Bounds for the zeros of unilateral octonionic polynomials

In the present work it is proved that the zeros of a unilateral octonionic polynomial belong to the conjugacy classes of the latent roots of an appropriate lambda-matrix. This allows the use of matricial norms, and matrix norms in particular, to obtain upper and lower bounds for the zeros of unilateral octonionic polynomials. Some results valid for complex and/or matrix polynomials are extended to octonionic polynomials.

Spectrahedral representation of polar orbitopes

Let K be a compact Lie group and V a finite-dimensional representation of K. The orbitope of a vector $$x\in V$$ x ∈ V is the convex hull $${\mathscr {O}}_x$$ O x of the orbit Kx in V. We show that if V is polar then $${\mathscr {O}}_x$$ O x is a spectrahedron, and we produce an explicit linear matrix inequality representation. We also consider the coorbitope $${\mathscr {O}}_x^o$$ O x o , which is the convex set polar to $${\mathscr {O}}_x$$ O x . We prove that $${\mathscr {O}}_x^o$$ O x o is the convex hull of finitely many K-orbits, and we identify the cases in which $${\mathscr {O}}_x^o$$ O x o is itself an orbitope. In these cases one has $${\mathscr {O}}_x^o=c\cdot {\mathscr {O}}_x$$ O x o = c · O x with $$c>0$$ c > 0 . Moreover we show that if x has “rational coefficients” then $${\mathscr {O}}_x^o$$ O x o is again a spectrahedron. This provides many new families of doubly spectrahedral orbitopes. All polar orbitopes that are derived from classical semisimple Lie algebras can be described in terms of conditions on singular values and Ky Fan matrix norms.

The Global Convergence of the Nonlinear Power Method for Mixed-Subordinate Matrix Norms

We analyze the global convergence of the power iterates for the computation of a general mixed-subordinate matrix norm. We prove a new global convergence theorem for a class of entrywise nonnegative matrices that generalizes and improves a well-known results for mixed-subordinate $$\ell ^p$$ ℓ p matrix norms. In particular, exploiting the Birkoff–Hopf contraction ratio of nonnegative matrices, we obtain novel and explicit global convergence guarantees for a range of matrix norms whose computation has been recently proven to be NP-hard in the general case, including the case of mixed-subordinate norms induced by the vector norms made by the sum of different $$\ell ^p$$ ℓ p -norms of subsets of entries.

One Type of Symmetric Matrix with Harmonic Pell Entries, Its Inversion, Permanents and Some Norms

The Pell numbers, named after the English diplomat and mathematician John Pell, are studied by many authors. At this work, by inspiring the definition harmonic numbers, we define harmonic Pell numbers. Moreover, we construct one type of symmetric matrix family whose elements are harmonic Pell numbers and its Hadamard exponential matrix. We investigate some linear algebraic properties and obtain inequalities by using matrix norms. Furthermore, some summation identities for harmonic Pell numbers are obtained. Finally, we give a MATLAB-R2016a code which writes the matrix with harmonic Pell entries and calculates some norms and bounds for the Hadamard exponential matrix.

Optimized Cramer’s Rule in WZ Factorization and Applications

In this paper, W Z factorization is optimized with a proposed Cramer’s rule and compared with classical Cramer’s rule to solve the linear systems of the factorization technique. The matrix norms and performance time of WZ factorization together with LU factorization are analyzed using sparse matrices on MATLAB via AMD and Intel processor to deduce that the optimized Cramer’s rule in the factorization algorithm yields accurate results than LU factorization and conventional W Z factorization. In all, the matrix group and Schur complement for every Zsystem (2×2 block triangular matrices from Z-matrix) are established.