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We give a unified proof of the interlacing properties of eigenvalues of principle submatrices of totally positive matrices.

This article considers two inverse eigenvalue problems for Lefkovitch and doubly Lefkovitch matrices. The problems consist of constructing these matrices from the maximal eigenvalues of all its leading principal submatrices, eigenvectors associated with some of these eigenvalues, and some diagonal entries. We give sufficient conditions for the existence of such matrices. In particular, when the matrices have constant diagonal entries. In addition, we provide numerical examples to show the results obtained.

Let A ∈ C n × n be normal with eigenvalues λ 1 , … , λ n . For a given z in the numerical range of A, consider the set B A , k ( z ) of k × k matrices W for which [ z ⁎ 0 W ] is a compression of A. Assuming that no three eigenvalues lie on the same line, we prove that for any compact subset K of the numerical range of A that avoids the eigenvalues, the mappings z ↦ B A , k ( z ) and z ↦ σ ( B A , k ( z ) ) are uniformly continuous on K.

In this paper, we study an inverse eigenvalue problem, referred to as the minimax inverse eigenvalue problem, of constructing matrices whose graph is a special type of tree called a generalized star of depth 2. A scheme of labelling the vertices of such a tree is introduced in order to express the corresponding matrices in a number of special forms. We then solve the minimax inverse eigenvalue problem of constructing such matrices from given eigen data consisting of the minimal and maximal eigenvalues of each of the leading principal submatrices of the required matrices. We pose an open problem of solving the minimax inverse eigenvalue problem for matrices described by arbitrary trees.

Let a norm on the set M n of real or complex n-by-n matrices be given. We investigate the question of finding the largest constants α n and β n such that for each A ∈ M n the average of the norms of its (n−1)-by-(n−1) principal submatrices is at least α n times the norm of A, and such that the maximum of the norms of those principal submatrices is at least β n times the norm of A. For a variety of classical norms including induced l p -norms, weakly unitarily invariant norms, and entrywise norms we give lower and upper bounds for α n and β n . In several cases α n and β n are explicitly determined.

We investigate the spectrum of the non-backtracking matrix of a graph. In particular, we show how to obtain eigenvectors of the non-backtracking matrix in terms of eigenvectors of a smaller matrix. In doing so, we create a block diagonal decomposition of the non-backtracking matrix, more clearly expressing its eigenvalues. Furthermore, we find an expression for the eigenvalues of the non-backtracking matrix in terms of eigenvalues of the adjacency matrix, and use this to upper-bound the spectral radius of the non-backtracking matrix, and to give a lower bound on the spectrum. Specifically, an upper-bound on the spectral radius is found in terms of the number of nodes and edges of the graph. We also investigate properties of a graph that can be determined by the spectrum. Specifically, we prove that the number of components, the number of degree 1 vertices, and whether or not the graph is bipartite are all determined by the spectrum of the non-backtracking matrix.

We show that a matrix A satisfying ‖ exp ⁡ ( t A ) ‖ ⩽ K for all t ⩾ 0 can be transformed to a block-diagonal form such that the condition numbers of all the diagonal blocks and the condition number of the transformation matrix depend only on K and the matrix size. This result is useful for the analysis of long-time simulation accuracy of numerical schemes.

It is shown that, if for some m, 2 ≤ m n all principal submatrices of an n-by-n P-matrix A are monotone, then A is monotone, generalizing a known result that also assumed symmetry. It is also shown that the P-matrix assumption cannot be entirely eliminated. However, in addition, A is doubly monotone ( n = m − 1 above, without the P-matrix assumption) if and only if A − 1 is invertible path product. It was known that inverse M-matrices are path product (they are more than doubly monotone). But (invertible) path product matrices are more general, and this is the first full and functional characterization of them. Several examples are given to illustrate the results and their generality.

In this paper, we deal with Leslie and doubly Leslie matrices of order n. In particular, with the companion and doubly companion matrices. We study three inverse eigenvalues problems which consist of constructing these matrices from the maximal eigenvalues of its all leading principal submatrices. For Leslie and doubly companion matrices, an eigenvector associated with the maximal eigenvalue of the matrix is additionally considered, and for the doubly Leslie matrix also an eigenvector associated with the maximal eigenvalue of leading principal submatrix of order n − 1 is required. We give necessary and sufficient conditions for the existence of a Leslie matrix and a companion matrix, and sufficient conditions for the existence of a doubly Leslie matrix and a doubly companion matrix. Our results are constructive and generate an algorithmic procedure to construct these special kinds of matrices.

A special two-by-two block matrix form arises in many important applications. Extending earlier results it is shown that parameter modified versions of a very efficient preconditioner does not improve its rate of convergence. This holds also for iterative refinement methods corresponding to a few fixed steps of the Chebyshev accelerated method. The parameter version can improve the defect-correction method but the convergence of this method is slower than an iterative refinement method with an optimal parameter. The paper includes also a discussion of how one can save computer elapsed times by avoiding use of global inner products such as by use of a Chebyshev accelerated method instead of a Krylov subspace method. Since accurate and even sharp eigenvalue bounds are available, the Chebyshev iteration method converges as fast as the Krylov subspace method.

Given an n by n matrix A over the complex numbers and an invariant subspace L, this paper gives a straightforward formula to construct an n by n matrix N that commutes with A and has L equal to the kernel of N. For Q a matrix putting A into Jordan canonical form J = RAQ with R the inverse of Q, we get N = RM$ where the kernel of M is an invariant subspace for J with M commuting with J. In the formula M = P ZVW with V the inverse of a constructed matrix T and W the transpose of P, the matrices Z and T are m by m and P is an n by m row selection matrix. If L is a marked subspace, m = n and Z is an n by n block diagonal matrix, and if L is not a marked subspace, then m > n and Z is an m by m near-diagonal block matrix. Strikingly, each block of Z is a monomial of a finite-dimensional backward shift. Each possible form of Z is easily arranged in a lattice structure isomorphic to and thereby displaying the complete invariant subspace lattice L(A) for A., 12 pages with two illustrations of invariant subspace lattice diagrams

Let M be a partitioned matrix and B be its equitable quotient matrix. In this paper, we prove the inclusion of their spectra σ ( B ) ⊆ σ ( M ) , and the equality of their spectral radii ρ ( B ) = ρ ( M ) if M is nonnegative. Moreover, σ ( M ) is shown to be completely determined by σ ( B ) if M is from a class of special matrices. Then we apply these results to obtain the spectra of some well-known matrices associated with graphs or digraphs, and then determine the extremal values of different spectral radii associated with connected graphs or strongly connected digraphs with given vertex connectivity.

We consider an m × m block matrix G with entries A i # A j where A 1 , … , A m are positive definite matrices of fixed size and A#B is the geometric mean of positive definite matrix A and B. We show that G is positive semidefinite if and only if the family of A 1 , … , A m is Γ-commuting; it can be transformed to a commuting family of positive definite matrices by a congruence transformation. This result via Γ-commuting families provides not only a kind of positive semidefinite block matrices but also a new extremal characterization of two variable geometric mean in terms of multivariate block matrices.

We study the Boolean rank of two families of binary matrices. The first is the binary matrix A k , t that represents the adjacency matrix of the intersection bipartite graph of all subsets of size t of { 1 , 2 , . . . , k } . We prove that its Boolean rank is k for every k ≥ 2 t . The second family is the family U s , m of submatrices of A k , t that is defined as U s , m = ( J m ⊗ I s ) + ( I ¯ m ⊗ J s ) , where I s is the identity matrix, J s is the all-ones matrix, s = k − 2 t + 2 and m = ( 2 t − 2 t − 1 ) . We prove that the Boolean rank of U s , m is also k for the following values of t and s: for s = 2 and any t ≥ 2 , that is k = 2 t ; for t = 3 and any s ≥ 2 ; and for any t ≥ 2 and s > 2 t − 2 , that is k > 4 t − 4 .

It is obtained that for positive semidefinite block matrix H = [ M K K ⁎ N ] if A , B ∈ M n ( C ) such that m a x ( ‖ A ‖ 2 , ‖ B ‖ 2 ) ≤ 2 and f is a nonnegative increasing convex function on [ 0 , ∞ ) satisfying f ( 0 ) = 0 , then 2 s j [ f ( | A ⁎ K B | ) ] ≤ m a x ( ‖ A ‖ 2 , ‖ B ‖ 2 ) s j [ f ( H ) ] for j = 1 , 2 , … , n . Also, it is shown that if H = [ M K K ⁎ N ] is a positive semidefinite 2 × 2 block matrix with square matrices M and N of the same size, then H r ≤ 2 ( a + b ) r − 1 ( M ⊕ N ) for r ≥ 1 , where a and b are the largest eigenvalues of the matrices M and N, respectively. In addition, if r ≥ 3 2 , then | | | H r | | | 2 ≤ 4 | | | M 2 ⊕ N 2 | | | ( | | | M ⊕ 0 | | | + | | | N ⊕ 0 | | | ) 2 r − 2 for every unitarily invariant norm.

Denote by B ( n 1 , … , n k ) the set of block matrices whose ( i , j ) -blocks are n i × n j complex matrices. Let A i ∈ B ( n 1 , … , n k ) be positive semidefinite and D i ∈ B ( n 1 , … , n k ) be block diagonal matrices for 1 ≤ i ≤ m . We obtain the following extension of Fischer's inequality: det ⁡ ( ∑ i = 1 m D i A i p i D i ⁎ ) ≤ ∏ j = 1 k det ⁡ ( ∑ i = 1 m [ D i ] j [ A i ] j p i [ D i ] j ⁎ ) , 0 ≤ p i ≤ 1 , where [ A i ] j is the j-th main diagonal block of A i . In addition, if A i and D i are nonsingular, the reverse inequality holds when − 1 ≤ p i ≤ 0 . We also extend these two results to a larger class of matrices, namely, matrices whose numerical ranges are contained in a sector.

A partial matrix over a field $\mathbb{F}$ is a matrix whose entries are either an element of $\mathbb{F}$ or an indeterminate and with each indeterminate only appearing once. A completion is an assignment of values in $\mathbb{F}$ to all indeterminates. Given a partial matrix, through elementary row operations and column permutation it can be decomposed into a block matrix of the form $\left[\begin{smallmatrix}{\bf W} & * & * \\ 0 & {\bf S} & * \\ 0 & 0 & {\bf T} \end{smallmatrix}\right]$ where ${\bf W}$ is wide (has more columns than rows), ${\bf S}$ is square, ${\bf T}$ is tall (has more rows than columns), and these three blocks have at least one completion with full rank. And importantly, each one of the blocks ${\bf W}$, ${\bf S}$ and ${\bf T}$ is unique up to elementary row operations and column permutation whenever ${\bf S}$ is required to be as large as possible. When this is the case $\left[\begin{smallmatrix}{\bf W} & * & * \\ 0 & {\bf S} & * \\ 0 & 0 & {\bf T} \end{smallmatrix}\right]$ will be called a WST-decomposition. With this decomposition it is trivial to compute maximum rank of a completion of the original partial matrix: $\#\mbox{rows}({\bf W})+\#\mbox{rows}({\bf S})+\#\mbox{cols}({\bf T})$. In fact we introduce the WST-decomposition for a broader class of matrices: the ACI-matrices.

The GM-switching, invented by Godsil and McKay, is a powerful method to construct cospectral graphs. In its simplest version, the method can be roughly described as follows: let A ( G ) be the adjacency matrix of a graph G on n vertices, let Q = diag ( R , I n − 2 m ) be a block diagonal matrix, where R = 1 m J − I 2 m is an orthogonal matrix of order 2m, J is the all-one matrix and I k denotes the identity matrix of order k. If G satisfies certain conditions, then Q T A ( G ) Q is an adjacency matrix of a graph G ′ which is cospectral with G (with cospectral complements). In this paper, we consider the problem of replacing the above R with another rational orthogonal matrix U of order 2p (p an odd prime). We characterize all graphs G admitting a GM-switching corresponding to Q = diag ( U , I n − 2 p ) , i.e., such that Q T A ( G ) Q is an adjacency matrix of a graph G ′ , and hence giving a counterpart of the GM-switching method for generating cospectral graphs (with cospectral complements), which can be used to construct pairs of non-isomorphic cospectral graphs that are not obtainable by the original GM-switching method. Next, we consider the problem of how to determine whether a given graph has a non-trivial GM-switching associated with some U. A simple algebraic condition is presented which partially answers this question.

We discuss two generalized inverse eigenvalue problems. The first one: For a given unreduced upper Hessenberg matrix H, find a nonsingular upper triangular matrix T such that all the pencils H k − λ T k have prescribed eigenvalues, where H k and T k are the leading k × k principal submatrices of H and T, respectively. The second one: For a given unitary unreduced upper Hessenberg matrix Q, find a nonsingular upper triangular matrix T such that all the pencils T k − θ Q k ⁎ have prescribed eigenvalues, where T k is the leading k × k principal submatrix of T, and Q k ⁎ is the conjugate transpose of the leading k × k principal submatrix of Q. We present the necessary and sufficient conditions for the solvability of the two problems. Our results lead to an alternative proof for the statement that any admissible Ritz value set or admissible harmonic Ritz value set is possible for the prescribed GMRES residual norms. Here, the term “admissible” means there are some restrictions on the sets if GMRES stagnates at some iterations.

Let M = [ A X X ⁎ B ] ∈ C 2 n × 2 n be positive semi-definite 2 × 2 block matrix, where A , B , X ∈ C n × n . A characterization for the matrix M with A + B = k I to be positive partial transpose is given in terms of its spectral norm. Using this result, a counter-example is constructed for the conjecture that ‖ M ‖ ≤ ‖ A + B ‖ when X is normal for all unitarily invariant norms.

For a class of matrices connected with Cauchon diagrams, Cauchon matrices, and the Cauchon algorithm, a method for determining the rank, and for checking a set of consecutive row (or column) vectors for linear independence is presented. Cauchon diagrams are also linked to the elementary bidiagonal factorization of a matrix and to certain types of rank conditions associated with submatrices called descending rank conditions.

In this paper, we introduce a new family of equations for matrix pencils that may be utilized for the construction of strong linearizations for any square or rectangular matrix polynomial. We provide a comprehensive characterization of the resulting vector spaces and show that almost every matrix pencil therein is a strong linearization regardless whether the matrix polynomial under consideration is regular or singular. These novel “ansatz spaces” cover all block Kronecker pencils as introduced in [6] as a subset and therefore contain all Fiedler pencils modulo permutations. The important case of square matrix polynomials is examined in greater depth. We prove that the intersection of any number of block Kronecker ansatz spaces is never empty and construct large subspaces of block-symmetric matrix pencils among which still almost every pencil is a strong linearization. Moreover, we show that the original ansatz spaces L 1 and L 2 may essentially be recovered from block Kronecker ansatz spaces via pre- and postmultiplication, respectively, of certain constant matrices.

In the spirit of our earlier articles [12] , [14] , [11] , and our work [13] , we define two dual q-Bernoulli polynomials, with corresponding vector and matrix forms. Following Aceto–Trigiante [1] , the q-L matrix, the indefinite q-integral of the q-Pascal matrix is the link between the q-Cauchy and the q-Bernoulli matrix. The q-analogue of the Bernoulli complementary argument theorem can be expressed in matrix form through the diagonal A n matrix. For the q-Euler polynomials corresponding results are obtained. The umbral calculus for generating functions of q-Appell polynomials is shown to be equivalent to a transform method, which maps polynomials to matrices, a true q-analogue of Arponen [6] . This is manifested by the Vein [21] matrix, which occurs as the transform of the q-difference operator. The Aceto–Trigiante shifted q-Bernoulli matrix has a simple connection to the q-Bernoulli Arponen matrix through the q-Pascal matrix. We reintroduce certain q-Stirling numbers ∈ Z ( q ) from [12] , which will be needed for the polynomial matrix definitions.

It is proved that regularity of an interval matrix implies singularity of four related interval matrices. The result is used to prove that for each nonsingular point matrix A, either A or A − 1 can be brought to a singular matrix by perturbing only the diagonal entries by an amount of at most 1 each. As a consequence, the notion of a diagonally singularizable matrix is introduced.

We introduce the notion of dual matrices of an infinite matrix A, which are defined by the dual sequences of the rows of A and naturally connected to the Pascal matrix P = [ ( i j ) ] ( i , j = 0 , 1 , 2 , … ) . We present the Cholesky decomposition of the symmetric Pascal matrix by means of its dual matrix. Decompositions of a Vandermonde matrix are used to obtain variants of the Lagrange interpolation polynomial of degree ≤n that passes through the n + 1 points ( i , q i ) for i = 0 , 1 , … , n .

The envelope of a square complex matrix is a spectrum encompassing region in the complex plane. It is contained in and is akin to the numerical range in the sense that the envelope is obtained as an infinite intersection of unbounded regions contiguous to cubic curves, rather than half-planes. In this article, the geometry and properties of the envelopes of special matrices are examined. In particular, symmetries of the envelope of a tridiagonal Toeplitz matrix are obtained, and the envelopes of block-shift matrices, Jordan blocks and 2 × 2 matrices are explicitly characterized.

The notions of weakly bipartitive and bipartitive families were introduced by Montgolfier (2003) as a general tool for studying some decomposition of graphs and other combinatorial structures. In this paper, we give a matrix description of these notions., 10 pages, 2 figures

If a 2 × 2 block matrix [ A B B ⁎ C ] is positive semidefinite, where each block is square, then the following trace inequality holds | tr A C − tr B ⁎ B | ≤ tr A tr C − | tr B | 2 . This improves a result of Besenyei [2] . Moreover, we show that | tr B m | ≤ tr A m / 2 C m / 2 , m = 2 , 3 , … .

Given matrices 𝐴 and 𝐵 such that 𝐵 = 𝑓(𝐴), where 𝑓(𝑧) is a holomorphic function, we analyze the relation between the singular values of the off-diagonal submatrices of 𝐴 and 𝐵. We provide a family of bounds which depend on the interplay between the spectrum of the argument 𝐴 and the singularities of the function. In particular, these bounds guarantee the numerical preservation of quasiseparable structures under mild hypotheses. We extend the Dunford-Cauchy integral formula to the case in which some poles are contained inside the contour of integration. We use this tool together with the technology of hierarchical matrices (H-matrices) for the effective computation of matrix functions with quasiseparable arguments. ispartof: Linear Algebra and Its Applications vol:516 pages:212-242 status: published

In this paper we introduce an isospectral matrix flow (Lax flow) that preserves some structures of an initial matrix. This flow is given by d A d t = [ A u − A l , A ] , A ( 0 ) = A 0 , where A is a real n × n matrix (not necessarily symmetric), [ A , B ] = A B − B A is the matrix commutator (also known as the Lie bracket), A u is the strictly upper triangular part of A and A l is the strictly lower triangular part of A. We prove that if the initial matrix A 0 is staircase, so is A ( t ) . Moreover, we prove that this flow preserves the certain positivity properties of A 0 . Also we prove that if the initial matrix A 0 is totally positive or totally nonnegative with non-zero codiagonal elements and distinct eigenvalues, then the solution A ( t ) converges to a diagonal matrix while preserving the spectrum of A 0 . Some simulations are provided to confirm the convergence properties.

A square matrix L is called a Laplacian-like matrix if L j = 0 and j T L = 0 . A square matrix D is left (or right) Laplacian expressible if there exist a number λ ≠ 0 , a column vector β satisfying β T j = 1 , and a square matrix L such that β T D = λ j T , L D + I = β j T and L j = 0 (or D β = λ j , D L + I = j β T and j T L = 0 ). We consider the generalized distance matrix D (see Definition 4.1 ) of a graph whose blocks correspond to left (or right) Laplacian expressible matrices. Then D is also left (or right) Laplacian expressible, and the inverse D − 1 , when it exists, can be expressed as the sum of a Laplacian-like matrix and a rank one matrix.

In this paper, we present inequalities related to partial transpose and partial trace for positive semidefinite matrices. Some interesting results involving traces and eigenvalues are also included.

The analysis of any Hessenberg matrix as a product of a companion matrix and a triangular matrix is presented in this paper. The factors are explicitly given on terms of the entries of the Hessenberg matrix and, further, various factorizations are undertaken. Applications to comrade and confederate matrices, as well as for the corresponding block cases, are included.

A square matrix is called stochastic (or row-stochastic) if it is non-negative and has each row sum equal to unity. Here, we constitute an eigenvalue localization theorem for a stochastic matrix, by using its principal submatrices. As an application, we provide a suitable bound for the eigenvalues, other than unity, of the Randic matrix of a connected graph.

Almost every nonsingular matrix A∈R2m,2m can be decomposed into the product of a symplectic matrix S and an upper J-triangular matrix R. This decomposition is not unique. In this paper we analyze the freedom of choice in the symplectic and the upper J-triangular factors and review several existing suggestions on how to choose the free parameters in the SR decomposition. In particular we consider two choices leading to the minimization of the condition number of the diagonal blocks in the upper J-triangular factor and to the minimization of the conditioning of the corresponding blocks in the symplectic factor. We develop bounds for the extremal singular values of the whole upper J-triangular factor and the whole symplectic factor in terms of the spectral properties of even-dimensioned principal submatrices of the skew-symmetric matrix associated with the SR decomposition. The theoretical results are illustrated on two small examples.

A matrix with a nonzero nonnegative vector in its null space is called central . We study classes of central matrices having zero column sums. The study is motivated by an engineering application concerning induction heating where central matrices provide a way to control the energy flow over time. A ( ± 1 ) -matrix A is called a ZSC-matrix (zero sum sign-central) if each matrix with the same sign pattern as A and having zero column sums is central. We establish several classes of ZSC-matrices, and give separate sufficient and necessary conditions for a matrix to be ZSC. Moreover, we give algorithms for finding central matrices that are used for power control in induction heating, and illustrate these by some numerical examples.

We present an algorithm for computing a symmetric rank-revealing decomposition of a symmetric n × n matrix A, as defined in the work of Hansen & Yalamov [9] : we factorize the original matrix into a product A = Q M Q T , with Q orthogonal and M symmetric and in block form, with one of the blocks containing the dominant information of A, such as its largest eigenvalues. Moreover, the matrix M is constructed in a form that is easy to update when adding to A a symmetric rank-one matrix or when appending a row and, symmetrically, a column to A: the cost of such an updating is O ( n 2 ) floating point operations. The proposed algorithm is based on the block anti-triangular form of the original matrix M, as introduced by the authors in [11] . Via successive orthogonal similarity transformations this form is then updated to a new form A = Q ˆ M ˆ Q ˆ T , whereby the first k rows and columns of M ˆ have elements bounded by a given threshold τ and the remaining bottom right part of M ˆ is maintained in block anti-triangular form. The updating transformations are all orthogonal, guaranteeing the backward stability of the algorithm, and the algorithm is very economical when the near rank deficiency is detected in some of the anti diagonal elements of the block anti-triangular form. Numerical results are also given showing the reliability of the proposed algorithm.

The paper presents a new algorithm to compute the LU-factorization of a matrix represented in a quasiseparable or semiseparable form (i.e., using generators). It obtains the quasiseparable representations of the factors L and U of an N × N block matrix via O ( N ) arithmetic operations on the block entries. The algorithm uses recursions based exclusively on unitary transformations which provide numerical stability even in singular cases. The method of the paper is based on the theory developed in [1] and provides an alternative to the approach proposed in [7] for strongly regular matrices. The algorithm presented here works also for some matrices with possibly singular principle submatrices. The results of numerical tests show that also for strongly regular matrices the new algorithm is comparable with the previous methods.

In this paper, we study the positive stability of P-matrices. We prove that a matrix A is positive stable if A is a P 2 -matrix and there is at least one nested sequence of principal submatrices of A each of which is also a P 2 -matrix. This result generalizes the result by Carlson which shows the positive stability of sign-symmetric P-matrices and the result by Tang, Simsek, Ozdaglar and Acemoglu which shows the positive stability of strictly row (column) square diagonally dominant for every order of minors P-matrices.

A new class of linearizations and l-ifications for m × m matrix polynomials P ( x ) of degree n is proposed. The l-ifications in this class have the form A ( x ) = D ( x ) + ( e ⊗ I m ) W ( x ) where D is a block diagonal matrix polynomial with blocks B i ( x ) of size m, W is an m × q m matrix polynomial and e = ( 1 , … , 1 ) t ∈ C q , for a suitable integer q. The blocks B i ( x ) can be chosen a priori, subjected to some restrictions. Under additional assumptions on the blocks B i ( x ) the matrix polynomial A ( x ) is a strong l-ification, i.e., the reversed polynomial of A ( x ) defined by A # ( x ) : = x deg ⁡ A ( x ) A ( x − 1 ) is an l-ification of P # ( x ) . The eigenvectors of the matrix polynomials P ( x ) and A ( x ) are related by means of explicit formulas. Some practical examples of l-ifications are provided. A strategy for choosing B i ( x ) in such a way that A ( x ) is a well conditioned linearization of P ( x ) is proposed. Some numerical experiments that validate the theoretical results are reported.

The nonexistence of a 2-resolvable balanced incomplete block design with parameters ( v , b , r , k , λ ) = ( 10 , 15 , 6 , 4 , 2 ) was established by Kadowaki in 2001 using a computer search and by Kadowaki and Kageyama in 2008 by a lengthy case-by-case analysis. We give a short proof based on properties of the row sum vectors of submatrices in an incidence matrix.

Arising from the classification of the matrix-roots of a nonnegative imprimitive irreducible matrix, we present results concerning the Jordan chains of an $h$-cyclic matrix. We also present ancillary results applicable to nonnegative imprimitive irreducible matrices and demonstrate these results via examples., Comment: To appear in the special issue "The Legacy of Hans Schneider" in Linear Algebra and its Applications

Let T be a tree with vertex set [ n ] = { 1 , 2 , … , n } . For each i ∈ [ n ] , let m i be a positive integer. An ordered pair of two adjacent vertices is called an arc. Each arc ( i , j ) of T has a weight W i , j which is an m i × m j matrix. For two vertices i , j ∈ [ n ] , let the unique directed path from i to j be P i , j = x 0 , x 1 , … , x d where d ⩾ 1 , x 0 = i and x d = j . Define the product distance from i to j to be the m i × m j matrix M i , j = W x 0 , x 1 W x 1 , x 2 ⋯ W x d − 1 , x d . Let N = ∑ i = 1 n m i . The N × N product distance matrix D of T is a partitioned matrix whose ( i , j ) -block is the matrix M i , j . We give a formula for det ⁡ ( D ) . When det ⁡ ( D ) ≠ 0 , the inverse of D is also obtained. These generalize known results for the product distance matrix when either the weights are real numbers, or m 1 = m 2 = ⋯ = m n = s and the weights W i , j = W j , i = W e for each edge e = { i , j } ∈ E ( T ) .

This paper deals with an inverse three spectra problem for Jacobi matrices, where the spectrum of the full N × N Jacobi matrix T is prescribed, together with the spectra of two matrices obtained from the principal submatrices of T, denoted by T 1 , n and T n + 1 , N , by modifying the lower right entry of T 1 , n and the upper left entry of T n + 1 , N . Here n satisfying 1 ≤ n N is fixed. Denote by T 1 , n − and T n + 1 , N + the modified principal submatrices. We give conditions for three given sets of points to be the spectra of a matrix T and of its two modified principal submatrices T 1 , n − and T n + 1 , N + to uniquely reconstruct the original matrix T, where the matrix T is a rank 2 perturbation of T 1 , n − ⊕ T n + 1 , N + .

Which assignments from 2 n − 1 arbitrary, distinct real numbers as eigenvalues of designated leading principal submatrices permit a real symmetric tridiagonal matrix? We raise this question, motivated both by known results and recent work on multiplicities and interlacing equalities in symmetric matrices whose graph is a given tree. Known results are reviewed, a general conjecture is given, and several new partial results are proved.

In this paper, we characterize the Drazin invertibility of a two by two matrix with zero (2,2) entry under some suitable similarity transformation. The results can be applied to obtain detailed Drazin inverse formulae of various structured matrices and some special cases are analyzed.

Matic (2014) [6] proved two inequalities regarding the ratio det(A+D)/det(A), where A is positive definite, and D is a positive definite block diagonal matrix. In this paper, we extend these results to a larger class of matrices, namely, matrices whose numerical ranges are contained in a sector.

E.M. Sa and R.C. Thompson proved that the invariant factors of a matrix over a commutative principal ideal domain and the invariant factors of its submatrices are related by a set of divisibility inequalities, called the interlacing inequalities for invariant factors. We extend this result to matrices over elementary divisor duo rings.

We give an operator inequality involving the off-diagonal block of a positive partial transpose matrix and the geometric mean of its diagonal blocks. Some new eigenvalue inequalities are derived, nicely completing the existing ones.

Cell matrix introduced by Jaklic and Modic [6] is a special Euclidean distance matrix (EDM) that has a quite attractive and simple structure. It is of interest to approximate an EDM by a cell matrix. In this paper, we consider the problem of finding the cell matrix that is closest to a given EDM with respect to the Frobenius norm. The majorization ordering of the eigenvalues of a cell matrix is also discussed.