Fuzhen Zhang

Quaternions and matrices of quaternions

Abstract We give a brief survey on quaternions and matrices of quaternions, present new proofs for certain known results, and discuss the quaternionic analogues of complex matrices. The methods of converting a quaternion matrix to a pair of complex matrices and homotopy theory are emphasized.

Eigenvalue Inequalities for Matrix Product

We present a family of eigenvalue inequalities for the product of a Hermitian matrix and a positive-semidefinite matrix. Our theorem contains or extends some existing results on trace and eigenvalues

On the Unitary Diagonalisation of a Special Class of Quaternion Matrices

Abstract We propose a unitary diagonalisation of a special class of quaternion matrices, the so-called η -Hermitian matrices A = A η H , η ∈ { i , j , κ } arising in widely linear modelling. In 1915, Autonne exploited the symmetric structure of a matrix A = A T to propose its corresponding factorisation (also known as the Takagi factorisation) in the complex domain C . Similarly, we address the factorisation of an ‘augmented’ class of quaternion matrices, by taking advantage of their structures unique to the quaternion domain H . Applications of such unitary diagonalisation include independent component analysis and convergence analysis in statistical signal processing.

On the eigenvalues of quaternion matrices

This article is a continuation of the article [F. Zhang, Geršgorin type theorems for quaternionic matrices, Linear Algebra Appl. 424 (2007), pp. 139–153] on the study of the eigenvalues of quaternion matrices. Profound differences in the eigenvalue problems for complex and quaternion matrices are discussed. We show that Brauers theorem for the inclusion of the eigenvalues of complex matrices cannot be extended to the right eigenvalues of quaternion matrices. We also provide necessary and sufficient conditions for a complex square matrix to have infinitely many left eigenvalues, and analyse the roots of the characteristic polynomials for 2 × 2 matrices. We establish a characterisation for the set of left eigenvalues to intersect or be part of the boundary of the quaternion balls of Geršgorin.

Some inequalities for the eigenvalues of the product of positive semidefinite Hermitian matrices

Let λ1(A)⩾⋯⩾λn(A) denote the eigenvalues of a Hermitian n by n matrix A, and let 1⩽i1< ⋯ <ik⩽n. Our main results are ∑t=1kλt(GH)⩽∑t=1kλit(G)λn−it+1(H) and ∑t=1kλit(GH)⩽∑t=1kλit(G)λn−t+1(H) . Here G and H are n by n positive semidefinite Hermitian matrices. These results extend Marshall and Olkins inequality ∑t=1kλt(GH)⩽∑t=1kλt(G)λn−t+1(H) . We also present analogous results for singular values.

Schur complements and matrix inequalities of hadamard products

Let A and B be n-square positive definite matrices. Denote the Hadamard product of A and B by A∘ B. The main results of the paper are: 1.For any matrices C and D of size m × n and 2.Let A/α be the Schur complement of A(α) in A. Then . Some other matrix inequalities of Schur complements and Hadamard products of positive definite matrices are also presented.

A generalization of the complex Autonne–Takagi factorization to quaternion matrices

A complex symmetric matrix A can always be factored as A = UΣU T , in which U is complex unitary and Σ is a real diagonal matrix whose diagonal entries are the singular values of A. This factorization may be thought of as a special singular value decomposition for complex symmetric matrices. We present an analogous special singular value decomposition for a class of quaternion matrices that includes complex matrices that are symmetric or Hermitian.

A matrix decomposition and its applications

We show the uniqueness and construction (of the Z matrix in Theorem 2.1, to be exact) of a matrix decomposition and give an affirmative answer to a question proposed in [J. Math. Anal. Appl. 407 (2013) 436-442].

Some inequalities on generalized Schur complements

Abstract This paper presents some inequalities on generalized Schur complements. Let A be an n×n (Hermitian) positive semidefinite matrix. Denote by A/α the generalized Schur complement of a principal submatrix indexed by a set α in A. Let A + be the Moore–Penrose inverse of A and λ(A) be the eigenvalue vector of A. The main results of this paper are: 1. λ(A + (α′))⩾λ((A/α) + ) , where α′ is the complement of α in {1,2,…,n} . 2. λ(A r /α)⩽λ r (A/α) for any real number r⩾1. 3. (C * AC)/α⩽C * /α A(α′) C/α for any matrix C of certain properties on partitioning.

Equivalence of the Wielandt Inequality and the Kantorovich Inequality

This note shows the equivalence of two well-known inequalities: the Wielandt inequality and the Kantorovich inequality.