Nonsingular Complex Matrices Research Articles

Investigation of positive definite solution of nonlinear matrix equation $$X^{p}=Q +\sum \nolimits _{i=1}^m A_i^*X^{\delta }A_i$$

In this paper, we consider the nonlinear matrix equation $$X^{p}=Q +\sum \nolimits _{i=1}^m A_i^*X^{\delta }A_i,$$ where $$A_i (i=1,2,\ldots ,m)$$ are $$n\times n$$ nonsingular complex matrices, Q is a $$n\times n$$ Hermitian positive definite (HPD) matrix, $$p\ge 1, m\ge 1$$ are positive integers, and $$\delta \in (0,1)$$ . We discuss the solution of this equation via properties of Thompson metric and two fixed point theorems in ordered Banach spaces and estimate the bounds of the HPD solution. Furthermore, perturbation analysis is investigated.

On the nonlinear matrix equation

We establish several necessary and sufficient conditions for the existence and uniqueness of Hermitian positive definite (HPD) solutions to the general matrix equation where p, m are positive integers, are nonsingular complex matrices, A and B are HPD matrices, and then give three algorithms to get the unique solution. Moreover, we give two numerical examples to illustrate the effectiveness of the theoretical results and the behavior of the considered methods.

Solvability of some constrained matrix approximation problems using core-EP inverses

Using the core-EP inverse, we obtain the unique solution to the constrained matrix minimization problem in the Euclidean norm: $$\mathrm{Minimize }\ \Vert Mx-b\Vert _2$$ , subject to the constraint $$x\in \mathcal{R}(M^k),$$ where $$M\in {\mathbb {C}}^{n\times n}$$ , $$k=\mathrm {Ind}(M)$$ and $$b\in {\mathbb {C}}^n$$ . This problem reduces to well-known results for complex matrices of index one and for nonsingular complex matrices. We present two kinds of Cramer’s rules for finding unique solution to the above mentioned problem, applying one well-known expression and one new expression for core-EP inverse. Also, we consider a corresponding constrained matrix approximation problem and its Cramer’s rules based on the W-weighted core-EP inverse. Numerical comparison with classical strategies for solving the least squares problems with linear equality constraints is presented. Particular cases of the considered constrained optimization problem are considered as well as application in solving constrained matrix equations.

A Study of the Second-Kind Multivariate Pseudo-Chebyshev Functions of Fractional Degree

Here, in this paper, the second-kind multivariate pseudo-Chebyshev functions of fractional degree are introduced by using the Dunford–Taylor integral. As an application, the problem of finding matrix roots for a wide class of non-singular complex matrices has been considered. The principal value of the fixed matrix root is determined. In general, by changing the determinations of the numerical roots involved, we could find n r roots for the n-th root of an r × r matrix. The exceptional cases for which there are infinitely many roots, or no roots at all, are obviously excluded.

Composition Identities of Chebyshev Polynomials, via 2 × 2 Matrix Powers

Starting from a representation formula for 2 × 2 non-singular complex matrices in terms of 2nd kind Chebyshev polynomials, a link is observed between the 1st kind Chebyshev polinomials and traces of matrix powers. Then, the standard composition of matrix powers is used in order to derive composition identities of 2nd and 1st kind Chebyshev polynomials. Before concluding the paper, the possibility to extend this procedure to the multivariate Chebyshev and Lucas polynomials is touched on.

On the Hermitian positive definite solutions of nonlinear matrix equation [formula omitted

In this paper, the Hermitian positive definite solutions of nonlinear matrix equation Xs+∑i=1mAi∗X-tiAi=Q are considered, where Q is a Hermitian positive definite matrix, Ai are nonsingular complex matrices, s,m are positive numbers, and 0<ti≤1,i=1,…,m. Necessary and sufficient conditions for the existence of Hermitian positive definite solutions are derived. A sufficient condition for the existence of a unique Hermitian positive definite solution is given. In addition, some necessary conditions and sufficient conditions for the existence of Hermitian positive definite solutions are presented. Finally, an iterative method is proposed to compute the maximal Hermitian positive definite solution, and some numerical examples are given to show the efficiency of the proposed iterative method.

On the Iterative Method for the System of Nonlinear Matrix Equations

The positive definite solutions for the system of nonlinear matrix equations are considered, where are two positive integers and A, B are nonsingular complex matrices. Some sufficient conditions for the existence of positive definite solutions for the system are derived. Under some conditions, an iterative algorithm for computing the positive definite solutions for the system is proposed. Also, the estimation of the error is obtained. Finally, some numerical examples are given to show the efficiency of the proposed iterative algorithm.

On the Hermitian Positive Definite Solutions of Nonlinear Matrix Equation Xs+A∗X−t1A+B∗X−t2B=Q

Nonlinear matrix equation has many applications in engineering; control theory; dynamic programming; ladder networks; stochastic filtering; statistics and so forth. In this paper, the Hermitian positive definite solutions of nonlinear matrix equation are considered, where Q is a Hermitian positive definite matrix, A, B are nonsingular complex matrices, s is a positive number, and 0 &lt; ti ≤ 1, i = 1, 2. Necessary and sufficient conditions for the existence of Hermitian positive definite solutions are derived. A sufficient condition for the existence of a unique Hermitian positive definite solution is given. In addition, some necessary conditions and sufficient conditions for the existence of Hermitian positive definite solutions are presented. Finally, an iterative method is proposed to compute the maximal Hermitian positive definite solution, and numerical example is given to show the efficiency of the proposed iterative method.

On the existence of extremal positive definite solutions of the nonlinear matrix equation [formula omitted

In the present paper, a necessary condition for the existence of positive definite solutions of the nonlinear matrix equation X r + ∑ i = 1 m A i ∗ X δ i A i = I is derived, where − 1 < δ i < 0 , I is an n × n identity matrix, A i are n × n nonsingular complex matrices and r , m are positive integers. Based on the Banach fixed point theorem, a sufficient condition for the existence of a unique positive definite solution of this equation is also derived. Iterative methods for obtaining the extremal (maximal–minimal) positive definite solutions of this equation are proposed. Furthermore, the rate of convergence of some proposed algorithms is proved. Finally, numerical examples are given to illustrate the performance and effectiveness of the proposed algorithms.

Congruence and $A^{-1}A^*$

A canonical form under congruence for nonsingular complex matrices may be deduced from the canonical pair form for Hermitian matrices in a straightforward way. This allows the study of A^{-1}A^* and what it says about the congruence class of A . In turn, alternate congruential forms may be given along with a further generalization of Sylvester's law of inertia and an application to C-S equivalence.

A Power Method for Computing Square Roots of Complex Matrices

In this paper higher order convergent methods for computing square roots of nonsingular complex matrices are derived. These methods are globally convergent and are based on eigenvalue shifting and powering. Specifically, it is shown for each positive integerr≥2, a convergent method of orderrcan be developed. These algorithms can be used to compute square roots of general nonsingular complex matrices such as computing square roots of matrices with negative eigenvalues.

Contragredient equivalence: A canonical form and some applications

Let A and C be m-by- n complex matrices, and let B and D be n-by- m complex matrices. The pair ( A, B) is contragrediently equivalent to the pair ( C, D) if there are square nonsingular complex matrices X and Y such that XAY −1 = C and YBX −1 = D. Contragredient equivalence is a common generalization of four basic equivalence relations: similarity, consimilarity, complex orthogonal equivalence, and unitary equivalence. We develop a complete set of invariants and an explicit canonical form for contragredient equivalence and show that ( A, A T ) is contragrediently equivalent to ( C, C T ) if and only if there are complex orthogonal matrices P and Q such that C = PAQ. Using this result, we show that the following are equivalent for a given n-by- n complex matrix A: 1. (1) A = QS for some complex orthogonal Q and some complex symmetric S; 2. (2) A T A is similar to AA T ; 3. (3) ( A, A T ) is contragrediently equivalent to ( A T , A); 4. (4) A = Q 1 A T Q 2 for some complex orthogonal Q 1, Q 2; 5. (5) A = PA T P for some complex orthogonal P. We then consider a linear operator φ on n-by- n complex matrices that shares the following properties with transpose operators: for every pair of n-by- n complex matrices A and B, (a) φ. preserves the spectrum of A, (b) φ( φ( A)) = A, and (c) φ( AB) = φ( B) φ( A). We show that ( A, φ( A)) is contragrediently similar to ( B, φ( B)) if and only if A = X 1 BX 2 for some nonsingular X 1, X 2 that satisfy X −1 1 = φ( X 1) and X −1 2 = φ( X 2). We also consider a factorization of the form A = XY, where X −1 = φ( X) and Y = φ( Y). We use the canonical form for the contragredient equivalence relation to give a new proof of a theorem of Flanders concerning the relative sizes of the nilpotent Jordan blocks of AB and BA. We present a sufficient condition for the existence of square roots of AB and BA and close with a canonical form for complex orthogonal equivalence.

A canonical form under ϕ-equivalence

Let M n denote the set of n-by- n complex matrices. Two matrices A, B ∈ M n are said to be ϕ-equivalent if there is a nonsingular R ∈ M n such that A = RBϕ( R), where ϕ: M k → M k satisfies: (1) ϕ( ϕ( X)) = X, (2) ϕ( XY) = ϕ( Y) ϕ( X), and (3) either ϕ( X⊕ Y) = ϕ( X)⊕ ϕ( Y) or ϕ( X⊕ Y) = ϕ( Y)⊕ ϕ( X) for all X, Y ∈ M k , k = 1,2,…, n. We show that an almost explicit canonical form of a nonsingular A ∈ M n under ϕ-equivalence can be obtained from a canonical solution of the matrix equation Xϕ(X −1)∈ O (J(Aϕ(A −1)) . We show how explicit canonical forms for nonsingular complex matrices under T-congruence, ∗-congruence, consimilarity, and AT-congruence can be obtained as special cases of this general principle.

Which Real Matrices Have Real Logarithms?

The exponential function exp(x) = L' OXn/n! can be defined in any domain in which the infinite series converges; thus, in particular, it makes sense in any Banach algebra. This note is concerned with characterizing the range of exp in Mn(R), the set of all n X n real matrices, which is the simplest nontrivial real Banach algebra. Let us agree to write log B = A whenever exp A = B. Then we wish to determine which real matrices possess a real logarithm. It is elementary that for the domain R the range of exp is R + = { y > 0) and for the domain the range is C {O}. For the domain Mn(C), consisting of all n X n complex matrices, it is not too difficult to show that the range is GLn(C), the set of all nonsingular complex matrices. That it can be no larger than GLn(C) follows from the Jacobi identity det(exp A) = exp(tr A), a classical result which is also a consequence of remarks below. Standard infinite series expansions for the logarithm can be used to construct log B for certain large open sets in GLn(C). For example, if IIB -II < 1, where denotes any operator norm, then a logarithm can be defined by the series

A determinant formulation of the Cauchy-Schwarz inequality

Let A, B, P, Q be n-square nonsingular complex matrices, and for 1 ⩽ m < n let U and V be n× m complex matrices. For n⩾3 necessary and sufficient conditions are given for the inequality det(U ∗AV) det(V ∗BU)⩽ det(U ∗PU) det(V ∗QV) to hold for all U and V.