two examples of this
                  phenomenon are the proof of the van der Waerden conjecture on per-
                  manents (see [47] or [69]) and the recent proof of Horn’s conjecture on
                  eigenvalues of sums of Hermitian matrices (see [11] and [32]). Difficult
                  matrix problems can also expose limits to the strength of existing math-
                  ematical tools.
                       We will describe the history and current state of some open prob-
                  lems in matrix theory, which we arrange chronologically in the following
                  sections.
                  1. Existence of Hadamard matrices
                       A Hadamard matrix is a square matrix with entries equal to ±1
                  whose rows and hence columns are mutually orthogonal. In other words,
                  a Hadamard matrix of order n is a {1,−1}-matrix A satisfying
                                                AAT =nI
                  where I is the identity matrix. In 1867 Sylvester proposed a recur-
                                                                              k
                  rent method for construction of Hadamard matrices of order 2 . In 1893
                     ∗Department of Mathematics, East China Normal University, Shanghai 200241,
                  China. e-mail: zhan@math.ecnu.edu.cn. The author’s research was supported by the
                  NSFC grant 10571060
              2                       X. Zhan
              Hadamard proved his famous determinantal inequality for a positive
              semidefinite matrix A :
                                    detA ≤ h(A)
              where h(A) is the product of the diagonal entries of A. It follows from
              this inequality that if A = (a ) is a real matrix of order n with |a | ≤ 1
                                   ij                         ij
              then
                                   | detA| ≤ nn/2;
              equality occurs if and only if A is a Hadamard matrix. This result
              gives rise to the term “Hadamard matrix”. In 1898 Scarpis proved that
              if p ≡ 3(mod4) or p ≡ 1(mod4) is a prime number then there is a
              Hadamard matrix of order p+1 and p+3 respectively.
                  In 1933 Paley stated that the order n (n ≥ 4) of any Hadamard
              matrix is divisible by 4. This is easy to prove. The converse has been a
              long-standing conjecture.
                  Conjecture1Foreverypositiveintegern, there exists a Hadamard
              matrix of order 4n.
                                               k       2   k
                  Conjecture 1 has been proved for 4n = 2 m with m ≤ 2 . Accord-
              ing to [68], the smallest unknown case is now 4n = 668. See [34, 57, 58,
              63, 64].
                  Hadamard matrices have applications in information theory and
              combinatorial designs. See [1].
                  Let k ≤ n be positive integers. A square matrix A of order n with
              entries in {0,−1,1} is called a weighted matrix with weight k if
                                    AAT =kI.
              GeramitaandWallisposedthefollowingmoregeneralconjecture in 1976
              [33].
                  Conjecture 2 If k ≤ n are positive integers with n ≡ 0(mod4),
              then there exists a weighted matrix of order n with weight k.
                  NotethatConjecture 1 corresponds to the case k = n of Conjecture
              2.
              2. Characterizationoftheeigenvaluesofnon-
                 negative matrices
                  In 1937 Kolmogorov asked the question: When is a given complex
              number an eigenvalue of some (entrywise) nonnegative matrix? The
              answer is: Every complex number is an eigenvalue of some nonnegative
              matrix [52, p.166]. Suleimanova [62] extended Kolmogorov’s question in
              1949 to the following problem which is called the nonnegative inverse
              eigenvalue problem.
                               Open problems in matrix theory            3
                    Problem3Determinenecessary and sufficient conditions for a set
                of n complex numbers to be the eigenvalues of a nonnegative matrix of
                order n.
                    Problem 3 is open for n ≥ 4. The case n = 2 is easy while the case
                n=3isdue to Loewy and London [48].
                    In the same paper [62] Suleimanova also considered the following
                real nonnegative inverse eigenvalue problem and gave a sufficient condi-
                tion.
                    Problem4Determinenecessary and sufficient conditions for a set
                of n real numbers to be the eigenvalues of a nonnegative matrix of order
                n.
                    Problem4isopenforn ≥ 5.In1974Fiedler[29]posedthefollowing
                symmetric nonnegative inverse eigenvalue problem.
                    Problem 5 Determine necessary and sufficient conditions for a
                set of n real numbers to be the eigenvalues of a symmetric nonnegative
                matrix of order n.
                    Problem 5 is open for n ≥ 5. There are some necessary conditions
                and many sufficient conditions for these three problems. See the survey
                paper [27] and the book [52, Chapter VII].
                3. The permanental dominance conjecture
                    Let S denote the symmetric group on {1,2,...,n} and M denote
                        n                                          n
                the set of complex matrices of order n. Suppose G is a subgroup of Sn
                and χ is a character of G. The generalized matrix function dχ : Mn → C
                is defined by
                                                 n
                                  dχ(A) = Xχ(σ)Ya      ,
                                                    iσ(i)
                                         σ∈G    i=1
                where A = (a ). Incidental to his work on group representation theory,
                          ij
                Schur introduced this notion. For G = Sn, if χ is the alternating charac-
                ter then dχ is the determinant while if χ is the principal character then
                dχ is the permanent
                                               n
                                    perA = X Ya      .
                                                  iσ(i)
                                          σ∈Sni=1
                When χ is the principal character of G = {e} where e is the identity
                permutation in Sn, dχ is Hadamard’s function h(A).
                    In 1907 Fischer proved that if the matrix
                                          µA B¶
                                      A=     1
                                            B∗ A2
                4                           X. Zhan
                is positive semidefinite with A and A square, then
                                          1      2
                                     detA ≤ (detA )(detA ).
                                                1      2
                Hadamard’s inequality follows from this inequality immediately. In 1918
                Schur obtained the following generalization of Fischer’s inequality:
                                       χ(e)detA ≤ dχ(A)
                for positive semidefinite A. Let G be a subgroup of S and let χ be an
                                                              n
                irreducible character of G. The normalized generalized matrix function
                is defined as
                                      ¯
                                      dχ(A) = dχ(A)/χ(e).
                Since any character of G is a sum of irreducible characters, Schur’s in-
                equality is equivalent to
                                                ¯
                                         detA ≤ dχ(A)
                for positive semidefinite A. In 1963, M. Marcus proved the permanental
                analog of Hadamard’s inequality
                                         perA ≥ h(A)
                and E.H. Lieb proved the permanental analog of Fischer’s inequality
                                     perA ≥ (perA1)(perA2)
                three years later, where A is positive semidefinite. These results natu-
                rally led to the following conjecture which was first published by Lieb
                [45] in 1966:
                     Conjecture 6 (The permanental dominance conjecture) Suppose
                Gis a subgroup of Sn and χ is an irreducible character of G. Then for
                any positive semidefinite matrix A of order n,
                                                ¯
                                         perA ≥ dχ(A).
                     A lot of work has been done on this conjecture. It has been con-
                firmed for every irreducible character of Sn with n ≤ 13. The reader is
                referred to [22 section 3] and the references therein for more details and
                recent progress.
                     Weorder the elements of Sn lexicographically to obtain a sequence
                L . For A = (a ) ∈ M the Schur power of A, denoted by Π(A), is
                  n           ij      n
                the matrix of order n! whose rows and columns are indexed by L and
                                   Q                                     n
                whose(σ,τ)−entryis   n  a      . Since Π(A) is a principal submatrix
                                     i=1 σ(i),τ(i)
                    n
                of ⊗ A, if A is positive semidefinite then so is Π(A). It is not difficult
                to see that both perA and detA are eigenvalues of Π(A). A result of
                Schur asserts that if A is positive semidefinite then detA is the smallest
                eigenvalue of Π(A). In 1966, Soules [61] posed the following
                     Conjecture 7 (The “permanent on top” conjecture) If the matrix
                Ais positive semidefinite, then perA is the largest eigenvalue of Π(A).
                     Conjecture 7, if true, implies Conjecture 6.
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...Iccm vol ii open problems in matrix theory xingzhi zhan abstract we survey some by briey describing their history and current state mathematics subject classication a b c keywords phrases problem sometimessolutions to challenging can reveal con nections between dierent parts of two examples this phenomenon are the proof van der waerden conjecture on per manents see or recent horn s eigenvalues sums hermitian matrices dicult also expose limits strength existing math ematical tools will describe prob lems which arrange chronologically following sections existence hadamard is square with entries equal whose rows hence columns mutually orthogonal other words order n satisfying aat ni where i identity sylvester proposed recur k rent method for construction department east china normal university shanghai e mail ecnu edu cn author research was supported nsfc grant x proved his famous determinantal inequality positive semidenite deta h product diagonal it follows from that if real ij then nn ...

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