Simultaneous diagonalization via congruence of Hermitian matrices: some equivalent conditions and a numerical solution Hieu Le Thanh* and Ngan Nguyen Thi† Department of Mathematics and Statistics, Quy Nhon University, Vietnam November 17, 2020 Abstract This paper aims at solving the Hermitian SDC problem, i.e., that of si- multaneously diagonalizing via -congruence a collection of finitely many (not need pairwise commute) Hermitian∗ matrices. Theoretically, we provide some equivalent conditions for that such a matrix collection can be simulta- neously diagonalized via ∗-congruence.Interestingly, one of such conditions leads to the existence of a positive definite solution to a semidefinite program (SDP). From practical point of view, we propose an algorithm for numeri- cally solving such problem. The proposed algorithm is a combination of (1) a positive semidefinite program detecting whether the initial Hermitian matri- ces are simultaneously diagonalizable via -congruence, and (2) a Jacobi-like ∗ algorithm for simultaneously diagonalizing via -congruence the commuting ∗ normal matrices derived from the previous stage. Illustrating examples by hand/coding in MATLAB are also presented. Keywords: Hermitian matrices, matrix ∗-congruence, matrix T-congruence, matrix similarity, simultaneous diagonalization via congruence (SDC), simultane- arXiv:2007.14034v2 [math.NA] 15 Nov 2020 ous diagonalization via similarity (SDS), semidefinite programming (SDP), maxi- mum rank linear combination of matrices. 1 Introduction Notations and preliminaries. Let F denote the field of real numbers R or complex n n n ones C, and F × denote the set of all n n matrices with entries in F. Let S (resp., n × n n H ) denote the set of real symmetric (resp., Hermitian) matrices in F × . By .∗ we denote the conjugate transpose of a matrix. For A Hn, by A 0 (resp., A 0) we mean A is positive semidefinite (resp., positive∈ definite) as usual . ≻ *Email: [email protected] †Email: [email protected] 1 n n From now on, without stated otherwise, C1,...,Cm F × are always Her- C ∈ C m mitian matrices. By (λ) we denote the Hermitian pencil (λ) =: i=1 λiCi, i.e., the parameters λ1,...,λm ranges only over real numbers. Finally, Hermitian P matrices C1,...,Cm are said to be (i) simultaneously diagonalizable via similarity on F, abbreviated F-SDS, if there n n 1 exists a nonsingular matrix P F × such that P − CiP ’s are all diagonal n n ∈ matrices in F × . When m = 1, we will say ”C1 is similar to a diagonal matrix” or “C1 is diagonalizable (via similarity)” as usual. (ii) simultaneously diagonalizable via -congruence on F, abbreviated F-SDC, ∗ n n if there exists a nonsingular matrix P F × such that P ∗CiP is diagonal ∈ for every i = 1, . , m. Here P ∗ means the conjugate transpose of P. When m =1, we will say “C1 is congruent to a diagonal matrix” as usual. It is worth mentioning that every diagonal matrix P ∗CiP is always real since Ci is Hermitian. Moreover, we will see in Theorems 4 and 7 below that P can be chosen to be real if Ci’s are all real. Additionally, one should distinguish that -congruence in this paper is different from T-one in [4]. These two types∗ of congruence will coincide only when the Hermitian matrices are real. (iii) commuting if they pairwise commute, i.e., CiCj = CjCi for every i, j = 1, . , m. Depend upon particular situation in the rest of this paper, the term “SDC” will mean either “simultaneous diagnalization” or “simultaneously diagnalizing”, or “simultaneously diagnalizable” via -congruence. It is analogously to the term “SDS”. ∗ We now recall some well-know results that will be frequently used in the rest of the paper. Fn n F Lemma 1. Let A, B × be A = diag(α1In1 ,...,αkInk ), αi’s are distinct. ∈ ni ni ∋ If AB = BA then B = diag(B ,...,Bk) with Bi F × for every i =1, . , k. 1 ∈ Furthermore, B is Hermitian (resp., symmetric) if and only if so are Bi’s. Proof. Partitioning B as B = [Bij]i,j=1,...,k, where Bii is a square submatrix of size ni ni, i = 1,...,k and off-diagonal blocks are of appropriate sizes. It then follows× from α1B11 ... α1B1k α1B11 ... αkB1k . .. .. . = AB = BA = . αkBk1 ... αkBkk α1Bk1 ... αkBkk that αiBij = αjBij, i = j. Thus Bij =0 for every i = j. The last claim is trivial.∀ 6 6 Lemma 2. (See, eg., in [13]) (i) Every A Hn can be diagonalized via similarity ∈ by a unitary matrix. That is, it can be written as A = UΛU ∗, where U is unitary, Λ is real diagonal and is uniquely defined up to a permutation of diagonal elements. Moreover, if A Sn then U can be picked to be real. ∈ n n (ii) Let C1,...,Cm F × such that each of which is similar to a diagonal matrix. They are then F-SDS∈ if and only if they are commuting. 2 n n m m (iii) Let A F × , B F × . The matrix M = diag(A, B) is diagonalizable via similarity if∈ and only if∈ so are both A and B. (iv) A complex symmetric matrix A is diagonalizable via similarity if and only 1 if it is complex orthogonally diagonalizable, i.e., Q− AQ is diagonal for some n n T complex orthogonal matrix Q C × : Q Q = I. ∈ Proof. The first and last parts can be found in [13]. We prove (ii) and (iii) for real matrices, the complex setting was proved in [13]. n n (ii) Suppose C1,...,Cm R × are commuting. We prove by induction on n. For n =1, there are nothing∈ to prove. Suppose that n 2 and that the assertion has been proved for all collections of m commuting real≥ matrices of size k k, × 1 k n 1. ≤ ≤ − By the hypothesis, C1 is diagonalized via similarity by a nonsingular matrix n n P R × , ∈ 1 P − C P = diag(α In1 ,...,αkInk ), n + ... + nk = n, αi = αj, i = j. 1 1 1 6 ∀ 6 1 1 The commutativity of C1,Ci implies that of P − C1P and P − CiP. By Lemma 1, 1 nt nt P − CiP = diag(Ci ,...,Cik), Cit R × , t =1, . , k. 1 ∈ ∀ 1 1 Furthermore, P − CiP and P − CjP commute due to the commutativity of Ci,Cj. So, Cit and Cjt commute for every t = 1, . , t. This means the matrices C2t, ...,Cmt are commuting for every t = 1, . , k. By the induction hypothesis, for each t = 1, . , k, the matrices C2t,...,Cmt are R-SDS by a nonsingular matrix nt nt Qt R × , and so are Int ,C t,...,Cmt. Let ∈ 2 n n Q = diag(Q ,...,Qk) R × , 1 ∈ and set U = P Q. One can check C1,...,Cm are R-SDS by U. The converse is trivial. (n+m) (n+m) (iii) Suppose M is DS in R × . Then there exists a nonsingular matrix (n+m) (n+m) S R × such that ∈ 1 S− MS =Λ= diag(α ,...,αn,...αn m), αi R, i =1,...,m + n. 1 + ∈ Let sj be the j-th column of S with ξj n m sj = , ξj R , νj R . νj ∈ ∈ n (n+m) m (n+m) Let E = [ξ1 ...ξn+m] R × , N = [ν1 ...νn+m] R × . If rank(E) < n or rank(N) < m then∈ ∈ m + n = rank(S) rank(E)+ rank(N) < m + n, ≤ which is impossible. Thus rank(E) = n and rank(N) = m. Moreover, it follows from MS = SΛ that the jth column ξj of E is the eigenvector of A w.r.t the eigenvalue λj. This means A has n linearly independent eigenvectors. By the same reasoning, B also has m linearly independent eigenvectors. Thus A and B are n n m m diagonalizable via similarity in R × and R × , respectively. 3 History of SDC problems. Matrix simultaneous diagonalizations, via similarity or congruence, appear in a number of research areas, for examples, quadratic equa- tions and optimization [12, 15], multi-linear algebra [6], signal processing, data analysis [6], quantum mechanics [22], ... The SDC problem, i.e., the one of finding a nonsingular matrix that simulta- neously diagonalizes a collection of matrices via congruence, can be dated back to 1868 by Weierstrass [25], 1930s by Finsler [8, 1, 11], and later studies devel- oped some conditions ensuring a collection of quadratic forms are SDC (see, eg., in [18, 20] and references therein). However, almost these works provide only sufficient (but not necessary) conditions, except for, eg., that in [23], [13, 14] and references therein dealing with two Hermitian/symmetric matrices; or that in [15] focusing on real symmetric matrices which have a positive definite linear combi- nation. Some other works focus on simultaneous block-diagonalizations of normal matrices, so does of Hermitian ones [13, 24]. An (equivalently) algorithmic con- dition for the real symmetric SDC problem is given in a preprint manuscript [19]. Very recently in the seminal paper [4], the authors develop an equivalent condition solving the complex symmetric SDC problem. Note that this admits T-congruence, since the matrices there are complex symmetric, instead of -one as in this paper. ∗ The simultaneous diagonalization via T-congruence for complex symmetric ma- trices does not guarantee for real setting. That is, the resulting nonsingular and diagonal matrices may not be real even thought the initial ones are real. An exam- ple given in [4] is that 0 1 1 1 C = , C = S2 1 1 1 2 1 0 ∈ are SDC via T-congruence. However, the resulting nonsingular matrix P and the T T corresponding diagonal ones P C1P,P C2P are properly complex. We will see in this paper that the simultaneous diagonalizableness via -congruence of Hermitian matrices will immediately holds true for real symmetric setting.∗ Turn- ing to the two matrices above, we will see from Theorem 4 below that they are not 1 0 1 R-SDC because C− C = is not similar to a real diagonal matrix (it has 1 2 1− 1 1 i√3 only complex eigenvalues ±2 ).