Topics in Quaternion Linear Algebra

Leiba Rodman
Pages: 384
https://www.jstor.org/stable/j.ctt6wpz0p

1. Front Matter
(pp. i-vi)
(pp. vii-x)
3. Preface
(pp. xi-xiv)
Leiba Rodman
4. Chapter One Introduction
(pp. 1-8)

Besides the introduction, front matter, back matter, and Appendix (Chapter 15), the book consists of two parts. The first part comprises Chapters 2–7. Here, fundamental properties and constructions of linear algebra are explored in the context of quaternions, such as matrix decompositions, numerical ranges, Jordan and Kronecker canonical forms, canonical forms under congruence, determinants, invariant subspaces, etc. The exposition in the first part is on the level of an upper undergraduate or graduate textbook. The second part comprises Chapters 8–14. Here, the emphasis is on canonical forms of quaternion matrix pencils with symmetries or, what is the same,...

5. Chapter Two The algebra of quaternions
(pp. 9-27)

In this chapter we introduce the quaternions and their algebra: multiplication, norm, automorphisms and antiautomorphisms, etc. We give matrix representations of various real linear maps associated with quaternion algebra. We also introduce representations of quaternions as real 4 ☓ 4 matrices and as complex 2 ☓ 2 matrices.

Fix an ordered basis {e, i, j, k} in a 4-dimensional real vector space H (we may take H = R⁴, the vector space of columns consisting of four real components), and introduce multiplication in H by the formulas

ei = ie = i, ej = je = j, ek = ke = k,...

6. Chapter Three Vector spaces and matrices: Basic theory
(pp. 28-63)

We introduce the basic structures in the quaternion vector space${{\text{H}}^{n \times 1}}$and in quaternion matrix algebras, including various types of matrix decompositions and factorizations. In particular, representations of quaternion matrix algebras in terms of real and complex matrix algebra are developed. Numerical ranges, joint numerical ranges, and their convexity properties are emphasized. We also provide an Appendix containing a few basic facts on analysis of sets and continuous multivariable functions that are used in the book.

Consider${{\text{H}}^{n \times 1}}$, the set of alln-component columns with quaternion components, as a right quaternion vector space; in other words, besides the standard addition,...

7. Chapter Four Symmetric matrices and congruence
(pp. 64-82)

In this chapter we develop canonical forms of hermitian and skewhermitian matrices under congruence, in the contexts of the conjugation and of nonstandard involutions. These canonical forms allow us to identify maximal neutral and maximal semidefinite subspaces, with respect to a given hermitian or skewhermitian matrix, in terms of their dimensions.

Definition 4.1.1. Two matrices$A,B \in {{\text{H}}^{n \times n}}$are said to becongruentif$A = S*BS$for some invertible$S \in {{\text{H}}^{n \times n}}$If$\phi$is a nonstandard involution, then we say that$A,B \in {{\text{H}}^{n \times n}}$are$\phi$-congruentif$A = {S_\phi }BS$for some invertibleS.

Clearly, congruence and$\phi$-congruence are equivalence relations. Also, ifAandB

To formulate...

8. Chapter Five Invariant subspaces and Jordan form
(pp. 83-130)

We start the chapter by introducing the notion of root subspaces for quaternion matrices. These are basic invariant subspaces, and we prove in particular that they enjoy the Lipschitz property with respect to perturbations of the matrix. Another important class of invariant subspaces are the 1-dimensional ones, i.e., generated by eigenvectors. Existence of quaternion eigenvalues and eigenvectors is proved, which leads to the Schur triangularization theorem (in the context of quaternion matrices) and its many consequences familiar for real and complex matrices. Jordan canonical form for quaternion matrices is stated and proved (both the existence and uniqueness parts) in full...

9. Chapter Six Invariant neutral and semidenite subspaces
(pp. 131-152)

In this chapter we study subspaces that are simultaneously neutral or semidefinite for one matrix and invariant for another. The presentation here does not use canonical forms for pairs of matrices, which is developed in later chapters.

In this and subsequent chapters we will often work with matrices that enjoy certain symmetry properties. For the reader’s convenience, we collect the nomenclature associated with classes of such matrices$A \in {{\text{H}}^{n \times n}}$, where F = R, F = C, or F = H, in three tables.

The notation$B \ge C$or$C \le B$, where$B,C \in {F^{n \times n}}$are hermitian matrices, indicates that the difference$B - C$is positive semidefinite....

10. Chapter Seven Smith form and Kronecker canonical form
(pp. 153-171)

In this chapter we study polynomials with quaternion matrix coeffcients. The exposition is focused on two major results. One is the Smith form, which asserts that every quaternion matrix polynomial can be brought to a diagonal form under pre- and postmultiplication by unimodular matrix polynomials, with the appropriate divisibility relations among the diagonal entries. The other is the Kronecker canonical form for quaternion matrix polynomials of first degree under pre- and postmultiplication by invertible constant matrices. The Kronecker form generalizes the Jordan canonical for matrices. Complete and detailed proofs are given for both the Smith form and the Kronecker form....

11. Chapter Eight Pencils of hermitian matrices
(pp. 172-193)

The main theme in this and the next chapter is the canonical forms of pairs of quaternion matrices, or matrix pencils, under simultaneous congruence, where the matrices are either hermitian or skewhermitian. For convenience of exposition, the material is divided between two chapters: In the current chapter the case of two hermitian matrices is studied, whereas the next chapter is devoted to pairs of skewhermitian matrices and to mixed pairs, when one of the matrices if hermitian and the other is skewhermitian. We also present here applications of the canonical form to the problems of existence of positive definite or...

12. Chapter Nine Skewhermitian and mixed pencils
(pp. 194-227)

Here we present canonical forms under strict equivalence and under congruence for pencils of quaternion matricesA+tB, where one of the matricesAandBis skewhermitian and the other can be hermitian or skewhermitian. We treat the case when bothAandBare skewhermitian in Section 9.1 (this is the relatively easy case). The case when one ofAandBis hermitian and the other is skewhermitian is treated in subsequent sections. Comparisons are made with the corresponding results for real and complex matrix pencils. We give an application to the canonical form of a quaternion matrix...

13. Chapter Ten Indefinite inner products: Conjugation
(pp. 228-260)

In this chapter we study inde_nite inner products de_ned on${H^{n \times 1}}$of the hermitianand skewhermitian-types and matrices having symmetry properties with respect to one of these indefinite inner products. The hermitian-type inner product is a function

$[ \cdot , \cdot ]:{H^{n \times 1}} \times {H^{n \times 1}} \to H$

with the following properties:

(1) Linearity in the _rst argument:$\left[ {{x_1}{\alpha _1} + {x_{_2}}{\alpha _2},y} \right] = \left[ {{x_1},y} \right]{\alpha _1} + \left[ {{x_2},y} \right]{\alpha _2}$for all${x_1},{x_2},y \in {H^{n \times 1}}$and all${\alpha _1},{\alpha _2} \in H.$

(2) Symmetry:$\left[ {x,y} \right] = {\left[ {y,x} \right]^*}$for all$x,y \in {H^{n \times 1}}$.

(3) Nondegeneracy: if${x_0} \in {H^{n \times 1}}$is such that$\left[ {{x_0},y} \right] = 0$for all$y \in {H^{n \times 1}}$, then${x_0} = 0$.

The skewhermitian-type inner product [.,.] is defined by properties (1), (3), and

(2´) antisymmetry:$\left[ {x,y} \right] = - {\left[ {y,x} \right]^*}$for all$x,y \in {H^{n \times 1}}$.

It follows from (1) and (2), or from (1)...

14. Chapter Eleven Matrix pencils with symmetries: Nonstandard involution
(pp. 261-278)

In this chapter the subject matter involves quaternion matrix pencils or, equivalently, pairs of quaternion matrices, with symmetries with respect to a fixed nonstandard involution$\phi$. Here, we provide canonical forms for$\phi$-hermitian pencils, i.e., pencils of the form$A + tB$, whereAandBare both$\phi$-hermitian. We also provide canonical forms for$\phi$-skewhermitian pencils. The canonical forms in question are with respect to either strict equivalence of pencils or to simultaneous$\phi$-congruence of matrices. Applications are made to joint$\phi$-numerical ranges of two$\phi$-skewhermitian matrices and to the corresponding joint$\phi$-numerical cones.

We fix a nonstandard involution$\phi$throughout...

15. Chapter Twelve Mixed matrix pencils: Nonstandard involutions
(pp. 279-299)

The canonical forms of mixed quaternion matrix pencils, i.e., such that one of the two matrices is$\phi$-hermitian and the other is$\phi$-skewhermitian, are also studied here with respect to simultaneous$\phi$-congruence. Other canonical forms of mixed matrix pencils are developed with respect to strict equivalence. As an application, we provide canonical forms of quaternion matrices under$\phi$-congruence.

As in the preceding chapter, we fix a nonstandard involution$\phi$throughout this chapter and a quaternion$\beta (\phi )$such that$\phi (\beta (\phi )) = - \beta (\phi )$and$|\beta (\phi )| = 1$.

Definition 12.1.1. A matrix pencilA+tB, where$A,B \in {H^{n \times n}}$, is said to be$\phi$-hermitian-skewhermitian,in short...

16. Chapter Thirteen Indefinite inner products: Nonstandard involution
(pp. 300-327)

In this chapter we fix a nonstandard involution$\phi$.

In parallel with Chapter 10, we introduce indefnite inner products defined on${H^{n \times 1}}$of the symmetric and skewsymmetric types associated with$\phi$and matrices having symmetry properties with respect to one of these indefinite inner products. The symmetric-type inner product is a function

${[ \cdot , \cdot ]^{(\phi )}}:{H^{n \times 1}} \times {H^{n \times 1}} \to H$

(the superscript$^{(\phi )}$indicates that the inner product is associated with$\phi$, in contrast with the inner product of Chapter 10) with the following properties:

$(1')$Linearity in the first argument:

${[{x_1}\alpha 1 + {x_2}{\alpha _2},y]^{(\phi )}} = {[{x_1},y]^{(\phi )}}\alpha 1 + {[{x_2},y]^{(\phi )}}{\alpha _2}$

for all${x_1},{x_2},y \in {H^{n \times 1}}$and all$\alpha 1,\alpha 2 \in H$.

$(2')$Symmetry:${[x,y]^{(\phi )}} = \phi ({[y,x]^{(\phi }})$for all$x,y \in {H^{n \times 1}}$.

$(3')$Nondegeneracy: if...

17. Chapter Fourteen Matrix equations
(pp. 328-338)

Here, we present applications to polynomial matrix equations, algebraic Riccati equations, and linear quadratic regulators. Without attempting to develop indepth exposition of the topics (this would take us too far afield), we present these applications in basic forms. Maximal invariant semidefinite or neutral subspaces will play a key role.

The approach to studying polynomial equations using companion matrices of Section 5.12 extends to polynomial matrix equations. Consider the matrix equation

${Z^n} + {A_{n - 1}}{Z^{n - 1}} - \cdots + {A_1}Z + {A_0} - 0$, (14.1.1)

where${A_0}, \ldots ,{A_{n - 1}} \in {H^{m \times m}}$are given and$Z \in {H^{m \times m}}$is the unknown matrix. LetCbe theblock companion matrixcorresponding to equation (14.1.1):

$\left( {\begin{array}{*{20}{c}} 0 & {{I_m}} & 0 & 0 & \cdots & 0 \\ 0 & 0 & {{I_m}} & 0 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \cdots & \vdots \\ 0 & 0 & 0 & 0 & \cdots & {{I_m}} \\ { - {A_0}} & { - {A_1}} & { - {A_2}} & { - {A_3}} & \cdots & { - {A_{n - 1}}} \\ \end{array}} \right) \in {H^{(mn) \times (mn)}}$

Theorem 14.1.1. There exists a one-to-one...

18. Chapter Fifteen Appendix: Real and complex canonical forms
(pp. 339-352)

For the reader’s convenience, we state here (without proof, but with references) canonical forms for real and complex matrices and for pairs of real and complex matrices, or matrix pencils, with symmetries. All these forms are known, and most are well-known. Our main sources for the material in this chapter are Gantmacher [48, 49] and expository papers by Thompson [151] and Lancaster and Rodman [93, 92]; Thompson [151] also contains an extensive bibliography.

For Jordan forms, we use the complex and real Jordan blocks (1.2.1) and (1.2.2).

Theorem 15.1.1.Let$F = R$or$F = C$.Let$A \in {F^{n \times n}}$.Then there exists an invertible...

19. Bibliography
(pp. 353-360)
20. Index
(pp. 361-363)
21. Back Matter
(pp. 364-364)