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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]

We introduced an efficient technique to suppress speckle noise in medical ultrasound images while maintaining low computational complexity. The medical ultrasound image is divided into overlapping subimages, and Lanczos decomposition is then applied to the average Hermitian covariance matrix of all subimages. The resultant orthonormal vectors are used for filtering speckle noise through orthogonal and oblique projections, i.e., by projecting noisy signal onto the signal subspace. After sorting the orthonormal vectors, an orthogonal projection matrix is formed by selecting the first K vectors contributing to the signal, whereas an oblique projection matrix is formed by selecting the first K vectors contributing to the signal and the last K vectors contributing to the noise. The procedure of Lanczos is also followed with singular value decomposition (SVD). Schemes are applied to real ultrasound images and two types of speckle noise simulations: fine and rough speckle noise. Numerical and visual results depict that proposed technique has outperformed various popular benchmark schemes, i.e., Frost, Lee, probabilistic nonlocal means, geometric nonlinear diffusion filter, guided speckle reducing bilateral filter, and SVD, while maintaining a competitive computational complexity. Lanczos-based scheme showed a relatively lagging performance in terms of resolution, which was always outshone by a leading performance in terms of key measures such as signal-to-noise ratio (SNR), peak signal-to-noise ratio (PSNR), feature similarity, and mean structural similarity. Orthogonal and oblique projections were found to perform the same except when speckle noise is fine, where insignificant differences were found in terms of SNR, PSNR, and resolution. Lanczos-based scheme tends to offer a better estimation of orthonormal vectors than SVD, consequently a better speckle noise suppression. Furthermore, it offers an efficient tuning parameter per block/subimage size to treat various speckle noise patterns. [ABSTRACT FROM AUTHOR]

We give a simple proof of a determinantal inequality due to Marcus (1976). [ABSTRACT FROM AUTHOR]

In this note we introduce a partial ordering in the set of complex Hermitian matrices which coincides with the well-known Löwner ordering when the considered matrices have the same number of negative eigenvalues. Some properties of the new ordering are investigated, and known results for shorted matrices are generalized. [ABSTRACT FROM AUTHOR]

A new method for adaptive beamforming in uniform linear arrays (ULAs) is introduced. The proposed method is based on generalised loading of the covariance matrix of the data vector received by the array. The covariance matrix is loaded with a Hermitian matrix which is chosen to shape the directional response of the array. Simulation results show that the proposed approach offers better handling of mismatches between presumed and actual array steering vectors. It produces output signal‐to‐interference‐plus‐noise ratio values higher than some well‐known approaches in the literature with much less complexity. [ABSTRACT FROM AUTHOR]

Let d , n ∈ Z + such that 1 ≤ d ≤ n. A d -code C ⊂ F q n × n is a subset of order n square matrices with the property that for all pairs of distinct elements in C , the rank of their difference is greater than or equal to d. A d -code with as many as possible elements is called a maximum d -code. The integer d is also called the minimum distance of the code. When d < n , a classical example of such an object is the so-called generalized Gabidulin code (Gabidulin and Kshevetskiy (2005)). In Delsarte and Goethals (1975), Schmidt (2015) and K.U. Schmidt (2018), several classes of maximum d -codes made up respectively of symmetric, alternating and Hermitian matrices were exhibited. In this article we focus on such examples. Precisely, we determine their automorphism groups and solve the equivalence issue for them. Finally, we exhibit a maximum symmetric 2-code which is not equivalent to the one with same parameters constructed in Schmidt (2015). [ABSTRACT FROM AUTHOR]

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.

In the paper, the authors establish some inequalities for generalized eigenvalues of perturbation problems on Hermitian matrices and modify shortcomings of some known inequalities for generalized eigenvalues in the related literature. [ABSTRACT FROM AUTHOR]

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

The application of the complex quadratic form as the decision boundary for complex‐valued data classification is described. This function is always real when its matrix is Hermitian. Thus, a simple sign function to classify the input data is used. This matrix is obtained through an iterative learning process similar to the Rosenblatt algorithm. The concept of the Frobenius matrix norm is used to prove that the proposed learning algorithm converges if a solution exists. This approach is different from other complex‐valued neural networks that use optimisation techniques or feature mapping. An artificial neuron that uses a complex quadratic form as the decision boundary is called a complex quadratic neural unit. [ABSTRACT FROM AUTHOR]