Let with 0 < 6"', < ~ 2 n) <. .. < 6'n" < 7r. Put cj,, (x) _ II(x-x á). 1 Suppose 0 <_ A < B < 7r. 'e denote by N"(A, B) the number of the z9j in (A, B). Let-1 <-a < b <_ 1. Then we denote by M,,,(a, b) the number of the x i in (a, b). It does not matter whether the intervals (A, B) and (a, b) are open or closed. In a previous paper2Turán and the author proved that if wn(x) be a triangular matr...

We derive a closed form for the Jordan decomposition of companion matrices including properties of generalized eigenvectors. As a consequence, we provide a formula for the inverse of confluent Vandermonde matrices and results on sensitivity of multiple roots of polynomials.

From (1) it follows tha tX = x^4x*. The set of all complex numbers zAz* where zz* — \ is called the field of values [25] of the matrix A. It follows that the characteristic roots of A belong to the field of values of A. Beginning with Bendixson [3] in 1900, many writers have obtained limits for the characteristic roots of a matrix. In many cases these were actually limits for the field of value...

It is the purpose of this note to point out that there may also exist bounded regions which exclude the latent roots of A. Let B: [&,-,•] be an nXn matrix with complex elements. Let (/i(l), • • ■ , n(n)) be a permutation of (1, • • • , n), and let B'\ [b'tj] be the matrix [b'v] = [&*.(,•)]. Thus£' is obtained from B by a permutation of its columns, and therefore B is nonsingular when B' is nons...

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Every n×n generalized K-centrosymmetric matrix A can be reduced into a 2× 2 block diagonal matrix (see [20]). This block diagonal matrix is called the reduced form of the matrix A. In this paper we further investigate some properties of the reduced form of these matrices and discuss the square roots of these matrices. Finally exploiting these properties, the development of structure-preserving ...

In this paper, we study the stochastic root matrices of stochastic matrices. All stochastic roots of 2¥2 stochastic matrices are found explicitly. A method based on characteristic polynomial of matrix is developed to find all real root matrices that are functions of the original 3¥3 matrix, including all possible (function) stochastic root matrices. In addition, we comment on some numerical met...

We propose new techniques and algorithms for the solution of a polynomial system of equations by matrix methods. For such a system, we seek its specified root, at which a fixed polynomial takes its maximum or minimum absolute value on the set of roots. We unify several known approaches and simplify the solution substantially, in particular in the case of an overconstrained polynomial system hav...