Quaternionic vector spaceFrom Wikipedia, the free encyclopedia

Contents

1 Quadratic eigenvalue problem 11.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Methods of solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Quadratic form 32.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Real quadratic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.4 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.4.1 Quadratic spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4.2 Further denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.5 Equivalence of forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.6 Geometric meaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.7 Integral quadratic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.7.1 Historical use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.7.2 Universal quadratic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.11 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.12 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Quadruple product 113.1 Scalar quadruple product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Vector quadruple product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4 Quasinorm 144.1 Related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

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ii CONTENTS

4.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5 Quaternionic matrix 155.1 Matrix operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.2 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

6 Quaternionic vector space 176.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.3 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 18

6.3.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.3.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.3.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Chapter 1

Quadratic eigenvalue problem

In mathematics, the quadratic eigenvalue problem[1] (QEP), is to nd scalar eigenvalues , left eigenvectors yand right eigenvectors x such that

Q()x = 0 and yQ() = 0;

where Q() = 2A2 + A1 +A0 , with matrix coecients A2; A1; A0 2 Cnn and we require that A2 6= 0 , (sothat we have a nonzero leading coecient). There are 2n eigenvalues that may be innite or nite, and possibly zero.This is a special case of a nonlinear eigenproblem. Q() is also known as a quadratic matrix polynomial.

1.1 ApplicationsA QEP can result in part of the dynamic analysis of structures discretized by the nite element method. In this casethe quadratic,Q() has the formQ() = 2M +C+K , whereM is the mass matrix, C is the damping matrixandK is the stiness matrix. Other applications include vibro-acoustics and uid dynamics.

1.2 Methods of solutionDirect methods for solving the standard or generalized eigenvalue problems Ax = x and Ax = Bx are based ontransforming the problem to Schur or Generalized Schur form. However, there is no analogous form for quadraticmatrix polynomials. One approach is to transform the quadratic matrix polynomial to a linear matrix pencil (AB), and solve a generalized eigenvalue problem. Once eigenvalues and eigenvectors of the linear problem have beendetermined, eigenvectors and eigenvalues of the quadratic can be determined.The most common linearization is the rst companion linearization

L() =

M 00 In

+

C KIn 0

;

where In is the n -by- n identity matrix, with corresponding eigenvector

z =

xx

:

We solve L()z = 0 for and z , for example by computing the Generalized Schur form. We can then take the rstn components of z as the eigenvector x of the original quadratic Q() .

1

2 CHAPTER 1. QUADRATIC EIGENVALUE PROBLEM

1.3 References[1] F. Tisseur and K. Meerbergen, The quadratic eigenvalue problem, SIAM Rev., 43 (2001), pp. 235286.

Chapter 2

Quadratic form

Inmathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,

4x2 + 2xy 3y2

is a quadratic form in the variables x and y.Quadratic forms occupy a central place in various branches of mathematics, including number theory, linear algebra,group theory (orthogonal group), dierential geometry (Riemannian metric), dierential topology (intersection formsof four-manifolds), and Lie theory (the Killing form).

2.1 IntroductionQuadratic forms are homogeneous quadratic polynomials in n variables. In the cases of one, two, and three variablesthey are called unary, binary, and ternary and have the following explicit form:

q(x) = ax2 (unary)

q(x; y) = ax2 + bxy + cy2 (binary)q(x; y; z) = ax2 + by2 + cz2 + dxy + exz + fyz (ternary)where a, ..., f are the coecients.[1] Note that quadratic functions, such as ax2 + bx + c in the one variable case, arenot quadratic forms, as they are typically not homogeneous (unless b and c are both 0).The theory of quadratic forms and methods used in their study depend in a large measure on the nature of the coef-cients, which may be real or complex numbers, rational numbers, or integers. In linear algebra, analytic geometry,and in the majority of applications of quadratic forms, the coecients are real or complex numbers. In the alge-braic theory of quadratic forms, the coecients are elements of a certain eld. In the arithmetic theory of quadraticforms, the coecients belong to a xed commutative ring, frequently the integers Z or the p-adic integers Zp.[2]Binary quadratic forms have been extensively studied in number theory, in particular, in the theory of quadraticelds, continued fractions, and modular forms. The theory of integral quadratic forms in n variables has importantapplications to algebraic topology.Using homogeneous coordinates, a non-zero quadratic form in n variables denes an (n2)-dimensional quadric inthe (n1)-dimensional projective space. This is a basic construction in projective geometry. In this way one mayvisualize 3-dimensional real quadratic forms as conic sections.A closely related notion with geometric overtones is a quadratic space, which is a pair (V,q), with V a vector spaceover a eld K, and q: V K a quadratic form on V. An example is given by the three-dimensional Euclidean spaceand the square of the Euclidean norm expressing the distance between a point with coordinates (x,y,z) and the origin:

q(x; y; z) = d((x; y; z); (0; 0; 0))2 = k(x; y; z)k2 = x2 + y2 + z2:

3

4 CHAPTER 2. QUADRATIC FORM

2.2 HistoryThe study of particular quadratic forms, in particular the question of whether a given integer can be the value ofa quadratic form over the integers, dates back many centuries. One such case is Fermats theorem on sums of twosquares, which determines when an integer may be expressed in the form x2 + y2, where x, y are integers. Thisproblem is related to the problem of nding Pythagorean triples, which appeared in the second millennium B.C.[3]

In 628, the Indian mathematician Brahmagupta wrote Brahmasphutasiddhanta which includes, among many otherthings, a study of equations of the form x2 ny2 = c. In particular he considered what is now called Pells equation,x2 ny2 = 1, and found a method for its solution.[4] In Europe this problem was studied by Brouncker, Euler andLagrange.In 1801 Gauss published Disquisitiones Arithmeticae, a major portion of which was devoted to a complete theory ofbinary quadratic forms over the integers. Since then, the concept has been generalized, and the connections withquadratic number elds, the modular group, and other areas of mathematics have been further elucidated.

2.3 Real quadratic formsSee also: Sylvesters law of inertia and Denite form

Any nn real symmetric matrix A determines a quadratic form qA in n variables by the formula

qA(x1; : : : ; xn) =

nXi=1

nXj=1

aijxixj :

Conversely, given a quadratic form in n variables, its coecients can be arranged into an nn symmetric matrix. Oneof the most important questions in the theory of quadratic forms is how much can one simplify a quadratic form qby a homogeneous linear change of variables. A fundamental theorem due to Jacobi asserts that q can be brought toa diagonal form

1~x21 + 2~x

22 + + n~x2n;

so that the corresponding symmetric matrix is diagonal, and this is even possible to accomplish with a change ofvariables given by an orthogonal matrix in this case the coecients 1, 2, , n are in fact determined uniquelyup to a permutation. If the change of variables is given by an invertible matrix, not necessarily orthogonal, thenthe coecients i can be made to be 0,1, and 1. Sylvesters law of inertia states that the numbers of 1 and 1 areinvariants of the quadratic form, in the sense that any other diagonalization will contain the same number of each.The signature of the quadratic form is the triple (n0, n, n) where n0 is the number 0s and n is the number of1s. Sylvesters law of inertia shows that this is a well-dened quantity attached to the quadratic form. The casewhen all i have the same sign is especially important: in this case the quadratic form is called positive denite(all 1) or negative denite (all 1); if none of the terms are 0 then the form is called nondegenerate; this includespositive denite, negative denite, and indenite (a mix of 1 and 1); equivalently, a nondegenerate quadratic formis one whose associated symmetric form is a nondegenerate bilinear form. A real vector space with an indenitenondegenerate quadratic form of index (p, q) (p 1s, q 1s) is often denoted as Rp,q particularly in the physical theoryof space-time.The discriminant of a quadratic form, concretely the class of the determinant of a representing matrix in K/(K*)2(up to non-zero squares) can also be dened, and for a real quadratic form is a cruder invariant than signature, takingvalues of only positive, zero, or negative. Zero corresponds to degenerate, while for a non-degenerate form it is theparity of the number of negative coecients, (1)n :These results are reformulated in a dierent way below.Let q be a quadratic form dened on an n-dimensional real vector space. Let A be the matrix of the quadratic formq in a given basis. This means that A is a symmetric nn matrix such that

q(v) = xTAx;

2.4. DEFINITIONS 5

where x is the column vector of coordinates of v in the chosen basis. Under a change of basis, the column x ismultiplied on the left by an nn invertible matrix S, and the symmetric square matrix A is transformed into anothersymmetric square matrix B of the same size according to the formula

A! B = SAST:Any symmetric matrix A can be transformed into a diagonal matrix

B =

0BBB@1 0 00 2 0... ... . . . 00 0 n

1CCCAby a suitable choice of an orthogonal matrix S, and the diagonal entries of B are uniquely determined this is Jacobistheorem. If S is allowed to be any invertible matrix then B can be made to have only 0,1, and 1 on the diagonal,and the number of the entries of each type (n0 for 0, n for 1, and n for 1) depends only on A. This is one of theformulations of Sylvesters law of inertia and the numbers n and n are called the positive and negative indices ofinertia. Although their denition involved a choice of basis and consideration of the corresponding real symmetricmatrix A, Sylvesters law of inertia means that they are invariants of the quadratic form q.The quadratic form q is positive denite (resp., negative denite) if q(v) > 0 (resp., q(v) < 0) for every nonzero vectorv.[5] When q(v) assumes both positive and negative values, q is an indenite quadratic form. The theorems of Jacobiand Sylvester show that any positive denite quadratic form in n variables can be brought to the sum of n squares by asuitable invertible linear transformation: geometrically, there is only one positive denite real quadratic form of everydimension. Its isometry group is a compact orthogonal group O(n). This stands in contrast with the case of indeniteforms, when the corresponding group, the indenite orthogonal group O(p, q), is non-compact. Further, the isometrygroups of Q and Q are the same (O(p, q) O(q, p)), but the associated Cliord algebras (and hence Pin groups) aredierent.

2.4 DenitionsAn n-ary quadratic form over a eld K is a homogeneous polynomial of degree 2 in n variables with coecients inK:

q(x1; : : : ; xn) =

nXi=1

nXj=1

aijxixj ; aij 2 K:

This formula may be rewritten using matrices: let x be the column vector with components x1, ..., xn and A = (aij)be the nn matrix over K whose entries are the coecients of q. Then

q(x) = xTAx:

Two n-ary quadratic forms and over K are equivalent if there exists a nonsingular linear transformation C GL(n, K) such that

(x) = '(Cx):

Let us assume that the characteristic of K is dierent from 2. (The theory of quadratic forms over a eld of charac-teristic 2 has important dierences and many denitions and theorems have to be modied.) The coecient matrixA of q may be replaced by the symmetric matrix (A + AT)/2 with the same quadratic form, so it may be assumedfrom the outset that A is symmetric. Moreover, a symmetric matrix A is uniquely determined by the correspondingquadratic form. Under an equivalence C, the symmetric matrix A of and the symmetric matrix B of are relatedas follows:

6 CHAPTER 2. QUADRATIC FORM

B = CTAC:

The associated bilinear form of a quadratic form q is dened by

bq(x; y) =12 (q(x+ y) q(x) q(y)) = xTAy = yTAx:

Thus, bq is a symmetric bilinear form over K with matrix A. Conversely, any symmetric bilinear form b denes aquadratic form

q(x) = b(x; x)

and these two processes are the inverses of one another. As a consequence, over a eld of characteristic not equal to2, the theories of symmetric bilinear forms and of quadratic forms in n variables are essentially the same.

2.4.1 Quadratic spaces

A quadratic form q in n variables over K induces a map from the n-dimensional coordinate space Kn into K:

Q(v) = q(v); v = [v1; : : : ; vn]T 2 Kn:

The map Q is a homogeneous function of degree 2, which means that it has the property that, for all a in K and v inV:

Q(av) = a2Q(v):

When the characteristic of K is not 2, the map B : V V K dened below is bilinear over K:

B(v; w) = 12 (Q(v + w)Q(v)Q(w)):

This bilinear form B has the properties that B(x, x) = Q(x) for all x in V and B(x, y) = B(y, x) for all x, y in V (it issymmetric).When the characteristic of K is 2, so that 2 is not a unit, it is still possible to use a quadratic form to dene a symmetricbilinear form B(x, y) = Q(x + y) Q(x) Q(y). However, Q(x) can no longer be recovered from this B in the sameway, since B(x, x) = 0 for all x. Alternately, there always exists a bilinear form B (not in general either unique orsymmetric) such that B(x, x) = Q(x).The pair (V, Q) consisting of a nite-dimensional vector space V over K and a quadratic map from V to K is calleda quadratic space, and B as dened here is the associated symmetric bilinear form of Q. The notion of a quadraticspace is a coordinate-free version of the notion of quadratic form. Sometimes, Q is also called a quadratic form.Two n-dimensional quadratic spaces (V, Q) and (V , Q ) are isometric if there exists an invertible linear transfor-mation T : V V (isometry) such that

Q(v) = Q0(Tv) all for v 2 V:

The isometry classes of n-dimensional quadratic spaces overK correspond to the equivalence classes of n-ary quadraticforms over K.

2.5. EQUIVALENCE OF FORMS 7

2.4.2 Further denitions

See also: Isotropic quadratic form

Two elements v and w of V are called orthogonal if B(v,w) = 0. The kernel of a bilinear form B consists of theelements that are orthogonal to all elements of V. Q is non-singular if the kernel of its associated bilinear form is 0.If there exists a non-zero v in V such that Q(v) = 0, the quadratic form Q is isotropic, otherwise it is anisotropic.This terminology also applies to vectors and subspaces of a quadratic space. If the restriction of Q to a subspace Uof V is identically zero, U is totally singular.The orthogonal group of a non-singular quadratic form Q is the group of the linear automorphisms of V that preserveQ, i.e. the group of isometries of (V, Q) into itself.

2.5 Equivalence of formsEvery quadratic form q in n variables over a eld of characteristic not equal to 2 is equivalent to a diagonal form

q(x) = a1x21 + a2x

22 + : : :+ anx

2n:

Such a diagonal form is often denoted by

ha1; : : : ; ani:

Classication of all quadratic forms up to equivalence can thus be reduced to the case of diagonal forms.

2.6 Geometric meaningUsing Cartesian coordinates in three dimensions, let x = (x; y; z)T , and let A be a symmetric 3-by-3 matrix. Thenthe geometric nature of the solution set of the equation xTAx + bTx = 1 depends on the eigenvalues of the matrixA .If all eigenvalues of A are non-zero, then the solution set is an ellipsoid or a hyperboloid. If all the eigenvaluesare positive, then it is an ellipsoid; if all the eigenvalues are negative, then it is an imaginary ellipsoid (we get theequation of an ellipsoid but with imaginary radii); if some eigenvalues are positive and some are negative, then it isa hyperboloid.If there exist one or more eigenvalues i = 0 , then the shape depends on the corresponding bi . If the correspondingbi 6= 0 , then the solution set is a paraboloid (either elliptic or hyperbolic); if the corresponding bi = 0 , then thedimension i degenerates and does not get into play, and the geometricmeaning will be determined by other eigenvaluesand other components of . When the solution set is a paraboloid, whether it is elliptic or hyperbolic is determined bywhether all other non-zero eigenvalues are of the same sign: if they are, then it is elliptic; otherwise, it is hyperbolic.

2.7 Integral quadratic formsQuadratic forms over the ring of integers are called integral quadratic forms, whereas the corresponding modulesare quadratic lattices (sometimes, simply lattices). They play an important role in number theory and topology.An integral quadratic form has integer coecients, such as x2 + xy + y2; equivalently, given a lattice in a vectorspace V (over a eld with characteristic 0, such as Q or R), a quadratic form Q is integral with respect to if andonly if it is integer-valued on , meaning Q(x,y) Z if x,y .This is the current use of the term; in the past it was sometimes used dierently, as detailed below.

8 CHAPTER 2. QUADRATIC FORM

2.7.1 Historical useHistorically there was some confusion and controversy over whether the notion of integral quadratic form shouldmean:

twos in the quadratic form associated to a symmetric matrix with integer coecientstwos out a polynomial with integer coecients (so the associated symmetricmatrixmay have half-integer coecients

o the diagonal)

This debate was due to the confusion of quadratic forms (represented by polynomials) and symmetric bilinear forms(represented by matrices), and twos out is now the accepted convention; twos in is instead the theory of integralsymmetric bilinear forms (integral symmetric matrices).In twos in, binary quadratic forms are of the form ax2 + 2bxy + cy2 , represented by the symmetric matrix

a bb c

this is the convention Gauss uses in Disquisitiones Arithmeticae.In twos out, binary quadratic forms are of the form ax2 + bxy + cy2 , represented by the symmetric matrix

a b/2b/2 c

:

Several points of view mean that twos out has been adopted as the standard convention. Those include:

better understanding of the 2-adic theory of quadratic forms, the 'local' source of the diculty; the lattice point of view, which was generally adopted by the experts in the arithmetic of quadratic forms duringthe 1950s;

the actual needs for integral quadratic form theory in topology for intersection theory; the Lie group and algebraic group aspects.

2.7.2 Universal quadratic formsAn integral quadratic form whose image consists of all the positive integers is sometimes called universal. Lagrangesfour-square theorem shows thatw2+x2+y2+z2 is universal. Ramanujan generalized this to aw2+bx2+cy2+dz2and found 54 multisets {a,b,c,d} that can each generate all positive integers, namely,

{1,1,1,d}, 1 d 7{1,1,2,d}, 2 d 14{1,1,3,d}, 3 d 6{1,2,2,d}, 2 d 7{1,2,3,d}, 3 d 10{1,2,4,d}, 4 d 14{1,2,5,d}, 6 d 10

There are also forms whose image consists of all but one of the positive integers. For example, {1,2,5,5} has 15 asthe exception. Recently, the 15 and 290 theorems have completely characterized universal integral quadratic forms:if all coecients are integers, then it represents all positive integers if and only if it represents all integers up through290; if it has an integral matrix, it represents all positive integers if and only if it represents all integers up through15.

2.8. SEE ALSO 9

2.8 See also -quadratic form Quadratic form (statistics) Quadric Discriminant of a quadratic form Cubic form Witt group Witts theorem HasseMinkowski theorem Orthogonal group Square class Ramanujans ternary quadratic form

2.9 Notes[1] A tradition going back to Gauss dictates the use of manifestly even coecients for the products of distinct variables, i.e.

2b in place of b in binary forms and 2d, 2e, 2f in place of d, e, f in ternary forms. Both conventions occur in the literature[2] away from 2, i. e. if 2 is invertible in the ring, quadratic forms are equivalent to symmetric bilinear forms (by the polarization

identities), but at 2 they are dierent concepts; this distinction is particularly important for quadratic forms over the integers.[3] Babylonian Pythagoras[4] Brahmagupta biography[5] If a non-strict inequality (with or ) holds then the quadratic form q is called semidenite.

2.10 References O'Meara, O.T. (2000), Introduction to Quadratic Forms, Berlin, New York: Springer-Verlag, ISBN 978-3-540-66564-9

Conway, John Horton; Fung, Francis Y. C. (1997), The Sensual (Quadratic) Form, Carus Mathematical Mono-graphs, The Mathematical Association of America, ISBN 978-0-88385-030-5

Shafarevich, I. R.; A. O. Remizov (2012). Linear Algebra and Geometry. Springer. ISBN 978-3-642-30993-9.

2.11 Further reading Cassels, J.W.S. (1978). Rational Quadratic Forms. London Mathematical Society Monographs 13. AcademicPress. ISBN 0-12-163260-1. Zbl 0395.10029.

Kitaoka, Yoshiyuki (1993). Arithmetic of quadratic forms. Cambridge Tracts in Mathematics 106. CambridgeUniversity Press. ISBN 0-521-40475-4. Zbl 0785.11021.

Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics 67.American Mathematical Society. ISBN 0-8218-1095-2. MR 2104929. Zbl 1068.11023.

Milnor, J.; Husemoller, D. (1973). Symmetric Bilinear Forms. Ergebnisse der Mathematik und ihrer Grenzge-biete 73. Springer-Verlag. ISBN 3-540-06009-X. Zbl 0292.10016.

O'Meara, O.T. (1973). Introduction to quadratic forms. Die Grundlehren der mathematischen Wissenschaften117. Springer-Verlag. ISBN 3-540-66564-1. Zbl 0259.10018.

10 CHAPTER 2. QUADRATIC FORM

2.12 External links A.V.Malyshev (2001), Quadratic form, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer,ISBN 978-1-55608-010-4

A.V.Malyshev (2001), Binary quadratic form, inHazewinkel, Michiel, Encyclopedia ofMathematics, Springer,ISBN 978-1-55608-010-4

Chapter 3

Quadruple product

See also: Vector algebra relations

In mathematics, the quadruple product is a product of four vectors in three-dimensional Euclidean space. Thename quadruple product is used for two dierent products,[1] the scalar-valued scalar quadruple product and thevector-valued vector quadruple product.

3.1 Scalar quadruple productThe scalar quadruple product is dened as the dot product of two cross products:

(a b)(c d) ;where a, b, c, d are vectors in three-dimensional Euclidean space.[2] It can be evaluated using the identity:[2]

(a b)(c d) = (a c)(b d) (a d)(b c) :or using the determinant:

(a b)(c d) =a c a db c b d

:3.2 Vector quadruple productThe vector quadruple product is dened as the cross product of two cross products:

(a b)(c d) ;where a, b, c, d are vectors in three-dimensional Euclidean space.[3] It can be evaluated using the identity:[4]

(a b)(c d) = [a; b; d]c [a; b; c]d ;This identity can also be written using tensor notation and the Einstein summation convention as follows:

(a b)(c d) = "ijkaicjdkbl "ijkbicjdkal = "ijkaibjdkcl "ijkaibjckdl

11

12 CHAPTER 3. QUADRUPLE PRODUCT

using the notation for the triple product:

[a; b; d] = (a b)d =a^i b^i d^ia^j b^j d^jak^ bk^ dk^

=a^i a^j ak^b^i b^j bk^d^i d^j dk^

;where the last two forms are determinants with i^; j^; k^ denoting unit vectors along three mutually orthogonal direc-tions.Equivalent forms can be obtained using the identity:[5]

[b; c; d]a [c; d; a]b+ [d; a; b]c [a; b; c]d = 0 :

3.3 ApplicationThe quadruple products are useful for deriving various formulas in spherical and plane geometry.[3] For example, iffour points are chosen on the unit sphere, A, B, C, D, and unit vectors drawn from the center of the sphere to the fourpoints, a, b, c, d respectively, the identity:

(a b)(c d) = (a c)(b d) (a d)(b c) ;in conjunction with the relation for the magnitude of the cross product:

ka bk = ab sin ab ;and the dot product:

ka bk = ab cos ab ;where a = b = 1 for the unit sphere, results in the identity among the angles attributed to Gauss:

sin ab sin cd cosx = cos ac cos bd cos ad cos bc ;where x is the angle between a b and c d, or equivalently, between the planes dened by these vectors.Josiah Willard Gibbs's pioneering work on vector calculus provides several other examples.[3]

3.4 Notes[1] Gibbs & Wilson 1901, 42 of section Direct and skew products of vectors, p.77

[2] Gibbs & Wilson 1901, p. 76[3] Gibbs & Wilson 1901, pp. 77

[4] Gibbs & Wilson 1901, p. 77[5] Gibbs Wilson, Equation 27, p. 77

3.5 References Gibbs, Josiah Willard; Wilson, Edwin Bidwell (1901). Vector analysis: a text-book for the use of students ofmathematics. Scribner.

3.6. SEE ALSO 13

3.6 See also BinetCauchy identity Lagranges identity

Chapter 4

Quasinorm

In linear algebra, functional analysis and related areas of mathematics, a quasinorm is similar to a norm in that itsatises the norm axioms, except that the triangle inequality is replaced by

kx+ yk K(kxk+ kyk)for someK > 1:This is not to be confused with a seminorm or pseudonorm, where the norm axioms are satised except for positivedeniteness.

4.1 Related conceptsA vector space with an associated quasinorm is called a quasinormed vector space.A complete quasinormed vector space is called a quasi-Banach space.A quasinormed space (A; k k) is called a quasinormed algebra if the vector space A is an algebra and there is aconstant K > 0 such that

kxyk Kkxk kykfor all x; y 2 A .A complete quasinormed algebra is called a quasi-Banach algebra.

4.2 See also Seminorm

4.3 References Aull, Charles E.; Robert Lowen (2001). Handbook of the History of General Topology. Springer. ISBN0-7923-6970-X.

Conway, John B. (1990). A Course in Functional Analysis. Springer. ISBN 0-387-97245-5. Nikolski, Nikola Kapitonovich (1992). Functional Analysis I: Linear Functional Analysis. Encyclopaedia ofMathematical Sciences 19. Springer. ISBN 3-540-50584-9.

Swartz, Charles (1992). An Introduction to Functional Analysis. CRC Press. ISBN 0-8247-8643-2.

14

Chapter 5

Quaternionic matrix

A quaternionic matrix is a matrix whose elements are quaternions.

5.1 Matrix operationsThe quaternions form a noncommutative ring, and therefore addition and multiplication can be dened for quater-nionic matrices as for matrices over any ring.Addition. The sum of two quaternionic matrices A and B is dened in the usual way by element-wise addition:

(A+B)ij = Aij +Bij :

Multiplication. The product of two quaternionic matrices A and B also follows the usual denition for matrixmultiplication. For it to be dened, the number of columns of Amust equal the number of rows of B. Then the entryin the ith row and jth column of the product is the dot product of the ith row of the rst matrix with the jth columnof the second matrix. Specically:

(AB)ij =Xs

AisBsj :

For example, for

U =

u11 u12u21 u22

; V =

v11 v12v21 v22

;

the product is

UV =

u11v11 + u12v21 u11v12 + u12v22u21v11 + u22v21 u21v12 + u22v22

:

Since quaternionic multiplication is noncommutative, care must be taken to preserve the order of the factors whencomputing the product of matrices.The identity for this multiplication is, as expected, the diagonal matrix I = diag(1, 1, ... , 1). Multiplication followsthe usual laws of associativity and distributivity. The trace of a matrix is dened as the sum of the diagonal elements,but in general

trace(AB) 6= trace(BA):

15

16 CHAPTER 5. QUATERNIONIC MATRIX

Left scalar multiplication is dened by

(cA)ij = cAij ; (Ac)ij = Aijc:

Again, since multiplication is not commutative some care must be taken in the order of the factors.[1]

5.2 DeterminantsThere is no natural way to dene a determinant for (square) quaternionic matrices so that the values of the determinantare quaternions.[2] Complex valued determinants can be dened however.[3] The quaternion a + bi + cj + dk can berepresented as the 22 complex matrix

a+ bi c+ di

c+ di a bi:

This denes a map mn from the m by n quaternionic matrices to the 2m by 2n complex matrices by replacing eachentry in the quaternionic matrix by its 2 by 2 complex representation. The complex valued determinant of a squarequaternionic matrix A is then dened as det((A)). Many of the usual laws for determinants hold; in particular, an nby n matrix is invertible if and only if its determinant is nonzero.

5.3 ApplicationsQuaternionic matrices are used in quantum mechanics[4] and in the treatment of multibody problems.[5]

5.4 References[1] Tapp, Kristopher (2005). Matrix groups for undergraduates. AMS Bookstore. pp. 11 . ISBN 0-8218-3785-0.

[2] HelmerAslaksen (1996). Quaternionic determinants. TheMathematical Intelligencer 18 (3): 5765. doi:10.1007/BF03024312.

[3] E. Study (1920). Zur Theorie der linearenGleichungen. ActaMathematica (inGerman) 42 (1): 161. doi:10.1007/BF02404401.

[4] N. Rsch (1983). Time-reversal symmetry, Kramers degeneracy and the algebraic eigenvalue problem. Chemical Physics80 (12): 15. doi:10.1016/0301-0104(83)85163-5.

[5] Klaus Grlebeck; Wolfgang Sprssig (1997). Quaternionic matrices. Quaternionic and Cliord calculus for physicistsand engineers. Wiley. pp. 3234. ISBN 978-0-471-96200-7.

Chapter 6

Quaternionic vector space

In mathematics, a left (or right) quaternionic vector space is a left (or right) H-module where H denotes thenoncommutative ring of the quaternions.The space Hn of n-tuples of quaternions is both a left and right H-module using the componentwise left and rightmultiplication:

q(q1; q2; : : : qn) = (qq1; qq2; : : : qqn)

(q1; q2; : : : qn)q = (q1q; q2q; : : : qnq)

for quaternions q and q1, q2, ... qn.Since H is a division algebra, every nitely generated (left or right) H-module has a basis, and hence is isomorphicto Hn for some n.

6.1 See also Vector space General linear group Special linear group SL(n,H) Symplectic group

6.2 References Harvey, F. Reese (1990). Spinors and Calibrations. San Diego: Academic Press. ISBN 0-12-329650-1.

17

18 CHAPTER 6. QUATERNIONIC VECTOR SPACE

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Quadratic eigenvalue problemApplicationsMethods of solutionReferences

Quadratic formIntroduction HistoryReal quadratic forms Definitions Quadratic spaces Further definitions

Equivalence of forms Geometric meaningIntegral quadratic forms Historical use Universal quadratic forms

See also NotesReferencesFurther readingExternal links

Quadruple productScalar quadruple product Vector quadruple product ApplicationNotesReferencesSee also

QuasinormRelated concepts See also References

Quaternionic matrixMatrix operationsDeterminantsApplicationsReferences

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