Iulian Haimovici

Complete Algorithm for Unitary Hessenberg Matrices

We study the divide and conquer method for computing the eigenstructure of unitary upper Hessenberg matrices using their quasiseparable representations. In doing so we use essentially the fact that in Theorem 25.1 we proved the results using r m × r m matrices α and β which may be different from the identity.

Multiplication of Matrices

This chapter considers the product A=A1A2 of block matrices \(A_{1}=\{A^{(1)}_{ij}\}^{N}_{i,j}=1\) and \(A_{1}=\{A^{(2)}_{ij}\}^{N}_{i,j}=1\) with block entries of compatible sizes m i× v j and v i× n j.One assumes that quasiseparable generators of the factors are given and one derives formulas and algorithms to compute quasiseparable generators of the product.

Matrices with Separable Representation and Low Complexity Algorithms

One of the simplest representations of matrices used for a reduction of complexity of algorithms is the separable representation. The term separable comes from the fact that the (block) entries A kj of such an N × N matrix A can be presented in a separated form \( A_{kj}=b_{K}.c_{j},\qquad\,j,k=1,....,N,\) where b k and C j are matrices of certain sizes.

The Reduction to Hessenberg Form

One of the methods used in the solution of the eigenvalue problem is to reduce at first a matrix to a unitarily similar one in the upper Hessenberg form. The subsequent QR iterations for an upper Hessenberg matrix can be performed in an efficient way.

Kernels of Quasiseparable of Order One Matrices

Next in this part we study the eigenspaces of quasiseparable of order one matrices. Notice that for a matrix A the matrix A – λ0 I has the same lower and upper quasiseparable generators. This is the reason we study the properties of kernels of the considered matrices.

The LDU Factorization and Inversion

Let A be a block matrix with block entries of sizes m i× m j and with invertible principal leading submatrices \( A(1:k,1:k),\quad k=1,....,N,\). Such matrix is called strongly regular. By Theorem 1.20, A admits the LDU factorization A=LDU where L,U,D are block matrices with the same sizes of blocks as A and L and U are block lower and upper triangular matrices with identities on the main diagonals, while D is a block diagonal matrix.

Completion to Unitary Matrices

In this chapter we study the problem of completion of a partially specified matrix with a given lower triangular part to a unitary matrix.

Unitary Matrices with Quasiseparable Representations

In this chapter we study in detail the quasiseparable representations of unitary matrices. We show that for unitary matrices the quasiseparable representations are closely connected with factorization representations of a matrix as a product of elementary unitary matrices.

Eigenvalues with Geometric Multiplicity One

In this chapter we study conditions under which the eigenspaces of quasiseparable of order one matrices are one-dimensional. We also derive explicit formulas for the corresponding eigenvectors.

Quasiseparable Representations and Descriptor Systems with Boundary Conditions

In this chapter we show that the quasiseparable representation of a matrix is closely connected with the treatment of this matrix as a matrix of the input-output operator of a discrete-time varying linear system with boundary conditions.