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In the field of radar target detection, the conventional approach is to employ the range profile energy accumulation method for detecting extended targets. However, this method becomes ineffective when dealing with non-stationary and non-uniform radar clutter scenarios, as well as long-distance targets with weak radar cross sections (RCSs). In such cases, the signal-to-noise ratio (SNR) of the target echo is severely degraded, rendering the energy accumulation detection algorithm unreliable. To address this issue, this paper presents a new extended target detection method based on the maximum eigenvalue of the Hermitian matrix. This method utilizes a detection model that incorporates observed data and employs the likelihood ratio test (LRT) theory to derive the maximum eigenvalue detector at low SNR. Specifically, the detector constructs a matrix using a sliding window block with the available data and then computes the maximum eigenvalue of the covariance matrix. Subsequently, the maximum eigenvalue matrix is transformed into a one-dimensional eigenvalue image, enabling extended target detection through analogy with the energy accumulation detection method. Furthermore, this paper analyzes the proposed extended target detection method from both theoretical and experimental perspectives, validating it through field-measured data. The results obtained from the measured data demonstrate that the method effectively enhances the SNR in low SNR conditions, thereby improving target detection performance. Additionally, the method exhibits robustness across different scattering center targets. [ABSTRACT FROM AUTHOR]

This paper considers the Hermitian solutions of a new system of commutative quaternion matrix equations, where we establish both necessary and sufficient conditions for the existence of solutions. Furthermore, we derive an explicit general expression when it is solvable. In addition, we also provide the least squares Hermitian solution in cases where the system of matrix equations is not consistent. To illustrate our main findings, in this paper we present two numerical algorithms and examples. [ABSTRACT FROM AUTHOR]

A mixed graph D is a graph that can be obtained from a graph by orienting some of its edges. Let α be a primitive nth root of unity, then the α−Hermitian adjacency matrix of a mixed graph is defined to be the matrix Hα = [hrs] where hrs = α if rs is an arc in D, hrs = α if sr is an arc in D, hrs = 1 if sr is a digon in D and hrs = 0 otherwise. In this paper we study the cospectrality of the Hermitian adjacency matrix of a mixed graph. [ABSTRACT FROM AUTHOR]

Matrices in R n × n with determinant equal to ∑ i ≤ n μ (i) , where μ represents the Möbius function, have been studied for decades. The motivation to study such matrices is the close connection that they have to the prime number theorem, Dirichlet convolution, and related concepts. We introduce two parameterized families of bordered Hermitian matrices that possess similar properties. Each family is comprised of matrices M s ∈ C (n − 1) × (n − 1) that satisfy det ⁡ M 0 = ∑ i ≤ n | μ (i) | , det ⁡ M 1 = (∑ i ≤ n μ (i)) 2 , and we show det ⁡ M s is a quadratic polynomial in s. We apply the Cauchy interlacing theorem to show that, for each matrix in one of the families, the product of all of the subdominant eigenvalues is bounded above by 6 / π 2 + O (n − 1 / 2). [ABSTRACT FROM AUTHOR]

Let D denote the smallest disc containing all eigenvalues of the matrix A. Without knowing the eigenvalues of A, we can estimate the spread of A and the radius of D. Some new bounds for the radius of D and the spread of A are given. These bounds involve the entries of A. Also sufficient conditions for equality are obtained for some inequalities. New proofs of some known results are presented, too. [ABSTRACT FROM AUTHOR]

Let S be an n-by-n, nonsingular, and Hermitian matrix. A square complex matrix Q is said to be S-unitary if Q*SQ = S. An S-unitary matrix Q is said to be elementary if rank(Q -- I) = 1. It is known what form every elementary S-unitary can take, and that every S-unitary can be written as a product of elementary S-unitaries. In this paper, we determine the Jordan canonical form of a product of two elementary S-unitaries. [ABSTRACT FROM AUTHOR]

We establish the converse of Weyl's eigenvalue inequality for additive Hermitian perturbations of a Hermitian matrix. [ABSTRACT FROM AUTHOR]

In 1893 Jacques Hadamard introduced the famous inequality concerning the determinant of the Gram matrix. The generalizations of this inequality are the main subject of considerations presented in this paper. Calculations executed for the gramian of order three revealed that the description of gramian hides a great potential enabling to strengthen essentially the Hadamard inequality. The research, undertaken in this direction, resulted in discovering the more subtle versions of the Hadamard inequality, also by involving the “language” of the positive semi-definite Hermitian matrices. [ABSTRACT FROM AUTHOR]

In this paper, we consider the eigenvalue variation for any perturbation of Hermitian matrices, and we obtain two perturbation bounds. The first bound always improves the existing bound, and the second bound also improves the existing one under a suitable condition. A simple example is given for comparing these bounds. [ABSTRACT FROM AUTHOR]

Let m , n ≥ 2 be positive integers and let H n be the set of n × n complex Hermitian matrices. We study linear maps ϕ : H m n → H m n satisfying det ⁡ ( A ⊗ B ) = det ⁡ ( ϕ ( A ⊗ B ) ) for all A ∈ H m , B ∈ H n . The connection of the problem to quantum information science is mentioned. [ABSTRACT FROM AUTHOR]

Maps that preserve adjacency on the set of all invertible hermitian matrices over a finite field are characterized. It is shown that such maps form a group that is generated by the maps A ↦ P A P ⁎ , A ↦ A σ , and A ↦ A − 1 , where P is an invertible matrix, P ⁎ is its conjugate transpose, and σ is an automorphism of the underlying field. Bijectivity of maps is not an assumption but a conclusion. Moreover, adjacency is assumed to be preserved in one direction only. [ABSTRACT FROM AUTHOR]

The enhanced principal rank characteristic sequence (epr-sequence) of a symmetric n × n matrix is a sequence ℓ 1 ℓ 2 ⋯ ℓ n where ℓ k is A , S , or N according as all, some, or none of its principal minors of order k are nonzero. Such sequences give more information than the (0,1) pr-sequences previously studied (where basically the k th entry is 0 or 1 according as none or at least one of its principal minors of order k is nonzero). Various techniques including the Schur complement are introduced to establish that certain subsequences such as NAN are forbidden in epr-sequences over fields of characteristic not two. Using probabilistic methods over fields of characteristic zero, it is shown that any sequence of A s and S s ending in A is attainable, and any sequence of A s and S s followed by one or more N s is attainable; additional families of attainable epr-sequences are constructed explicitly by other methods. For real symmetric matrices of orders 2, 3, 4, and 5, all attainable epr-sequences are listed with justifications. [ABSTRACT FROM AUTHOR]

The inverse eigenvalue problem and the associated optimal approximation problem for Hermitian reflexive matrices with respect to a normal { k + 1 } -potent matrix are considered. First, we study the existence of the solutions of the associated inverse eigenvalue problem and present an explicit form for them. Then, when such a solution exists, an expression for the solution to the corresponding optimal approximation problem is obtained. [ABSTRACT FROM AUTHOR]

For a positive integer $n\geq 2$, let $M_{n}$ be the set of $n\times n$ complex matrices and $H_{n}$ the set of Hermitian matrices in $M_{n}$. We characterize injective linear maps ${\it\phi}:H_{m_{1}\cdots m_{l}}\rightarrow H_{n}$ satisfying $$\begin{eqnarray}\text{rank}(A_{1}\otimes \cdots \otimes A_{l})=1\Longrightarrow \text{rank}({\it\phi}(A_{1}\otimes \cdots \otimes A_{l}))=1\end{eqnarray}$$ for all $A_{k}\in H_{m_{k}}$, $k=1,\dots ,l$, where $l,m_{1},\dots ,m_{l}\geq 2$ are positive integers. The necessity of the injectivity assumption is shown. Moreover, the connection of the problem to quantum information science is mentioned. [ABSTRACT FROM AUTHOR]

Without 'positive definiteness' demanded in the present papers, the forward and reverse inequalities for Hadamard products of any number of invertible Hermitian matrices are obtained, and the sufficient and necessary conditions for the equations in these inequalities are given. As Hermitian positive matrices naturally satisfy the added constraints, these results generalize and improve the corresponding results in the present papers. Beyond that, with no demand of 'positive definiteness', these forward and backward inequalities are not determined mutually any longer. [ABSTRACT FROM AUTHOR]

We consider the problem of determining those undirected n -vertex graphs with a corresponding Hermitian matrix that admits only two distinct eigenvalues, with multiplicities k and n--k. After giving some general algebraic characterizations of these dual multiplicity graphs, we then prove two major graph theoretic necessary conditions on such graphs. Construction techniques are then developed, and these lead to a characterization of dual multiplicity graphs for which the lesser multiplicity is two. [ABSTRACT FROM AUTHOR]

Abstract: Let be a positive semidefinite block matrix with square matrices M and N of the same order and denote . The main results are the following eigenvalue majorization inequalities: for any complex number z of modulus 1, If, in addition, K is Hermitian, then for any real number , while if K is skew-Hermitian, then for any real number , where O is the zero matrix of compatible size. These majorization inequalities generalize some results due to Furuichi and Lin, Turkmen, Paksoy and Zhang, Lin and Wolkowicz. [Copyright &y& Elsevier]

Abstract: We consider the conditions under which the Cayley transform of the Kronecker product of two Hermitian matrices can be again presented as a Kronecker product of two matrices and, if so, if it is a product of the Cayley transforms of the two Hermitian matrices. We also study the related question: given two matrices, which matrix under the Cayley transform yields the Kronecker product of their Cayley transforms. [Copyright &y& Elsevier]

We prove that a continuous map $\phi $ defined on the set of all $n\times n$ Hermitian matrices preserving order in both directions is up to a translation a congruence transformation or a congruence transformation composed with the transposition. [ABSTRACT FROM PUBLISHER]

If a convex body C in …n has modular and irreducible face lattice and C is not strictly convex, there is a face-preserving homeomorphism from C to a set of positive-semidefinite Hermitian matrices of trace 1 over …, … or …, or C has dimension 8, 14 or 26. [ABSTRACT FROM AUTHOR]

Necessary and sufficient conditions for the Stieltjes moment problem to have a unique solution and for the Hamburger moment problem with the same moments to have infinitely many solutions were obtained in Hamburger's papers on the classical moment problem. In this paper, we obtain an analog of Hamburger's criterion for the matrix moment problem. [ABSTRACT FROM AUTHOR]

Abstract: We investigate constant rank subspaces of symmetric and hermitian matrices over finite fields, using a double counting method related to the number of common zeros of the corresponding subspaces of symmetric bilinear and hermitian forms. We obtain optimal bounds for the dimensions of constant rank subspaces of hermitian matrices, and good bounds for the dimensions of subspaces of symmetric and hermitian matrices whose non-zero elements all have odd rank. [Copyright &y& Elsevier]

Abstract: Estimating upper bounds of the spectrum of large Hermitian matrices has long been a problem with both theoretical and practical significance. Algorithms that can compute tight upper bounds with minimum computational cost will have applications in a variety of areas. We present a practical algorithm that exploits k-step Lanczos iteration with a safeguard step. The k is generally very small, say 5–8, regardless of the large dimension of the matrices. This makes the Lanczos iteration economical. The safeguard step can be realized with marginal cost by utilizing the theoretical bounds developed in this paper. The bounds establish the theoretical validity of a previous bound estimator that has been successfully used in various applications. Moreover, we improve the bound estimator which can now provide tighter upper bounds with negligible additional cost. [Copyright &y& Elsevier]

Abstract: Eigenvalue problems arise in many application areas ranging from computational fluid dynamics to information retrieval. In these fields we are often interested in only a few eigenvalues and corresponding eigenvectors of a sparse matrix. In this paper, we comment on the modifications of the eigenvalue problem that can simplify the computation of those eigenpairs. These transformations allow us to avoid difficulties associated with non-Hermitian eigenvalue problems, such as the lack of reliable non-Hermitian eigenvalue solvers, by mapping them into generalized Hermitian eigenvalue problems. Also, they allow us to expose and explore parallelism. They require knowledge of a selected eigenvalue and preserve its eigenspace. The positive definiteness of the Hermitian part is inherited by the matrices in the generalized Hermitian eigenvalue problem. The position of the selected eigenspace in the ordering of the eigenvalues is also preserved under certain conditions. The effect of using approximate eigenvalues in the transformation is analyzed and numerical experiments are presented. [Copyright &y& Elsevier]

Abstract: When an edge is removed from an undirected graph, there is a limited change that can occur in the multiplicity of an eigenvalue of a Hermitian matrix with that graph. Primarily for trees, we identify the changes that can occur and characterize the circumstances under which they occur. This extends known results for the removal of vertices. A catalog of examples is given to illustrate the possibilities that can occur and to contrast the case of trees with that of general graphs. [ABSTRACT FROM AUTHOR]

Abstract: Polygon transformations based on taking the apices of similar triangles constructed on the sides of an initial polygon are analyzed as well as the limit polygons obtained by iteratively applying such transformations. In contrast to other approaches, this is done with respect to two construction parameters representing a base angle and an apex perpendicular subdivision ratio. Furthermore, a combined transformation leading to circulant Hermitian matrices is proposed, which eliminates the rotational effect of the basic transformation. A finite set of characteristic parameter subdomains is derived for which the sequence converges to specific eigenpolygons. Otherwise, limit polygons turn out to be linear combinations of up to three eigenpolygons. This leads to a full classification of circulant Hermitian similar triangles based polygon transformations and their limit polygons. As a byproduct classical results as Napoleon’s theorem and the Petr–Douglas–Neumann theorem can be easily deduced. [Copyright &y& Elsevier]

Abstract: The inertia of a Hermitian matrix is defined to be a triplet composed of the numbers of the positive, negative and zero eigenvalues of the matrix counted with multiplicities, respectively. In this paper, we show some basic formulas for inertias of block Hermitian matrices. From these formulas, we derive various equalities and inequalities for inertias of sums, parallel sums, products of Hermitian matrices, submatrices in block Hermitian matrices, differences of outer inverses of Hermitian matrices. As applications, we derive the extremal inertias of the linear matrix expression with respect to a variable Hermitian matrix X. In addition, we give some results on the extremal inertias of Hermitian solutions to the matrix equation , as well as the extremal inertias of a partial block Hermitian matrix. [Copyright &y& Elsevier]

Abstract: Denote by a graph which is the vertex set and is an adjacency relation on a subset of . In this paper, the good distance graph is defined. Let and be two good distance graphs, and be a map. The following theorem is proved: is a graph isomorphism is a bounded distance preserving surjective map in both directions is a distance preserving surjective map in both directions (where is a positive integer), etc. Let be a division ring with an involution such that both and is not a field of characteristic 2 with , where and is the center of . Let be the set of Hermitian matrices over . It is proved that is a good distance graph, where for all . [Copyright &y& Elsevier]

Abstract: It is shown that any map which preserves adjacency on hermitian matrices over a finite field is necessary bijective and hence of the standard form. [Copyright &y& Elsevier]

Abstract: A formula for the inverse of any nonsingular matrix partitioned into two-by-two blocks is derived through a decomposition of the matrix itself and generalized inverses of the submatrices in the matrix. The formula is then applied to three matrix inverse completion problems to obtain their complete solutions. [Copyright &y& Elsevier]

Abstract: In this paper, by using normal maps originated by Lewis [A.S. Lewis, Group invariance and convex matrix analysis, SIAM J. Matrix Anal. Appl. 17 (1996) 927–949], we present a method for producing majorization inequalities on eigenvalues and singular values of matrices. Some classical and new inequalities are derived. [Copyright &y& Elsevier]

Abstract: We study theoretical properties of two inexact Hermitian/skew-Hermitian splitting (IHSS) iteration methods for the large sparse non-Hermitian positive definite system of linear equations. In the inner iteration processes, we employ the conjugate gradient (CG) method to solve the linear systems associated with the Hermitian part, and the Lanczos or conjugate gradient for normal equations (CGNE) method to solve the linear systems associated with the skew-Hermitian part, respectively, resulting in IHSS(CG, Lanczos) and IHSS(CG, CGNE) iteration methods, correspondingly. Theoretical analyses show that both IHSS(CG, Lanczos) and IHSS(CG, CGNE) converge unconditionally to the exact solution of the non-Hermitian positive definite linear system. Moreover, their contraction factors and asymptotic convergence rates are dominantly dependent on the spectrum of the Hermitian part, but are less dependent on the spectrum of the skew-Hermitian part, and are independent of the eigenvectors of the matrices involved. Optimal choices of the inner iteration steps in the IHSS(CG, Lanczos) and IHSS(CG, CGNE) iterations are discussed in detail by considering both global convergence speed and overall computation workload, and computational efficiencies of both inexact iterations are analyzed and compared deliberately. [Copyright &y& Elsevier]

We show that any n × n conjugate-normal matrix can be brought by a unitary congruence transformation to block-tridiagonal form with the orders of the consecutive diagonal blocks not exceeding 1, 2, 3, ..., respectively. The proof is constructive; namely, a finite process is described that implements the reduction to the desired form. Sufficient conditions are indicated for the orders of the diagonal blocks to stabilize. In this case, the condensed form is a band matrix. [ABSTRACT FROM AUTHOR]

Abstract: Two Hermitian matrices are said to be Hermitian-congruent if there exists a nonsingular Hermitian matrix such that . In this paper, we give necessary and sufficient conditions for two nonsingular simultaneously unitarily diagonalizable Hermitian matrices A and B to be Hermitian-congruent. Moreover, when A and B are Hermitian-congruent, we describe the possible inertias of the Hermitian matrices C that carry the congruence. We also give necessary and sufficient conditions for any 2-by-2 nonsingular Hermitian matrices to be Hermitian-congruent. In both of the studied cases, we show that if A and B are real and Hermitian-congruent, then they are congruent by a real symmetric matrix. Finally we note that if A and B are 2-by-2 nonsingular real symmetric matrices having the same sign pattern, then there is always a real symmetric matrix C satisfying . Moreover, if both matrices are positive, then C can be picked with arbitrary inertia. [Copyright &y& Elsevier]

Abstract: For a self-adjoint linear operator with a discrete spectrum or a Hermitian matrix, the “extreme” eigenvalues define the boundaries of clusters in the spectrum of real eigenvalues. The outer extreme ones are the largest and the smallest eigenvalues. If there are extended intervals in the spectrum in which no eigenvalues are present, the eigenvalues bounding these gaps are the inner extreme eigenvalues. We will describe a procedure for detecting the extreme eigenvalues that relies on the relationship between the acceleration rate of polynomial acceleration iteration and the norm of the matrix via the spectral theorem, applicable to normal matrices. The strategy makes use of the fast growth rate of Chebyshev polynomials to distinguish ranges in the spectrum of the matrix which are devoid of eigenvalues. The method is numerically stable with regard to the dimension of the matrix problem and is thus capable of handling matrices of large dimension. The overall computational cost is quadratic in the size of a dense matrix; linear in the size of a sparse matrix. We verify computationally that the algorithm is accurate and efficient, even on large matrices. [Copyright &y& Elsevier]

Abstract: In this paper, using a minimum principle for Schur complements of positive semidefinite Hermitian matrices and some estimates of the eigenvalues and the singular values, we obtain some inequalities for the eigenvalues of the Schur complement of the matrix product in terms of the eigenvalues of the Schur complements of and A. [Copyright &y& Elsevier]

Abstract: We propose a new QR-like algorithm, symmetric squared QR (SSQR) method, that can be readily parallelized using commonly available parallel computational primitives such as matrix–matrix multiplication and QR decomposition. The algorithm converges quadratically and the quadratic convergence is achieved through a squaring technique without utilizing any kind of shifts. We provide a rigorous convergence analysis of SSQR and derive structures for several of the important quantities generated by the algorithm. We also discuss various practical implementation issues such as stopping criteria and deflation techniques. We demonstrate the convergence behavior of SSQR using several numerical examples. [Copyright &y& Elsevier]

Abstract: Let K be a field of characteristic not two or three with an automorphism − of order two. Let . We characterize additive mappings T from one F-vector space of Hermitian matrices over K to another that preserve rank less than or equal to one for the following cases: (i) the image of T contains a matrix of rank at least three, (ii) every nonzero endomorphism of F is surjective. [Copyright &y& Elsevier]

Abstract: In this paper, an extended projection method in the complex number field is presented. We consider two classes of complex linear matrix inequalities and then derive the corresponding projection operators. Applications to the control system with pole assignment problem and the robust stability of linear descriptor systems, which are described in complex linear matrix inequalities, are given. Based on the numerical algorithms, some examples are illustrated for the merits of the proposed method. [Copyright &y& Elsevier]

Groß [Linear Algebra Appl. 326 (2001) 215] developed characterizations of the minus and star partial orders between the squares of Hermitian nonnegative definite matrices referring to the concept of the space preordering. In the present paper, his results are generalized by deleting the nonnegative definiteness assumption and supplemented by alternative characterizations. [Copyright &y& Elsevier]

Baksalary and Pukelsheim [Linear Algebra Appl. 151 (1991) 135] considered the problem of how an order between two Hermitian nonnegative definite matrices A and B is related to the corresponding order between the squares A2 and B2, in the sense of the star partial ordering, the minus partial ordering, and the Lo¨wner partial ordering. In the present paper, possibilities of generalizing and strengthening their results are studied from two points of view: by widening the class of matrices considered and by replacing the squares by arbitrary powers. [Copyright &y& Elsevier]

Certain classes of matrices are indicated for which the star, left-star, right-star, and minus partial orderings, or some of them, are equivalent. Characterizations of the left-star and right-star orderings, similar to those devised by Hartwig and Styan [Linear Algebra Appl. 82 (1986) 145] for the star and minus orderings, are established along with other auxiliary results, which are of independent interest as well. Some inheritance-type properties of matrices are also given. The class of EP matrices appears to be essential in several points of our considerations. [Copyright &y& Elsevier]

We describe the set of all (α,n), for which the scalar complex matrix αIn is a sum of k idempotent Hermitian matrices, and get the minimal number of summands for each αIn. [Copyright &y& Elsevier]

If −∞<α<β<∞ let mid(α,x,β)=α if x<α, x if α&les;x&les;β, β if x>β. Let An=Bn+Pn where Bn and Pn are n×n Hermitian matrices. We show that if &par;Pn&par;F2=o(n) then, for any [α,β], (A) ∑i=1n|F(mid(α,λi(An),β))−F(mid(α,λi(Bn),β))|=o(n) if F∈C[α,β]. (Eigenvalues numbered in nondecreasing order.) We consider the special case where {Pn} are real Hankel matrices. We also show that if rank(Pn)=o(n) then (A) holds for every [α,β] and F∈C[α,β]. Combining these results yields a result concerning Cn=Bn+En+Rn, where &par;En&par;2F=o(n) and rank(Rn)=o(n). We also consider the case where the conditions on {En} are stated in terms of Schatten p-norms. Finally, we show that if {Tn} are Hermitian Toeplitz matrices generated by f∈C[−π,π] with minimum mf and maximum Mf, (2(i−1)−n)π/n&les;ξin&les;(2i−n)π/n, 1&les;i&les;n, and τn is a permutation of {1,2,…,n} such that f(ξτn(1),n)&les;f(ξτn(2),n)&les;⋯&les;f(ξτn(n),n), then ∑i=1n|F(λi(Tn))−F(f(ξτn(i),n))|=o(n) if F∈C[mf,Mf]. [Copyright &y& Elsevier]

For Hermitian matrices, whose graph is a given tree, the relationships among vertex degrees, multiple eigenvalues and the relative position of the underlying eigenvalue in the ordered spectrum are discussed in detail. In the process, certain aspects of special vertices, whose removal results in an increase in multiplicity are investigated. [Copyright &y& Elsevier]

The article discusses the measurement of the orthogonal matrix polynomials with periodic coefficients of recurrence relations. It uses the diagonalization of the matrix transform to determine the matrices. It presents mathematical theorems that explain the problem of reconstructing the measure of a periodic sequence of coefficients.

The article discusses the proof of Thompson's determinantal inequality and tackles the conjugate transpose of a matrix or a vector X.