Narrow Results

We show that for any two $n\times n$ matrices $X$ and $Y$ we have the inequality $s_j^2(I+XY)\le {\lambda }_j((I+X^{\ast }X)(I+Y^{\ast }Y)),$ where $s_j(T)$ and ${\lambda }_j(T)$ denote the decreasingly ordered singular values and eigenvalues of $T$. As an application, we show that for $2n\times 2n$ real positive definite matrices the symplectic eigenvalues $d_j,$ under some special conditions, satisfy the inequality $d_j(A+B)\ge d_j(A)+d_1(B)$.

Finding new bounds for norms and the numerical radius of certain matrix forms is a renowned topic that has attracted numerous researchers. In this paper, we show some new bounds for unitarily invariant norms and the numerical radius of certain matrix forms that involve products of matrices and sums of products.

New arithmetic-geometric mean-type inequalities and a refinement of the celebrated Cauchy–Schwarz inequality are shown among the obtained results.

Our results are compared with those in the literature through numerical examples and rigorous approaches.

The empirical distribution of the eigenvalues of the matrix HH^T divided by its trace is considered, where H is a Hankel random matrix. The normal distribution with different parameters are considered and the effect of scale and shape parameters are evaluated. The correlation among eigenvalues are assessed using parametric and non-parametric association criteria.

In this paper, we introduce a sparse recovery strategy for analytic signals in Hardy space H²(𝔻), where 𝔻 denotes the unit disk of the complex plane. The representation strategy is based on the optimization technique. We investigate the asymptotic singular values distribution of the dictionary matrix and give an estimation of the number of rows of the random matrix. To the best of our knowledge, this is the first time that such result is given. This result demonstrates that the dictionary of the normalized Szegö kernels (or reproducing kernels) is perfect for decompositions of analytic signals. A numerical example is presented exhibiting the theory. As applications, we still work on time-frequency analysis and propose a new type of non-negative time-frequency distribution associated with mono-components in the periodic case.

Following a recent attempt by Waziri et al. [2019] to find an appropriate choice for the nonnegative parameter of the Hager–Zhang conjugate gradient method, we have proposed two adaptive options for the Hager–Zhang nonnegative parameter by analyzing the search direction matrix. We also used the proposed parameters with the projection technique to solve convex constraint monotone equations. Furthermore, the global convergence of the methods is proved using some proper assumptions. Finally, the efficacy of the proposed methods is demonstrated using a number of numerical examples.

Literature suggests that distinguishing chaos from noise continues to remain a highly contentious issue in the modern age as it has been historically. This is because chaos and noise share common properties which in turn makes it indistinguishable. In this paper, we seek to provide a viable solution to this problem by presenting a novel approach for the differentiating and identifying chaos from noise. The proposed approach is one that is based on dynamical systems, embedding theorem, matrix algebra and statistical theory. To achieve our objective, the distribution, pattern and behaviour of eigenvalues are analysed in-depth. This yields several important properties with broad application, enabling the distinction between chaos and noise in time series analysis. The applicability of the proposed approach is examined using WTI Spot Price time series.