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3 Special Matrices

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Elementary triangular matrices and inverses of k-Hessenberg and triangular matrices

100%

Verde-Star L.

tom 3

nr 1

We use elementary triangular matrices to obtain some factorization, multiplication, and inversion properties of triangular matrices. We also obtain explicit expressions for the inverses of strict k-Hessenberg matrices and banded matrices. Our results can be extended to the cases of block triangular and block Hessenberg matrices. An n × n lower triangular matrix is called elementary if it is of the form I + C, where I is the identity matrix and C is lower triangular and has all of its nonzero entries in the k-th column,where 1 ≤ k ≤ n.

Polynomial sequences generated by infinite Hessenberg matrices

100%

Verde-Star L.

nr 1

64-72

We show that an infinite lower Hessenberg matrix generates polynomial sequences that correspond to the rows of infinite lower triangular invertible matrices. Orthogonal polynomial sequences are obtained when the Hessenberg matrix is tridiagonal. We study properties of the polynomial sequences and their corresponding matrices which are related to recurrence relations, companion matrices, matrix similarity, construction algorithms, and generating functions. When the Hessenberg matrix is also Toeplitz the polynomial sequences turn out to be of interpolatory type and we obtain additional results. For example, we show that every nonderogative finite square matrix is similar to a unique Toeplitz-Hessenberg matrix.

Representation of doubly infinite matrices as non-commutative Laurent series

63%

Arenas-Herrera M. I. , Verde-Star L.

nr 1

250-257

We present a new way to deal with doubly infinite lower Hessenberg matrices based on the representation of the matrices as the sum of their diagonal submatrices. We show that such representation is a simple and useful tool for computation purposes and also to obtain general properties of the matrices related with inversion, similarity, commutativity, and Pincherle derivatives. The diagonal representation allows us to consider the ring of doubly infinite lower Hessenberg matrices over a ring R as a ring of Laurent series in one indeterminate, with coefficients in the ring of R-valued sequences that don’t commute with the indeterminate.