A simple proof of polar decomposition in pseudo-Euclidean geometry
We give a simple direct proof of the polar decomposition for separated linear maps in pseudo-Euclidean geometry.
A simple proof of the classification of normal Toeplitz matrices.
A Simple Proof of the Convexity of the Field of Values Defined by Two Hermitian Forms.
A simple proof of Valiant's lemma
A solution of the continuous Lyapunov equation by means of power series
A sparsity result on nonnegative real matrices with given spectrum
Let σ=(λ1,...,λn) be the spectrum of a nonnegative real n × n matrix. It is shown that σ is the spectrum of a nonnegative real n × n matrix having at most nonzero entries.
A spectral theorem for matrices over fields of power series
A Spectral Theory for Tensors
In this paper we propose a general spectral theory for tensors. Our proposed factorization decomposes a tensor into a product of orthogonal and scaling tensors. At the same time, our factorization yields an expansion of a tensor as a summation of outer products of lower order tensors. Our proposed factorization shows the relationship between the eigen-objects and the generalised characteristic polynomials. Our framework is based on a consistent multilinear algebra which explains how to generalise...
A strategy for proving Riemann hypothesis.
A strong complement property of Dedekind domains
A stronger version of matrix convexity as applied to functions of Hermitian matrices.
A structured staircase algorithm for skew-symmetric/symmetric pencils.
A study on new right/left inverses of nonsquare polynomial matrices
This paper presents several new results on the inversion of full normal rank nonsquare polynomial matrices. New analytical right/left inverses of polynomial matrices are introduced, including the so-called τ-inverses, σ-inverses and, in particular, S-inverses, the latter providing the most general tool for the design of various polynomial matrix inverses. The applicationoriented problem of selecting stable inverses is also solved. Applications in inverse-model control, in particular robust minimum...
A subclass of strongly clean rings
In this paper, we introduce a subclass of strongly clean rings. Let be a ring with identity, be the Jacobson radical of , and let denote the set of all elements of which are nilpotent in . An element is called very -clean provided that there exists an idempotent such that and or is an element of . A ring is said to be very -clean in case every element in is very -clean. We prove that every very -clean ring is strongly -rad clean and has stable range one. It is shown...
A sufficient condition for Hurwitz stability of the convex combination of two matrices
A survey of certain trace inequalities
This paper concerns inequalities like TrA ≤ TrB, where A and B are certain Hermitian complex matrices and Tr stands for the trace. In most cases A and B will be exponential or logarithmic expressions of some other matrices. Due to the interest of the author in quantum statistical mechanics, the possible applications of the trace inequalities will be commented from time to time. Several inequalities treated below have been established in the context of Hilbert space operators or operator algebras....
A survey of generalized matrix inverses
A survey of some recent results on Clifford algebras in
We will study applications of numerical methods in Clifford algebras in , in particular in the skew field of quaternions, in the algebra of coquaternions and in the other nondivision algebras in . In order to gain insight into the multidimensional case, we first consider linear equations in quaternions and coquaternions. Then we will search for zeros of one-sided (simple) quaternion polynomials. Three different classes of zeros can be distinguished. In general, the quaternionic coefficients can...
A survey of the Kreiss matrix theorem for power bounded families of matrices and its extensions
We survey results related to the Kreiss Matrix Theorem, especially examining extensions of this theorem to Banach space and Hilbert space. The survey includes recent and established results together with proofs of many of the interesting facts concerning the Kreiss Matrix Theorem.
A Symmetric Numerical Range for Matrices.