Yoopyo Hong

A canonical form for matrices under consimilarity

Abstract Square complex matrices A , B are said to be consimilar if A=SB S −1 for some nonsingular matrix S . Consimilarity is an equivalence relation that is a natural matrix generalization of rotation of scalars in the complex plane. We survey the known forms to which a given complex matrix may be reduced by unitary consimilarity and describe a canonical form to which it may be reduced by a general consimilarity. We derive a useful criterion for two matrices to be consimilar and show that every matrix is consimilar to its own conjugate, transpose, and adjoint, to a real matrix, and to a Hermitian matrix.

A lower bound for the smallest singular value

Abstract A lower bound for the smallest singular value of A ϵ n x n is given in terms of the determinant and the 2-norm of the columns and the rows of A . Examples are given to show that this bound performs well for a large class of matrices.

A new block algorithm for full-rank solution of the Sylvester-observer equation

A new block algorithm for computing a full rank solution of the Sylvester-observer equation arising in state estimation is proposed. The major computational kernels of this algorithm are: 1) solutions of standard Sylvester equations, in each case of which one of the matrices is of much smaller order than that of the system matrix and (furthermore, this small matrix can be chosen arbitrarily), and 2) orthogonal reduction of small order matrices. There are numerically stable algorithms for performing these tasks including the Krylov-subspace methods for solving large and sparse Sylvester equations. The proposed algorithm is also rich in Level 3 Basic Linear Algebra Subroutine (BLAS-3) computations and is thus suitable for high performance computing. Furthermore, the results on numerical experiments on some benchmark examples show that the algorithm has better accuracy than that of some of the existing block algorithms for this problem.

A characterization of unitary congruence

Two square complex matrices A B are said to be unitarily congruent if there is a unitary matrix U of the same size such that A=UBUT . We show that A and B are unitarily congruent if and only if there is a unitary V such that . As an application of this characterization, we give simple derivations of several canonical forms for special classes of matrices under unitary congruence.

The Jordan cononical form of a product of a Hermitian and a positive semidefinite matrix

Abstract A given square complex matrix C is the product of a positive semidefinite matrix A and a Hermitian matrix B if and only if C 2 is diagonalizable and has nonnegative eigenvalues. This condition is equivalent to requiring that C have real eigenvalues and a Jordan canonical form that is diagonal except for r copies of a 2-by-2 nilpotent Jordan block. We show that r is bounded from above by the rank of A , the nullity of A , and both the positive and negative inertia of B . It follows that a product of two positive semidefinite matrices is diagonalizable and has nonnegative eigenvalues, a result that leads to a characterization of the possible concanonical forms of a positive semidefinite matrix.

A Hermitian canonical form for complex matrices under consimilarity

Abstract We produce an explicit Hermitian canonical form for complex square matrices under consimilarity. We apply a simple algorithmic procedure to a concanonical form for complex matrices to construct a form that is not only canonical but also Hermitian. We also show that a similar algorithmic procedure can be used to produce an explicit real canonical form for complex matrices under consimilarity.

Parametrized Newton’s Iteration for Computing an Eigenpair of a Real Symmetric Matrix in an Interval

Abstract A parameterized Newton’s method to guarantee convergence to an eigenpair of a real symmetric matrix in a designated interval has been developed. The method is parametric in nature and with appropriate choices of parameters, the classical methods such as Newton’s method and the Rayleigh quotient method can easily be recovered.

Applications of linear transformations to matrix equations

Abstract We consider the linear transformation T ( X ) = AX − CXB where A , C ∈ M n , B ∈ M s . We show a new approach to obtaining conditions for the existence and uniqueness of the solution X of the matrix equation T ( X ) = R . As a consequence of our approach we present a simple characterization of a full-rank solution to the matrix equation. We apply the existence theorem to a general form of the observer matrix equation and characterize the existence of a full-rank solution.

Linear operators preserving t-congruence on matrices

Abstract Two complex or real square matrices A and B are said to be t -congruent if there exists a nonsingular matrix S such that A = SBS t . If A and B are complex (respectively, real) and S is unitary (respectively, real orthogonal), we say that they are unitarily t -congruent (respectively, real orthogonally congruent). In this note we obtain characterizations of linear operators on various matrix spaces that preserve t -congruence. Results and problems concerning unitary t -congruence or orthogonal t -congruence are discussed.

Computing for eigenpairs on globally convergent iterative method for hermitian matrices

Let A = A* ∈ Mn and