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We present a procedure which enables the computation and the description of structures of isotropy subgroups of the group of complex orthogonal matrices with respect to the action of *congruence on Hermitian matrices. A key ingredient in our proof is an algorithm giving solutions of a certain rectangular block (complex-alternating) upper triangular Toeplitz matrix equation. [ABSTRACT FROM AUTHOR]

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]

Let A and B be $ n\times n $ n × n positive semidefinite matrices, $ f:[0,\infty)\rightarrow \lbrack 0,\infty) $ f : [ 0 , ∞) → [ 0 , ∞) be an increasing convex function with $ f\left (0\right) =0 $ f (0) = 0 , and let $ \Phi : \mathbb {M}_{n}(\mathbb {C})\rightarrow \mathbb {M}_{n}(\mathbb {C}) $ Φ : M n (C) → M n (C) be a positive unital linear map. It is shown that $$\begin{align*} & \prod_{j=1}^{k} s_{j}\left(f^{4}\left(\Phi \left(\frac{A+B}{2}\right) \right) \right)\\ & \quad \leq 2^{-4} \prod_{j=1}^{k} \left(s_{j}\left(\Phi^{2}\left(f^{2}\left(A\right) \right) +\Phi \left(f^{2}\left(A\right) \right) \right) +\left\Vert \Phi \left(f^{2}\left(B\right) \right) \right\Vert +1\right) \\ & \quad\quad \times \left(s_{j}\left(\Phi^{2}\left(f^{2}\left(B\right) \right) +\Phi \left(f^{2}\left(B\right) \right) \right) +\left\Vert \Phi \left(f^{2}\left(A\right) \right) \right\Vert +1\right) \end{align*}$$ ∏ j = 1 k s j (f 4 (Φ (A + B 2))) ≤ 2 − 4 ∏ j = 1 k (s j (Φ 2 (f 2 (A)) + Φ (f 2 (A))) + ‖ Φ (f 2 (B)) ‖ + 1) × (s j (Φ 2 (f 2 (B)) + Φ (f 2 (B))) + ‖ Φ (f 2 (A)) ‖ + 1) for $ k=1,2,\ldots,n $ k = 1 , 2 , ... , n , where $ s_{j}\left (T\right) $ s j (T) and $ \left \Vert T\right \Vert $ ‖ T ‖ are the $ j{\rm th} $ j th singular value and the spectral norm of T, respectively. Applications of our results to sectorial matrices are given. [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]

Summary: The hybrid precoding problem is considered a Frobenius norm reduction problem for the narrowband channel in a millimeter wave (mmWave) with multiple inputs and outputs (mmWave MIMO). This work proposes a hybrid distributed online and alternating convex optimization (HDO‐ACO) algorithm to improve hybrid precoding (HP), interpreted as a trace minimization problem. HDO‐ACO alternately determines the digital and analog precoders to reduce the trace by keeping the other constant. Initially, HDO‐ACO uses Lagrange's method to determine the digital precoding subproblem. Then, it uses the integrated distributed online convex optimization and alternating minimization algorithm in the analog precoder design. In the HP method, the digital and analog precoders are iteratively updated until the highest number of iterations or convergence is reached. But this hybrid precoding method requires an initial analog precoding matrix input to begin the iteration. The algorithms converge gradually and fall into a suboptimal solution when the initial analog precoding matrix is set randomly. Hence, an initial value acceleration‐based heuristic approach is used in the HDO‐ACO alternating minimization algorithm that calculates the initial feasible value of an alternating minimization method using channel conditions. The simulation results of the proposed HDO‐ACO algorithm are presented under bit error rate (BER), spectral efficiency (SE), and convergence behavior by comparing it with modified block coordinate descent–HP (MBCD‐HP), manifold optimization–alternating minimization (MO‐AltMin), and semi‐definite relaxation‐based alternating optimization (SDR‐AO). The proposed HDO‐ACO attains maximum SE and less BER than other hyper‐precoding designs. [ABSTRACT FROM AUTHOR]

The Cayley transform of a matrix A , C (A) = (I + A) − 1 (I − A) , has been investigated in the contexts of skew-symmetry, eigenvalue mapping and as a factorization technique for several matrix positivity classes. Here, we extend the discussion to Cayley transforms of almost skew-Hermitian matrices, positive definite matrices, semipositive matrices, as well as subclasses of the M -matrices, H -matrices and their inverses. [ABSTRACT FROM AUTHOR]

In this paper, by using gradients of Gateaux differentiable G -increasing functions, we establish some interpolation results for the following G -majorization triangle inequality: (x + y) ↓ ≺ G x ↓ + y ↓ for vectors x , y of an Eaton triple with normal map (⋅) ↓. In particular, we show refinements for the majorization Ky Fan's eigenvalue inequality λ (x + y) ≺ m λ (x) + λ (y) for x , y ∈ H n , where H n is the space of n × n Hermitian matrices and λ (⋅) is the eigenvalue operator on H n. We derive analogous results for an inner product space with its standard norm, and for real skew-symmetric matrices with their singular values. [ABSTRACT FROM AUTHOR]