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    Topics in Quaternion Linear Algebra
    Topics in Quaternion Linear Algebra
    Topics in Quaternion Linear Algebra
    Ebook763 pages5 hours

    Topics in Quaternion Linear Algebra

    By Leiba Rodman

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    Quaternions are a number system that has become increasingly useful for representing the rotations of objects in three-dimensional space and has important applications in theoretical and applied mathematics, physics, computer science, and engineering. This is the first book to provide a systematic, accessible, and self-contained exposition of quaternion linear algebra. It features previously unpublished research results with complete proofs and many open problems at various levels, as well as more than 200 exercises to facilitate use by students and instructors. Applications presented in the book include numerical ranges, invariant semidefinite subspaces, differential equations with symmetries, and matrix equations.

    Designed for researchers and students across a variety of disciplines, the book can be read by anyone with a background in linear algebra, rudimentary complex analysis, and some multivariable calculus. Instructors will find it useful as a complementary text for undergraduate linear algebra courses or as a basis for a graduate course in linear algebra. The open problems can serve as research projects for undergraduates, topics for graduate students, or problems to be tackled by professional research mathematicians. The book is also an invaluable reference tool for researchers in fields where techniques based on quaternion analysis are used.

    LanguageEnglish
    PublisherPrinceton University Press
    Release dateAug 24, 2014
    ISBN9781400852741
    Topics in Quaternion Linear Algebra
    Author

    Leiba Rodman

    Leiba Rodman is professor of mathematics at the College of William & Mary. His books include Matrix Polynomials, Algebraic Riccati Equations, and Indefinite Linear Algebra and Applications.

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      Topics in Quaternion Linear Algebra - Leiba Rodman

      Algebra

      Chapter One

      Introduction

      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, pairs of matrices with symmetries, and the exposition approaches that of a research monograph. Applications are given to systems of linear differential equations with symmetries, and matrix equations.

      The mathematical tools used in the book are easily accessible to undergraduates with a background in linear algebra and rudiments of complex analysis and, on occasion, multivariable calculus. The exposition is largely based on tools of matrix analysis. The author strived to make the book self-contained and inclusive of complete proofs as much as possible, at the same time keeping the size of the book within reasonable limits. However, some compromises were inevitable here. Thus, proofs are often omitted for many linear algebra results that are standard for real and complex matrices, are often presented in textbooks, and are valid for quaternion matrices as well with essentially the same proofs.

      The book can be used in a variety of ways. More than 200 exercises are provided, on various levels of difficulty, ranging from routine verification of facts and numerical examples designed to illustrate the results to open-ended questions. The exercises and detailed exposition make the book suitable in teaching as supplementary material for undergraduate courses in linear algebra, as well as for students’ independent study or reading courses. For students’ benefit, several appendices are included that contain background material used in the main text. The book can serve as a basis for a graduate course named advanced linear algebra, topics in linear algebra, or (for those who want to keep the narrower focus) quaternion linear algebra. For example, one can build a graduate course based on Chapters 2–8 and selections from later chapters.

      Open problems presented in the book provide an opportunity to do original research. The open problems are on various levels: open-ended problems that may serve as subject for research by mathematicians and concrete, more-specific problems that are perhaps more suited for undergraduate research work under faculty supervision, honors theses, and the like.

      For working mathematicians in both theoretical and applied areas, the book may serve as a reference source. Such areas include, in particular, vector calculus, ordinary and partial differential equations, and boundary value problems (see, e.g., Gürlebeck and Sprössig [60]), and numerical analysis (Bunse-Gerstner et al. [22]). The accessibility and importance of the mathematics should make this book a widely useful work not only for mathematicians, but also for scientists and engineers.

      Quaternions have become increasingly useful for practitioners in research, both in theory and applications. For example, a significant number of research papers on quaternions, perhaps even most of them, appear regularly in mathematical physics journals, and quantum mechanics based on quaternion analysis is mainstream physics. In engineering, quaternions are often used in control systems, and in computer science they play a role in computer graphics. Quaternion formalism is also used in studies of molecular symmetry. For practitioners in these areas, the book can serve as a valuable reference tool.

      New, previously unpublished results presented in the book with complete proofs will hopefully be useful for experts in linear algebra and matrix analysis. Much of the material appears in a book for the first time; this is true for Chapters 5–14, most of Chapter 4, and a substantial part of Chapter 3.

      As far as the author is aware, this is the first book dedicated to systematic exposition of quaternion linear algebra. So far, there are only a few expository papers and chapters in books on the subject (for example, Chapter 1 in Gürlebeck and Sprössig [60], Brieskorn [20], Zhang [164], or Farenick and Pidkowich [38]) as well as algebraic treatises on skew fields (e.g., Cohn [29] or Wan [156]).

      It is inevitable that many parts of quaternion linear algebra are not reflected in the book, most notably those parts pertaining to numerical analysis (Bunse-Gerstner et al. [22] and Faßbender et al. [40]). Also, the important classes of orthogonal, unitary, and symplectic quaternion matrices are given only brief exposure.

      We now describe briefly the contents of the book chapter by chapter.

      Chapter 2 concerns (scalar) quaternions and the basic properties of quaternion algebra, with emphasis on solution of equations such as axb = c and ax − xb = c. Description of all automorphisms and antiautomoprhisms of quaternions is given, and representations of quaternions in terms of 2 × 2 complex matrices and 4 × 4 real matrices are introduced. These representations will play an important role throughout the book.

      Chapter 3 covers basics on the vector space of columns with quaternion components, matrix algebra, and various matrix decomposition. The real and complex representations of quaternions are extended to vectors and matrices. Various matrix decompositions are studied; in particular, Cholesky factorization is proved for matrices that are hermitian with respect to involutions other than the conjugation. A large part of this chapter is devoted to numerical ranges of quaternion matrices with respect to conjugation as well as with respect to other involutions. Finally, a brief exposition is given for the set of quaternion subspaces, understood as a metric space with respect to the metric induced by the gap function.

      In a short Chapter 4 we develop diagonal canonical forms and prove inertia theorems for hermitian and skewhermitian matrices with respect to involutions (including the conjugation). We also identify dimensions of subspaces that are neutral or semidefinite relative to a given hermitian matrix and are maximal with respect to this property. The material in Chapters 3 and 4 does not depend on the more involved constructions such as the Jordan form and its proof.

      Chapter 5 is a key chapter in the book. Root subspaces of quaternion matrices are introduced and studied. The Jordan form of a quaternion matrix is presented in full detail, including a complete proof. The complex matrix representation plays a crucial role here. Although the standard definition of a determinant is not very useful when applied to quaternion matrices, nevertheless several notions of determinant-like functions for matrices over quaternions have been defined and used in the literature; a few of these are explored in this chapter as well. Several applications of the Jordan form are treated. These include matrix equations of the form AX − XB = C, functions of matrices, and stability of systems of differential equations of the form

      with constant quaternion matrix coefficients Aℓ, …, A0. Stability of an analogous system of difference equations is studied as well.

      The main theme of Chapter 6 concerns subspaces that are simultaneously invariant for one matrix and semidefinite (or neutral) with respect to another. Such subspaces show up in many applications, some of them presented later in Chapter 13. For a given invertible plus-matrix A, the main result here asserts that any subspace which is A-invariant and at the same time nonnegative with respect to the underlying indefinite inner product can be extended to an A-invariant subspace which is maximal nonnegative. Analogous results are proved for related classes of matrices, such as unitary and dissipative, as well as in the context of indefinite inner products induced by involutions other than the conjugation.

      Chapter 7 treats matrix polynomials with quaternion coefficients. A diagonal form (known as the Smith form) is proved for such polynomials. In contrast to matrix polynomials with real or complex coefficients, a Smith form is generally not unique. For matrix polynomials of first degree, a Kronecker form—the canonical form under strict equivalence—is available, which is presented with a complete proof. Furthermore, a comparison is given for the Kronecker forms of complex or real matrix polynomials with the Kronecker forms of such matrix polynomials under strict equivalence using quaternion matrices.

      In Chapters 8, 9, and 10 we develop canonical forms of quaternion matrix pencils A + tB in which the matrices A and B are either hermitian or skewhermitian and their applications. Chapter 8 is concerned with the case when both matrices A and B are hermitian. Full and detailed proofs of the canonical forms under strict equivalence and simultaneous congruence are provided, based on the Kronecker form of the pencil A + tB. Several variations of the canonical forms are included as well. Among applications here: criteria for existence of a nontrivial positive semidefinite real linear combination and sufficient conditions for simultaneous diagonalizability of two hermitian matrices under simultaneous congruence. A comparison is made with pencils of real symmetric or complex hermitian matrices. It turns out that two pencils of real symmetric matrices are simultaneously congruent over the reals if and only if they are simultaneously congruent over the quaternions. An analogous statement holds true for two pencils of complex hermitian matrices.

      The subject matter of Chapter 9 is concerned mainly with matrix pencils of the form A + tB, where one of the matrices A or B is skewhermitian and the other may be hermitian or skewhermitian. Canonical forms of such matrix pencils are given under strict equivalence and under simultaneous congruence, with full detailed proofs, again based on the Kronecker forms. Comparisons with real and complex matrix pencils are presented. In contrast to hermitian matrix pencils, two complex skewhermitian matrix pencils that are simultaneously congruent under quaternions need not be simultaneously congruent under the complex field, although results hold for real or complex matrix pencils A+tB, where A is real symmetric or complex hermitian and B is real skewsymmetric or complex skewhermitian. In each case, we sort out the relationships of simultaneous congruence over the complex field of complex matrix pencils where one matrix is hermitian and the other is skewhermitian versus simultaneous congruence over the skew field of quaternions for such pencils. As an applications we obtain a canonical form for quaternion matrices under (quaternion) congruence.

      In Chapter 10 we study matrices (or linear transformations) that are selfadjoint or skewadjoint with respect to a nondegenerate hermitian or skewhermitian inner product. As an application of the canonical forms obtained in Chapters 8 and 9, canonical forms for such matrices are derived. Matrices that are skewadjoint with respect to skewhermitian inner products are known as Hamiltonian matrices; they play a key role in many applications such as linear control systems (see Chapter 14). The canonical forms allow us to study invariant Lagrangian subspaces; in particular, they give criteria for existence of such subspaces. Another application involves boundedness and stable boundedness of linear systems of differential equations with constant coefficients under suitable symmetry requirements.

      The development of material in Chapters 11, 12, and 13 is largely parallel to that in Chapters 8, 9, and 10, but with respect to an involution of the quaternions other than the conjugation and with respect to indefinite inner products induced by matrices that are hermitian or skewhermitian with respect to such involutions. Thus, letting ϕ be a fixed involution of the quaternions which is different from the conjugation, the canonical forms (under both strict equivalence and simultaneous ϕ-congruence) of quaternion matrix pencils A+tB, where each of A and B is either ϕ-hermitian or ϕ-skewhermitian, are given in Chapters 11 and 12. As before, full and detailed proofs are supplied.

      Applications are made in Chapter 14 to various types of matrix equations over quaternions, such as

      where A0, …, Am−1 are given n × n quaternion matrices,

      where A, B, C, and D are given quaternion matrices of suitable sizes, and the symmetric version of the latter equation,

      where D and C are assumed to be hermitian. The theory of invariant subspaces of quaternion matrices—and for equation (1.0.1) also of subspaces that are simultaneously invariant and semidefinite—plays a crucial role in study of these matrix equations. Equation (1.0.1) and its solutions, especially hermitian solutions, are important in linear control systems. A brief description of such systems and their relation to equations of the type of (1.0.1) is also provided.

      For the readers’ benefit, in Chapter 15 we bring several well-known canonical forms for real and complex matrices that are used extensively in the text. No proofs are given; instead we supply references that contain full proofs and further bibliographical information.

      1.1 NOTATION AND CONVENTIONS

      Numbers, sets, spaces

      A := B—the expression or item A is defined by the expression or item B

      R—the real field

      ⌊x⌋—the greatest integer not exceeding x ∈ R

      C—the complex field

      I(z)=(z )/(2i) ∈ R—the imaginary part of a complex number z

      C+ — closed upper complex half-plane

      D∊(λ) := {z ∈ C+ : |z − λ| < ∊}—part of the open circular disk centered at λ with radius ∊ that lies in C+

      C+,0—open upper complex half-plane

      H—the skew field of the quaternions

      i, j, k—the standard quaternion imaginary units

      R(x)= x0 and V(x)= x1i + x2j + x3k—the real and the vector part of x, respectively, for x = x0 + x1i + x2j + x3k ∈ H, where x0, x1, x2, x3 ∈ R

      —the length of x ∈ H

      Inv (ϕ) := {x ∈ H : ϕ(x) = x}—the set (real vector space) of quaternions invariant under an involution ϕ of H

      β(ϕ) ∈ H—quaternion with the properties that ϕ(β(ϕ)) = −β(ϕ) and |β(ϕ)| = 1, where ϕ is an involution of H that is different from the quaternion conjugation; for a given ϕ, the quaternion β(ϕ) ∈ H is unique up to negation

      Con (α)= {y*αy : y ∈ H \{0}}—the congruence orbit of α ∈ H

      Sim (α)= {y−1αy : y ∈ H \{0}}—the similarity orbit of α ∈ H

      Fn×1—the vector space of n-components columns with components in F, where F = R, F = C, or F = H; Hn×1 is understood as a right quaternion vector space

      ej ∈ Hn×1—the vector with 1 in the jth component and zero elsewhere; n is understood from context

      〈x, y〉 := y*x, x, y ∈ Hn×1—the standard inner product defined on Hn×1

      or ∥x∥—the norm of x ∈ Hn×1

      Fm×n—the vector space of m×n matrices with entries in F, where F = R, F = C, or F = H, and Hm×n is understood as a left quaternion vector space; if m = n, then Cn×n is a complex algebra, whereas Rn×n and Hn×n are real algebras

      Subspaces

      PM ∈ Fn×n—the orthogonal projection onto the subspace M ⊆ Fn×1; here F = R, F = C, or F = H

      SM := {x ∈M; ∥x∥ = 1}—the unit sphere of a nonzero subspace M

      A|R—restriction of a square-size matrix A to its invariant subspace R (we represent A|R as a matrix with respect to some basis in R)

      N—direct sum of subspaces M and N

      Span {x1, …, xp} or SpanH {x1, …, xp}—the quaternion subspace spanned by vectors x1, …, xp ∈ Hn×1

      SpanR {x1, …, xp}—the real subspace spanned by vectors x1, …, xp ∈ Hn×1

      dimH M or dim M—the (quaternion) dimension of a quaternion vector space M

      Grassn—the set of all (quaternion) subspaces in Hn×1

      θ(M, N)= ∥PM − PN ∥—the gap between subspaces M and N

      Matrix-related notation

      In or I (with n understood from context)—n × n identity matrix

      0u×v, abbreviated to 0u, if u = v—the u × v zero matrix, also 0 (with u and v understood from context)

      C-eigenvalues of A—for a square-size complex matrix A, defined as the (complex) roots of the characteristic polynomial of A, and σC(Awe have

      AT—transposed matrix

      A*—conjugate transposed matrix

      —the matrix obtained from A ∈ Cm×n or A ∈ Hm×n by replacing each entry with its complex or quaternion conjugate

      Aϕ ∈ Hn×m, where ϕ is an involution of H

      Ran A = {Ax : x ∈ Fn×1} ⊆ Fm×1—the image or range of A ∈ Fm×n; here F ∈{R, C, H} (understood from context)

      Ker A = {x ∈ Fn×1 : Ax = 0}—the kernel of A ∈ Fm×n; here F ∈{R, C, H}(understood from context)

      ∥A∥H or ∥A∥—the norm of A ∈ Hm×n; it is taken to be the largest singular value of A

      rank A—the (quaternion) rank of a matrix A ∈ Hm×n; if A is real or complex, then rank A coincides with the rank of A as a real or complex matrix

      B ≥ C or C ≤ B—for hermitian matrices B, C ∈ Hn×n, indicates that the difference B − C is positive semidefinite

      In+(A), In−(A), In0(A)—the number of positive, negative, or zero eigenvalues of a quaternion hermitian matrix A, respectively, counted with algebraic multiplicities

      (In+ (H), In− (H), In0 (H))—the β(ϕ)-inertia, or the β(ϕ)-signature, of a ϕ-skewhermitian matrix H ∈ Hn×n; here ϕ is an involution of H different from the quaternion conjugation

      —block diagonal matrix with the diagonal blocks X1, X2, …, Xk (in that order)

      —block row matrix

      —block column matrix

      X⊕m—X ∈ Hδ1×δ2, the mδ1 × mδ2 matrix X ⊕…⊕ X, where X is repeated m times

      1.2 STANDARD MATRICES

      In this section we collect matrices in standard forms and fixed notation that will be used throughout the book, sometimes without reference. The subscript in notation for a square-size matrix will always denote the size of the matrix.

      Ir or I (with r understood from context)—the r × r identity matrix

      0u×v—often abbreviated to 0u; if u = v or 0 (with u and v understood from context), the u × v zero matrix

      Jordan blocks:

      real Jordan blocks:

      symmetric matrices:

      We also define

      where α ∈ H, and

      where β ∈ H. Note that

      in particular Ξm(α)=(−1)m(Ξm(α))* if and only if the real part of α is zero.

      Note that Y2m is real symmetric.

      Real matrix pencils—

      where µ ∈ R, ν ∈ R \ {0}.

      Singular matrix pencils—

      Here ε is a positive integer; Lε×(ε+1)(t) is of size ε × (ε + 1).

      Chapter Two

      The algebra of quaternions

      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.

      2.1 BASIC DEFINITIONS AND PROPERTIES

      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

      and by the requirement that the multiplication of elements of H is distributive with respect to addition and commutes with scalar multiplication:

      for all x, y, z ∈ H and all λ ∈ R.

      Definition 2.1.1. The elements of H with the algebraic operations of H as a real vector space and with the multiplication introduced as above are called the (real) quaternions.

      The letter H stands for William Rowan Hamilton (1805–1865), inventor of quaternions. Clearly, the multiplication in the algebra H is noncommutative.

      Proposition 2.1.2. H is a unital associative algebra with the unity e:

      for all x, y, z ∈ H.

      In the sequel we identify the real number λ with the quaternion λe; in particular, 1 stands for 1e. Also, it is easy to see that the real span of 1 and i is isomorphic (as a subalgebra of H) to C; thus, we identify, when convenient, C with the subalgebra of H spanned (as a real vector space) by 1 and i.

      Definition 2.1.3. For a quaternion x = x0 + x1i + x2j + x3k, where x0, x1, x2, x3 ∈ R, we define R(x)= x0, the real part of x, and V(x)= x1i + x2j + x3k, the vector part (or imaginary part) of x. The conjugate quaternion of x is defined by x0 − x1i − x2j − x3k = R(x) − V(xor x*. The norm of x is

      . We say that x ∈ H is a unit quaternion if |x| = 1.

      Some elementary properties of the algebra of quaternions are listed below.

      Proposition 2.1.4. Let x, y ∈ H. Then:

      1. x*x = xx*;

      2. |x| = |x*|;

      3. | · | is indeed a norm on H; in more detail, for all x, y ∈ H we have:

      |x|≥ 0 with equality if and only if x = 0;

      for every c ∈ C;

      5. (xy)* = y*x*;

      6. x = x* if and only if x ∈ R;

      7. if a ∈ H, then ax = xa for every x ∈ H if and only if a ∈ R;

      8. every x ∈ H \{0} has an inverse x= x*/|x|² ∈ H; in more detail,

      9. |x−1| = |x|−1 for every x ∈ H \{0};

      10. x ∈ H and x* are solutions of the following quadratic equation with real coefficients: t² − 2R(x)t + |x|² = 0;

      11. Cauchy-Schwarz-type inequality is max{|R(xy)|, |V(xy)|} ≤ |x| · |y|;

      12. R(xy)= R(yx) for all x, y ∈ H;

      13. if R(x) = 0, then x= −|x|².

      We indicate a proof of |xy| = |x| · |y|:

      Thus, H is a division ring, i.e., a unital ring in which every nonzero element has a multiplicative inverse, and also a 4-dimensional algebra over the field of real numbers R.

      Note that the multiplication of quaternions with zero real parts can be expressed in terms of the usual inner product and cross product of vectors in R³, namely, if x = x1i + x2j + x3k, y = y1i + y2j + y3k ∈ H, where xℓ, yℓ ∈ R, then

      where

      and where in the right-hand side of (2.1.1) × denotes the cross product (also known as vector product) of vectors in R³×¹:

      The verification of (2.1.1) is straightforward. More generally, let

      and define px, py by (2.1.2). Then

      2.2 REAL LINEAR TRANSFORMATIONS AND EQUATIONS

      For fixed a, b ∈ H, the map x ↦ axb is obviously a real linear transformation on H. We give the matrix form of this transformation with respect to the (ordered) basis 1, i, j, k of H as a real vector space.

      Theorem 2.2.1. Let

      where aj, bj ∈ R for j = 0, 1, 2, 3. Let

      be a real linear transformation. Then Ta, b is given by the following matrix with respect to the ordered real basis {1,i, i, k} in H:

      The proof is obtained by tedious but straightforward computation.

      The following particular cases are of interest.

      Corollary 2.2.2. The real linear transformations T1, b, Ta,1, Ta, a*, and Ta, a−1 (in the latter case it is assumed a ≠ 0)are given by the following matrices, respectively, with respect to the ordered real basis {1, i, j, k} of H and using the notation of Theorem 2.2.1:

      and , where X is the matrix (2.2.2).

      The statement of Corollary 2.2.2 concerning Ta, a−1 follows from the observation that Ta, a−1 = |a|−2Ta, a*. Note that the matrices corresponding to Tb,1, resp. to T1, a, are skewsymmetric if and only if R(b) = 0, resp. R(a) = 0, whereas the matrices corresponding to Ta, a* and Ta, a−1 are symmetric if and only if R(a) = 0 or V(a) = 0.

      Corollary 2.2.3. Let a = a0 + a1i + a2j + a3k ∈ H, a0, a1, a2, a3 ∈ R. Then the real linear transformation T1, a − Ta,1 that maps x ∈ H to xa − ax is given by the skewsymmetric matrix

      with respect to the ordered real basis {1, i, j, k}.

      Observe that for a = a0 + a1i + a2j + a3k ∈ H \{0}, where a0, a1, a2, a3 are real, the matrix

      is orthogonal, i.e., UT U = I. A straightforward computation will verify this assertion. Moreover, det U = 1. Indeed, the set of all nonzero quaternions is connected, and det U is a continuous function of the components of a. Therefore, the values of det U also form a connected set (Theorem 3.10.7). But determinants of real orthogonal matrices can be only 1 or −1. It follows that either det U = 1 for all U, or det U = −1 for all U. Since for a = 1 we have U = I, the second possibility cannot happen.

      We obtain that 1 is an eigenvalue of U, and the corresponding eigenvector is unique up to scaling (apart from the case U = I). So, in a suitable orthonormal basis in R³, the matrix U has the form

      Comparing with Corollary 2.2.2, the following geometric description of the transformation Ta, a–1 is obtained. In this description, H0 stands for the real vector space of all quaternions with zero real parts, and orthogonality in H0 is understood in the sense of the real-valued inner product that has i, j, k as an orthonormal basis.

      Corollary 2.2.4. Let a ∈ H \{0}, and assume Ta, a–1≠I. Then Ta, a–1 maps H0 onto itself. Moreover, there is a unique (up to scaling) nonzero x0 ∈ H0 such that Ta, a−1 x0 = x0, and denoting by SpanR{x0} the 2-dimensional plane in H0 orthogonal to x0, we have that Ta, a–1 acts as a rotation through a fixed angle µ, 0 < µ < 2π, in SpanR{x0}.

      Definition 2.2.5. We say that two quaternions x, y are similar if axa−1 = y for some a ∈ H \{0} and congruent if axa* = y for some a ∈ H\{0}.

      Clearly, both similarity and congruence are equivalence relations. Denote by

      and

      the similarity orbit and the congruence orbit of x ∈ H, respectively.

      We have

      Indeed, this follows from the formula a* = |a|²a–1, a∈ H \{0}, with λ = |a|².

      Theorem 2.2.6. Fix x = x0 + x1i + x2j + x3k ∈ H, where xj ∈ R. The following statements are equivalent for y = y0 + y1i + y2j + y3k ∈ H, yj ∈ R:

      (1)  y ∈ Sim (x);

      (2)  y = axa* for some unit quaternion a;

      (3)  

      for some 3 × 3 real orthogonal matrix Q;

      (4)  

      for some 3 × 3 real orthogonal matrix Q′ having determinant 1;

      (5)  R(y)= R(x) and |V(y)| = |V(x)|.

      Proof. (1) ⇒ (2): If y = bxb−1, b ∈ H \{0}, then (2) holds with a = b/|b|.

      (2) ⇒ (4): Follows from Corollary 2.2.2 and Ex. 2.7.7.

      (4) ⇒ (3): Obvious.

      (3) ⇒ (5): Follows from the isometric property of real orthogonal matrices, i.e., ∥Qu∥ = ∥u∥ for all real orthogonal Q ∈ Rn×n and all u ∈ Rn×1.

      (5) ⇒ (1): By hypothesis, y0 = x. Consider the equation

      with real unknowns z0, z1, z2, z3. Equating the coefficients of each of 1, i, j, k in the left and the right sides of (2.2.3), we see that (2.2.3), after some simple algebra, boils down to the system of equations

      We claim that the matrix in the left-hand side of (2.2.4), call it X, so, unless

      the matrix X is singular. But if (2.2.5) holds, X is easily seen to be singular as well. Thus, (2.2.3) has a nontrivial solution, and (1) follows.

      Theorem 2.2.6 allows us to express the similarity orbits of quaternions in a more detailed way:

      where

      Geometrically, S is the unit sphere in R³×¹

      2.3 THE SYLVESTER EQUATION

      Let a, b ∈ H. In this section we study the Sylvester equation

      and the corresponding real linear transformation

      In what follows, we make use of the inner product.

      Definition 2.3.1. The real-valued inner product of two quaternions is defined by

      Note that \x, x/ = |x|² for every x ∈ H. Also, for x, y ∈ H with zero real parts, we have \x, xy/ = \y, xy/ = 0, as can be easily verified using formula (2.1.1).

      For given a, b ∈ H, define the quaternions {x+, y+, x−, y−} as follows:

      (i)  If V(a) and V(b) are linearly independent over R, we set

      where

      Note that in view of the Cauchy-Schwarz inequality applied to V(a) and V(b) (interpreted as vectors in R³) and the linear independence of V(a) and V(b), we have |V(a)||V(b)|±\V(a), V(b)/ > 0.

      (ii) Suppose V(a) and V(b) are linearly dependent over R. Then there exists q ∈ H with R(q) = 0, |q| = 1, V(a)= |V(a)|q, and V(b)= |V(b)|q or V(b)= −|V(b)|q| = 1 and 〈q〉 = 0. If V(b)= |V(b)|q, we define

      If V(b)= −|V(b)|q, we define

      Note that q are not unique (for given a and b); more precisely, q is unique if V(a) ≠ 0 and is unique up to negation if V(b) ≠V(a) = 0.

      Furthermore, we define the subspaces

      are uniquely determined by a and b (unless V(a)= V(b) = 0), see Ex. 2.7.25. Moreover, we have

      if at least one of a and b is nonreal.

      Finally, we set

      for all ξ ∈ R²×¹, where we have used the euclidean norm in R²×¹.

      The main result of this section gives explicitly the orthonormal basis that reduces all three linear transformations Ta,1, T1, b (defined in (2.2.1)) and Sa, b to a real Jordan form.

      Theorem 2.3.2. (a) The vectors x+, y+, x−, y− form an orthonormal basis (with respect to 〈·, ·〉) of H.

      (b) The equalities

      and

      hold true.

      (c) The subspaces and are both Ta,1- and T1, b-invariant.

      (d) The equality

      holds true.

      Proof. Part (a) is established by a straightforward but tedious computation. Parts (c) and (d) are immediate from (b) as Sa, b = Ta,1 − T1, b. The verification of (b) is straightforward (Proposition 2.1.4(13) is used repeatedly).

      We can read off many important properties of Sa, b from Theorem 2.3.2, such as the following.

      Theorem 2.3.3. Let a, b ∈ H, and Sa, b defined by (2.3.1). Then:

      (1)  the four singular values of Sa, b are

      moreover, |Sa, b(x)| = σ4|x| for , and |Sa, b(x)| = σ1|x| for ;

      (2) Sa, b is singular if and only if R(a)= R(b) and |V(a)| = |V(b)|. If these conditions hold and a, b ∉R, then

      (3)  Sa, b has a real eigenvalue (which then is R(a) −R(b)) if and only if |V(a)| = |V(b)| and the associated eigenspace is ;

      (4)  the centralizer of a ∈ H is

      we have

      In the case a and b are similar (by Theorem 2.2.6 this happens if and only if R(a)= R(b) and |V(a)| = |V(b)|), the kernel and image of Sa, b have alternative descriptions.

      Theorem 2.3.4. Assume a, b ∈ H\R are similar, so that b = z–1az, z ∈ H\{0}. Then:

      (a)  Ran Sa, b = Ker Sa, b* . In other words, the equation ax −xb = y has a solution x if and only if ay = yb*;

      (b)  Ker Sa, b = Cen (a) z = SpanR {z, az}.

      Proof. Part (a) follows from Theorem 2.3.3(2). Part (b) is a consequence of the identity

      We conclude this section with formulas for the unique solution of the Sylvester equation, provided Sa, b is invertible. For a, b ∈ H, define

      Proposition 2.3.5. The following statements are equivalent:

      (1)  f1(a, b) = 0;

      (2)  f2(a, b) = 0;

      (3)  a and b are similar;

      (4)  R(a)= R(b) and |a| = |b|.

      Proof. Equivalence of (3) and (4) follows from equivalence of (1) and (5) in Theorem 2.2.6. The implications (3) ⇒ (1) and (3) ⇒ (2) follow easily from Proposition 2.1.4(10). Suppose (1) holds true. Subtracting the equality b² − 2R(b)b + |b|² = 0 from f1(a, b) = 0, we obtain 2(R(a) − R(b))b = |a|² −|b|². If R(a) ≠R(b), then b must be real. Subtracting a² − 2R(a)a + |a|² = 0 from f1(a, b) = 0 yields a = b. If R(a)= R(b), then |a|² −|b|² = 0, and (4) follows. Analogously, one proves that (2) implies (4).

      Theorem 2.3.6. If Sa, b is nonsingular, then the unique solution to the equation Sa, b(x)= y satisfies

      The proof follows from the equalities

      for all z ∈ H, which can be verified without difficulty.

      2.4 AUTOMORPHISMS AND INVOLUTIONS

      Definition 2.4.1. An ordered triple of quaternions (q1, q2, q3) is said to be a units triple if

      For example, {i, j, k} is a units triple.

      Proposition 2.4.2. An ordered triple (q1, q2, q3), qj ∈ H, is a units triple if and only if there exists a 3 × 3 real orthogonal matrix with determinant 1 such that

      Proof. A straightforward computation verifies that x ∈ H satisfies x² = −1 if and only if

      where a1, a2, a. Thus, we may assume that qα are given by (2.4.2) with the vectors pα := (p1,α, p2,α, p3,α)T ∈ R³×¹ having euclidean norm 1, for α = 1, 2, 3. Next, in view of (2.1.1) we have

      The result of Proposition 2.4.2 now follows easily.

      In particular, for every units triple (q1, q2, q3) the quaternions 1, q1, q2, q3 form a basis of the real vector space H.

      Next, we consider endomorphisms and antiendomorphisms of quaternions.

      Definition 2.4.3. A map

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