Asymmetric Norm 2

Asymmetric normFrom Wikipedia, the free encyclopedia

Contents

1 Absolutely convex set 11.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Absolutely convex hull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Adjugate matrix 32.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2.1 2 2 generic matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2.2 3 3 generic matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2.3 3 3 numeric matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3.1 Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3.2 Characteristic polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3.3 Jacobis formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3.4 CayleyHamilton formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Ane space 83.1 Informal descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2.1 Subtraction and Weyls axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2.2 Ane combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.4 Ane subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.5 Ane combinations and ane dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.6 Geometric objects as points and vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.7 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.8 Relation to projective spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

i

ii CONTENTS

3.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4 AmitsurLevitzki theorem 154.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5 Angles between ats 175.1 Jordans denition[1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.2 Angles between subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.3 Variational characterisation[3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5.3.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.3.3 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

6 Antilinear map 206.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

7 Antiunitary operator 217.1 Invariance transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

7.1.1 Geometric Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.4 Decomposition of an antiunitary operator into a direct sum of elementary Wigner antiunitaries . . . 227.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

8 Asymmetric norm 248.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248.4 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 25

8.4.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258.4.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258.4.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Chapter 1

Absolutely convex set

A set C in a real or complex vector space is said to be absolutely convex if it is convex and balanced.

1.1 PropertiesA setC is absolutely convex if and only if for any pointsx1; x2 inC and any numbers1; 2 satisfying j1j+j2j 1the sum 1x1 + 2x2 belongs to C .Since the intersection of any collection of absolutely convex sets is absolutely convex then for any subset A of a vectorspace one can dene its absolutely convex hull to be the intersection of all absolutely convex sets containing A.

1.2 Absolutely convex hullThe absolutely convex hull of the set A assumes the following representationabsconvA = fPni=1 ixi : n 2 N; xi 2 A; Pni=1 jij 1g .1.3 References

Robertson, A.P.; W.J. Robertson (1964). Topological vector spaces. Cambridge Tracts in Mathematics 53.Cambridge University Press. pp. 46.

1.4 See also vector (geometric), for vectors in physics Vector eld

1

2 CHAPTER 1. ABSOLUTELY CONVEX SET

The light gray area is the Absolutely convex hull of the cross.

Chapter 2

Adjugate matrix

In linear algebra, the adjugate, classical adjoint, or adjunct of a square matrix is the transpose of the cofactormatrix.The adjugate has sometimes been called the adjoint, but today the adjoint of a matrix normally refers to itscorresponding adjoint operator, which is its conjugate transpose.

2.1 DenitionThe adjugate of A is the transpose of the cofactor matrix C of A:

adj(A) = CT

In more detail: suppose R is a commutative ring and A is an nn matrix with entries from R.

The (i,j) minor of A, denoted Mij, is the determinant of the (n 1)(n 1) matrix that results from deletingrow i and column j of A.

The cofactor matrix of A is the nn matrix C whose (i,j) entry is the (i,j) cofactor of A:

Cij = (1)i+jMij

The adjugate of A is the transpose of C, that is, the nn matrix whose (i,j) entry is the (j,i) cofactor of A:

adj(A)ij = Cji = (1)i+jMji

The adjugate is dened as it is so that the product of A and its adjugate yields a diagonal matrix whose diagonal entriesare det(A):

A adj(A) = det(A) IA is invertible if and only if det(A) is an invertible element of R, and in that case the equation above yields:

adj(A) = det(A)A1

A1 = 1det(A) adj(A)

3

4 CHAPTER 2. ADJUGATE MATRIX

2.2 Examples

2.2.1 2 2 generic matrixThe adjugate of the 2 2 matrix

A =a bc d

is

adj(A) =

d bc a

It is seen that det(adj(A)) = det(A) and adj(adj(A)) = A.

2.2.2 3 3 generic matrixConsider the 3 3 matrix

A =

0@a11 a12 a13a21 a22 a23a31 a32 a33

1A =0@1 2 34 5 67 8 9

1AIts adjugate is the transpose of the cofactor matrix

C =

0BBBBBBBBBB@

+

a22 a23a32 a33 a21 a23a31 a33

+ a21 a22a31 a32

a12 a13a32 a33

+ a11 a13a31 a33 a11 a12a31 a32

+

a12 a13a22 a23 a11 a13a21 a23

+ a11 a12a21 a22

1CCCCCCCCCCA=

0BBBBBBBBBB@

+

5 68 9 4 67 9

+ 4 57 8

2 38 9

+ 1 37 9 1 27 8

+

2 35 6 1 34 6

+ 1 24 5

1CCCCCCCCCCASo that we have

adj(A) =

0BBBBBBBBBB@

+

a22 a23a32 a33 a12 a13a32 a33

+ a12 a13a22 a23

a21 a23a31 a33

+ a11 a13a31 a33 a11 a13a21 a23

+

a21 a22a31 a32 a11 a12a31 a32

+ a11 a12a21 a22

1CCCCCCCCCCA=

0BBBBBBBBBB@

+

5 68 9 2 38 9

+ 2 35 6

4 67 9

+ 1 37 9 1 34 6

+

4 57 8 1 27 8

+ 1 24 5

1CCCCCCCCCCAwhere

aim ainajm ajn = det aim ainajm ajn

Therefore C, the matrix of cofactors for A, is

2.3. PROPERTIES 5

C =

0@3 6 36 12 63 6 3

1AThe adjugate is the transpose of the cofactor matrix. Thus, for instance, the (3,2) entry of the adjugate is the (2,3)cofactor of A. (In this example, C happens to be its own transpose, so adj(A) = C.)

2.2.3 3 3 numeric matrixAs a specic example, we have

adj

0@3 2 51 0 23 4 1

1A =0@8 18 45 12 1

4 6 2

1AThe 6 in the third row, second column of the adjugate was computed as follows:

(1)2+3 det3 2

3 4

= ((3)(4) (3)(2)) = 6:

Again, the (3,2) entry of the adjugate is the (2,3) cofactor of A. Thus, the submatrix

3 23 4

was obtained by deleting the second row and third column of the original matrix A.

2.3 PropertiesThe adjugate has the properties

adj(I) = I;

adj(AB) = adj(B) adj(A);adj(cA) = cn1adj(A)for nnmatricesA andB. The second line follows from equations adj(B)adj(A) = det(B)B1 det(A)A1 = det(AB)(AB)1.Substituting in the second line B = Am 1 and performing the recursion, one gets for all integer m

adj(Am) = adj(A)m:

The adjugate preserves transposition:

adj(AT) = adj(A)T:

Furthermore,

detadj(A)

= det(A)n1;

6 CHAPTER 2. ADJUGATE MATRIX

adj(adj(A)) = det(A)n2Aso if n = 2 and A is invertible, then det(adj(A)) = det(A) and adj(adj(A)) = A.Taking the adjugate k times of an invertible matrix A yields:

adjk(A) = det(A)(n1)k(1)k

n A(1)k

detadjk(A)

= det(A)(n1)k

2.3.1 InversesAs a consequence of Laplaces formula for the determinant of an nn matrix A, we have

A adj(A) = adj(A)A = det(A) In ()

where In is the nn identity matrix. Indeed, the (i,i) entry of the product A adj(A) is the scalar product of row i ofA with row i of the cofactor matrix C, which is simply the Laplace formula for det(A) expanded by row i. Moreover,for i j the (i,j) entry of the product is the scalar product of row i of A with row j of C, which is the Laplace formulafor the determinant of a matrix whose i and j rows are equal and is therefore zero.From this formula follows one of the most important results in matrix algebra: A matrix A over a commutative ringR is invertible if and only if det(A) is invertible in R.For if A is an invertible matrix then

1 = det(In) = det(AA1) = det(A) det(A1);

and equation (*) above shows that

A1 = det(A)1 adj(A):

See also Cramers rule.

2.3.2 Characteristic polynomialIf p(t) = det(A t I) is the characteristic polynomial of A and we dene the polynomial q(t) = (p(0) p(t))/t, then

adj(A) = q(A) = (p1I+ p2A+ p3A2 + + pnAn1);

where pj are the coecients of p(t),

p(t) = p0 + p1t+ p2t2 + + pntn:

2.3.3 Jacobis formulaThe adjugate also appears in Jacobis formula for the derivative of the determinant:

dd det(A) = tr

adj(A)dAd

:

2.4. SEE ALSO 7

2.3.4 CayleyHamilton formulaCayleyHamilton theorem allows the adjugate of A to be represented in terms of traces and powers of A:

adj(A) =n1Xs=0

AsX

k1;k2;:::;kn1

n1Yl=1

(1)kl+1lklkl!

tr(Al)kl ;

where n is the dimension ofA, and the sum is taken over s and all sequences of kl 0 satisfying the linear Diophantineequation

s+

n1Xl=1

lkl = n 1:

For the 22 case this gives

adj(A) = I2trA A:


adj(A) = 12

(trA)2 trA2 I3 AtrA+ A2:


adj(A) = 16

(trA)3 3trAtrA2 + 2trA3 I4 1

2A(trA)2 trA2+ A2trA A3:

2.4 See also Trace diagram

2.5 References Strang, Gilbert (1988). Section 4.4: Applications of determinants. Linear Algebra and its Applications (3rded.). Harcourt Brace Jovanovich. pp. 231232. ISBN 0-15-551005-3.

2.6 External links Matrix Reference Manual Online matrix calculator (determinant, track, inverse, adjoint, transpose) Compute Adjugate matrix up to order8

adjugate of { { a, b, c }, { d, e, f }, { g, h, i } }". Wolfram Alpha.

Chapter 3

Ane space

Not to be confused with anity space.For a concept in algebraic geometry, see ane space (algebraic geometry).In mathematics, an ane space is a geometric structure that generalizes certain properties of parallel lines in

Line segments on a two-dimensional ane space

8

3.1. INFORMAL DESCRIPTIONS 9

Euclidean space. In an ane space, there is no distinguished point that serves as an origin. Hence, no vector has axed origin and no vector can be uniquely associated to a point. In an ane space, there are instead displacementvectors between two points of the space. Thus it makes sense to subtract two points of the space, giving a vector, butit does not make sense to add two points of the space. Likewise, it makes sense to add a vector to a point of an anespace, resulting in a new point displaced from the starting point by that vector.The simplest example of an ane space is a linear subspace of a vector space that has been translated away fromthe origin. In nite dimensions, such an ane subspace corresponds to the solution set of an inhomogeneous linearsystem. The displacement vectors for that ane space live in the solution set of the corresponding homogeneous linearsystem, which is a linear subspace. Linear subspaces, in contrast, always contain the origin of the vector space.

3.1 Informal descriptionsThe following characterization may be easier to understand than the usual formal denition: an ane space is whatis left of a vector space after you've forgotten which point is the origin (or, in the words of the French mathematicianMarcel Berger, An ane space is nothing more than a vector space whose origin we try to forget about, by addingtranslations to the linear maps[1]). Imagine that Alice knows that a certain point is the actual origin, but Bob believesthat another point call it p is the origin. Two vectors, a and b, are to be added. Bob draws an arrow from pointp to point a and another arrow from point p to point b, and completes the parallelogram to nd what Bob thinks is a+ b, but Alice knows that he has actually computed

p + (a p) + (b p).

Similarly, Alice and Bob may evaluate any linear combination of a and b, or of any nite set of vectors, and willgenerally get dierent answers. However, if the sum of the coecients in a linear combination is 1, then Alice andBob will arrive at the same answer.If Bob travels to

a + (1 )b

then Alice can similarly travel to

p + (a p) + (1 )(b p) = a + (1 )b.

Then, for all coecients + (1 ) = 1, Alice and Bob describe the same point with the same linear combination,starting from dierent origins.While Alice knows the linear structure, both Alice and Bob know the ane structurei.e. the values of anecombinations, dened as linear combinations in which the sum of the coecients is 1. An underlying set with anane structure is an ane space.

3.2 DenitionAn ane space[2] is a set A together with a vector space V over a eld F and a transitive and free group action of V(with addition of vectors as group action) on A. (That is, an ane space is a principal homogeneous space for theaction of V.)Explicitly, an ane space is a point set A together with a map

l : V A! A; (v; a) 7! v+ awith the following properties:.[3][4][5]

1. Left identity8a 2 A; 0+ a = a

10 CHAPTER 3. AFFINE SPACE

2. Associativity

8v;w 2 V; 8a 2 A; v+ (w+ a) = (v+ w) + a

3. Uniqueness8a 2 A; V ! A : v 7! v+ a is a bijection.

(Since the group V is abelian, there is no dierence between its left and right actions, so it is also permissible to placevectors on the right.)By choosing an origin, o, one can thus identify A with V, hence turn A into a vector space. Conversely, any vectorspace, V, is an ane space over itself.

3.2.1 Subtraction and Weyls axiomsThe uniqueness property ensures that subtraction of any two elements of A is well dened, producing a vector of V:

a b is the unique vector in V such that (a b) + b = a .

One can equivalently dene an ane space as a point set A, together with a vector space V, and a subtraction map

: A A ! V; (a; b) 7! b a !abwith the following properties:[6]

1. 8p 2 A; 8v 2 V there is a unique point q 2 A such that q p = v and2. 8p; q; r 2 A; (q p) + (r q) = r p .

These two properties are called Weyl's axioms.

3.2.2 Ane combinationsFor any choice of origin o, and two points a, b in A and scalar , there is a unique element of A, denoted by a +(1 )b such that

(a+ (1 )b) o = (a o) + (1 )(b o):This element can be shown to be independent of the choice of origin o. Instead of arbitrary linear combinations, onlysuch ane combinations of points have meaning.

3.3 Examples When children nd the answers to sums such as 4 + 3 or 4 2 by counting right or left on a number line, theyare treating the number line as a one-dimensional ane space.

Any coset of a subspace V of a vector space is an ane space over that subspace. If T is a matrix and b lies in its column space, the set of solutions of the equation T x = b is an ane spaceover the subspace of solutions of T x = 0.

The solutions of an inhomogeneous linear dierential equation form an ane space over the solutions of thecorresponding homogeneous linear equation.

Generalizing all of the above, if T : V W is a linear mapping and y lies in its image, the set of solutions x V to the equation T x = y is a coset of the kernel of T , and is therefore an ane space over Ker T .

3.4. AFFINE SUBSPACES 11

3.4 Ane subspacesAn ane subspace (sometimes called a linear manifold, linear variety, or a at) of a vector space V is a subset closedunder ane combinations of vectors in the space. For example, the set

A =n NX

i

ivi NX

i

i = 1o

is an ane space, where fvigi2I is a family of vectors in V; this space is the ane span of these points. To see thatthis is indeed an ane space, observe that this set carries a transitive action of the vector subspace W of V

W =n NX

i

ivi NX

i

i = 0o:

This ane subspace can be equivalently described as the coset of the W-action

S = p+W;where p is any element of A, or equivalently as any level set of the quotient map V V/W. A choice of p gives abase point of A and an identication of W with A, but there is no natural choice, nor a natural identication of Wwith A.A linear transformation is a function that preserves all linear combinations; an ane transformation is a function thatpreserves all ane combinations. A linear subspace is an ane subspace containing the origin, or, equivalently, asubspace that is closed under linear combinations.For example, in R3 , the origin, lines and planes through the origin and the whole space are linear subspaces, whilepoints, lines and planes in general as well as the whole space are the ane subspaces.

3.5 Ane combinations and ane dependenceMain article: Ane combination

An ane combination is a linear combination in which the sum of the coecients is 1. Just as members of a set ofvectors are linearly independent if none is a linear combination of the others, so also they are anely independent ifnone is an ane combination of the others. The set of linear combinations of a set of vectors is their linear span andis always a linear subspace; the set of all ane combinations is their ane span and is always an ane subspace.For example, the ane span of a set of two points is the line that contains both; the ane span of a set of threenon-collinear points is the plane that contains all three.Vectors

v1, v2, , vn

are linearly dependent if there exist scalars a1, a2, , an, not all zero, for which

Similarly they are anely dependent if in addition the sum of coecients is zero:

nXi=1

ai = 0

a condition that ensures that the combination (1) makes sense as a displacement vector.


3.6 Geometric objects as points and vectorsIn an ane space, geometric objects have two dierent (although related) descriptions on languages of points (ele-ments of A) and vectors (elements of V ). A vector description can specify an object only up to translations.

3.7 AxiomsAne space is usually studied as analytic geometry using coordinates, or equivalently vector spaces. It can also bestudied as synthetic geometry by writing down axioms, though this approach is much less common. There are severaldierent systems of axioms for ane space.Coxeter (1969, p. 192) axiomatizes ane geometry (over the reals) as ordered geometry together with an ane formof Desarguess theorem and an axiom stating that in a plane there is at most one line through a given point not meetinga given line.Ane planes satisfy the following axioms (Cameron 1991, chapter 2): (in which two lines are called parallel if theyare equal or disjoint):

Any two distinct points lie on a unique line. Given a point and line there is a unique line which contains the point and is parallel to the line There exist three non-collinear points.

As well as ane planes over elds (or division rings), there are also many non-Desarguesian planes satisfying theseaxioms. (Cameron 1991, chapter 3) gives axioms for higher-dimensional ane spaces.

3.8 Relation to projective spacesSee also: ane space (algebraic geometry)Ane spaces are subspaces of projective spaces: an ane plane can be obtained from any projective plane by

An ane space is a subspace of projective space, which is in turn a quotient of a vector space.

3.9. SEE ALSO 13

removing a line and all the points on it, and conversely any ane plane can be used to construct a projective plane asa closure by adding a line at innity whose points correspond to equivalence classes of parallel lines.Further, transformations of projective space that preserve ane space (equivalently, that leave the hyperplane at in-nity invariant as a set) yield transformations of ane space. Conversely, any ane linear transformation extendsuniquely to a projective linear transformation, so the ane group is a subgroup of the projective group. For instance,Mbius transformations (transformations of the complex projective line, or Riemann sphere) are ane (transforma-tions of the complex plane) if and only if they x the point at innity.However, one cannot take the projectivization of an ane space, so projective spaces are not naturally quotients ofane spaces: one can only take the projectivization of a vector space, since the projective space is lines through agiven point, and there is no distinguished point in an ane space. If one chooses a base point (as zero), then an anespace becomes a vector space, which one may then projectivize, but this requires a choice.

3.9 See also Space (mathematics) Ane geometry Ane group Ane transformation Ane variety Ane hull Heap (mathematics) Equipollence (geometry) Interval measurement, an ane observable in statistics Exotic ane space Complex ane space

3.10 Notes[1] Berger 1987, p. 32

[2] Berger, Marcel (1984), Ane spaces, Problems in Geometry, p. 11, ISBN 9780387909714

[3] Berger 1987, p. 33

[4] Snapper, Ernst; Troyer, Robert J. (1989), Metric Ane Geometry, p. 6

[5] Tarrida, Agusti R. (2011), Ane spaces, Ane Maps, Euclidean Motions and Quadrics, pp. 12, ISBN 9780857297105

[6] Nomizu & Sasaki 1994, p. 7

3.11 References Berger, Marcel (1984), Ane spaces, Problems in Geometry, Springer-Verlag, ISBN 978-0-387-90971-4 Berger, Marcel (1987), Geometry I, Berlin: Springer, ISBN 3-540-11658-3 Cameron, Peter J. (1991), Projective and polar spaces, QMW Maths Notes 13, London: Queen Mary andWesteld College School of Mathematical Sciences, MR 1153019


Coxeter, Harold Scott MacDonald (1969), Introduction to Geometry (2nd ed.), New York: JohnWiley & Sons,ISBN 978-0-471-50458-0, MR 123930

Dolgachev, I.V.; Shirokov, A.P. (2001), A/a011100, in Hazewinkel, Michiel, Encyclopedia of Mathematics,Springer, ISBN 978-1-55608-010-4

Snapper, Ernst; Troyer, Robert J. (1989), Metric Ane Geometry (Dover edition, rst published in 1989 ed.),Dover Publications, ISBN 0-486-66108-3

Nomizu, K.; Sasaki, S. (1994), Ane Dierential Geometry (New ed.), Cambridge University Press, ISBN978-0-521-44177-3

Tarrida, Agusti R. (2011), Ane spaces, Ane Maps, Euclidean Motions and Quadrics, Springer, ISBN978-0-85729-709-9

Chapter 4

AmitsurLevitzki theorem

In algebra, the AmitsurLevitzki theorem states that the algebra of n by n matrices satises a certain identity ofdegree 2n. It was proved by Amitsur and Levitsky (1950). In particular matrix rings are polynomial identity ringssuch that the smallest identity they satisfy has degree exactly 2n.

4.1 StatementThe standard polynomial of degree n is

Sn(x1; : : : ; xn) =X2Sn

sgn()x1 xn

in non-commutative polynomials x1,...,xn, where the sum is taken over all n! elements of the symmetric group Sn.The AmitsurLevitzki theorem states that for n by n matrices A1,...,An then

S2n(A1; : : : ; A2n) = 0 :

4.2 ProofsAmitsur and Levitzki (1950) gave the rst proof.Kostant (1958) deduced the AmitsurLevitzki theorem from the KoszulSamelson theorem about primitive coho-mology of Lie algebras.Swan (1963) and Swan (1969) gave a simple combinatorial proof as follows. By linearity it is enough to prove thetheorem when each matrix has only one nonzero entry, which is 1. In this case each matrix can be encoded as adirected edge of a graph with n vertices. So all matrices together give a graph on n vertices with 2n directed edges.The identity holds provided that for any two vertices A and B of the graph, the number of odd Eulerian paths from Ato B is the same as the number of even ones. (Here a path is called odd or even depending on whether its edges takenin order give an odd or even permutation of the 2n edges.) Swan showed that this was the case provided the numberof edges in the graph is at least 2n, thus proving the AmitsurLevitzki theorem.Razmyslov (1974) gave a proof related to the CayleyHamilton theorem.Rosset (1976) gave a short proof using the exterior algebra of a vector space of dimension 2n.

4.3 References Amitsur, A. S.; Levitzki, Jakob (1950), Minimal identities for algebras, Proceedings of the American Mathe-matical Society 1: 449463, doi:10.1090/S0002-9939-1950-0036751-9, ISSN 0002-9939, JSTOR 2032312,

15

16 CHAPTER 4. AMITSURLEVITZKI THEOREM

MR 0036751

Amitsur, A. S.; Levitzki, Jakob (1951), Remarks on Minimal identities for algebras, Proceedings of theAmerican Mathematical Society 2: 320327, ISSN 0002-9939, JSTOR 2032509, MR ?

Formanek, E. (2001), AmitsurLevitzki theorem, in Hazewinkel, Michiel, Encyclopedia of Mathematics,Springer, ISBN 978-1-55608-010-4

Formanek, Edward (1991), The polynomial identities and invariants of nnmatrices, Regional Conference Se-ries inMathematics 78, Providence, RI: AmericanMathematical Society, ISBN0-8218-0730-7, Zbl 0714.16001

Kostant, Bertram (1958), A theorem of Frobenius, a theorem of AmitsurLevitski and cohomology theory,J. Math. Mech. 7: 237264, doi:10.1512/iumj.1958.7.07019, MR 0092755

Razmyslov, Ju. P. (1974), Identities with trace in full matrix algebras over a eld of characteristic zero,Mathematics of the USSR-Izvestiya 8 (4): 727, doi:10.1070/IM1974v008n04ABEH002126, ISSN 0373-2436,MR 0506414

Rosset, Shmuel (1976), A new proof of the AmitsurLevitski identity, Israel Journal of Mathematics 23 (2):187188, doi:10.1007/BF02756797, ISSN 0021-2172, MR 0401804

Swan, Richard G. (1963), An application of graph theory to algebra, Proceedings of the American Mathe-matical Society 14: 367373, ISSN 0002-9939, JSTOR 2033801, MR 0149468

Swan, Richard G. (1969), Correction to An application of graph theory to algebra"", Proceedings of theAmerican Mathematical Society 21: 379380, doi:10.2307/2037008, ISSN 0002-9939, JSTOR 2037008, MR0255439

Chapter 5

Angles between ats

The concept of angles between lines in the plane and between pairs of two lines, two planes or a line and a planein space can be generalised to arbitrary dimension. This generalisation was rst discussed by Jordan.[1] For any pairof ats in a Euclidean space of arbitrary dimension one can dene a set of mutual angles which are invariant underisometric transformation of the Euclidean space. If the ats do not intersect, their shortest distance is one moreinvariant.[1] These angles are called canonical[2] or principal.[3] The concept of angles can be generalised to pairsof ats in a nite-dimensional inner product space over the complex numbers.

5.1 Jordans denition[1]

Let F and G be ats of dimensions k and l in the n -dimensional Euclidean space En . By denition, a translationof F or G does not alter their mutual angles. If F and G do not intersect, they will do so upon any translation of Gwhich maps some point in G to some point in F . It can therefore be assumed without loss of generality that F andG intersect.Jordan shows that Cartesian coordinates x1; : : : ; x; y1; : : : ; y; z1; : : : ; z ; u1; : : : ; u; v1; : : : ; x; w1; : : : ; w inEn can then be dened such that F and G are described, respectively, by the sets of equations

x1 = 0; : : : ; x = 0;

u1 = 0; : : : ; u = 0;

v1 = 0; : : : ; v = 0

and

x1 = 0; : : : ; x = 0;

z1 = 0; : : : ; z = 0;

v1 cos 1 + w1 sin 1 = 0; : : : ; v cos + w sin = 0with 0 < i < /2; i = 1; : : : ; . Jordan calls these coordinates canonical. By denition, the angles i are theangles between F and G .The non-negative integers ; ; ; ; are constrained by

+ + + + 2 = n;

+ + = k;

+ + = l:

17

18 CHAPTER 5. ANGLES BETWEEN FLATS

For these equations to determine the ve non-negative integers completely, besides the dimensions n; k and l andthe number of angles i , the non-negative integer must be given. This is the number of coordinates yi , whosecorresponding axes are those lying entirely within both F and G . The integer is thus the dimension of F \ G .The set of angles i may be supplemented with angles 0 to indicate that F \G has that dimension.Jordans proof applies essentially unaltered whenEn is replaced with the n -dimensional inner product spaceCn overthe complex numbers. (For angles between subspaces, the generalisation toCn is discussed by Galntai and Hegedsin terms of the below variational characterisation.[4])

5.2 Angles between subspacesNow let F andG be subspaces of the n -dimensional inner product space over the real or complex numbers. Geomet-rically, F and G are ats, so Jordans denition of mutual angles applies. When for any canonical coordinate thesymbol ^ denotes the unit vector of the axis, the vectors y^1; : : : ; y^; w^1; : : : ; w^; z^1; : : : ; z^ form an orthonormalbasis for F and the vectors y^1; : : : ; y^; w^01; : : : ; w^0; u^1; : : : ; u^ form an orthonormal basis for G , where

w^0i = wî cos i vî sin i; i = 1; : : : ; :

Being related to canonical coordinates, these basic vectors may be called canonical.When ai; i = 1; : : : ; k denote the canonical basic vectors for F and bi; i = 1; : : : ; l the canonical basic vectors forG then the inner product hai; bji vanishes for any pair of i and j except the following ones.

hyî; yîi = 1; i = 1; : : : ; ;

hwî; w^0ii = cos i; i = 1; : : : ; :With the above ordering of the basic vectors, the matrix of the inner products hai; bji is thus diagonal. In other words,if (a0i; i = 1; : : : ; k) and (b0i; i = 1; : : : ; l) are arbitrary orthonormal bases in F and G then the real, orthogonal orunitary transformations from the basis (a0i) to the basis (ai) and from the basis (b0i) to the basis (bi) realise a singularvalue decomposition of the matrix of inner products ha0i; b0ji . The diagonal matrix elements hai; bii are the singularvalues of the latter matrix. By the uniqueness of the singular value decomposition, the vectors yî are then unique upto a real, orthogonal or unitary transformation among them, and the vectors wî and w^0i (and hence vî ) are unique upto equal real, orthogonal or unitary transformations applied simultaneously to the sets of the vectors wî associatedwith a common value of i and to the corresponding sets of vectors w^0i (and hence to the corresponding sets of vî ).A singular value 1 can be interpreted as cos 0 corresponding to the angles 0 introduced above and associated withF \ G and a singular value 0 can be interpreted as cos/2 corresponding to right angles between the orthogonalspaces F \G? and F? \G , where superscript ? denotes the orthogonal complement.

5.3 Variational characterisation[3]

The variational characterisation of singular vaues and vectors implies as a special case a variational characterisationof the angles between subspaces and their associated canonical vectors. This characterisation includes the angles 0and /2 introduced above and orders the angles by increasing value. It can be given the form of the below alternativedenition. In this context, it is customary to talk of principal angles and vectors.

5.3.1 DenitionLet V be an inner product space. Given two subspaces U ;W with dim(U) = k dim(W) := l , there exists then asequence of k angles 0 1 2 : : : k /2 called the principal angles, the rst one dened as

1 := minarccos

jhu;wijkukkwk

u 2 U ; w 2 W = \(u1; w1);

5.4. REFERENCES 19

where h; i is the inner product and k k the induced norm. The vectors u1 and w1 are the corresponding principalvectors.The other principal angles and vectors are then dened recursively via

i := minarccos

jhu;wijkukkwk

u 2 U ; w 2 W; u ? uj ; w ? wj 8j 2 f1; : : : ; i 1g :This means that the principal angles (1; : : : k) form a set of minimized angles between the two subspaces, and theprincipal vectors in each subspace are orthogonal to each other.

5.3.2 ExamplesGeometric example

Geometrically, subspaces are ats (points, lines, planes etc.) that include the origin, thus any two subspaces intersectat least in the origin. Two two-dimensional subspaces U andW generate a set of two angles. In a three-dimensionalEuclidean space, the subspaces U andW are either identical, or their intersection forms a line. In the former case,both 1 = 2 = 0 . In the latter case, only 1 = 0 , where vectors u1 andw1 are on the line of the intersection U \Wand have the same direction. The angle 2 > 0 will be the angle between the subspaces U andW in the orthogonalcomplement to U \ W . Imagining the angle between two planes in 3D, one intuitively thinks of the largest angle,2 > 0 .

Algebraic example

In 4-dimensional real coordinate space R4, let the two-dimensional subspace U be spanned by u1 = (1; 0; 0; 0)and u2 = (0; 1; 0; 0) , while the two-dimensional subspace W be spanned by w1 = (1; 0; 0; a)/

p1 + a2 and

w2 = (0; 1; b; 0)/p1 + b2 with some real a and b such that jaj < jbj . Then u1 and w1 are, in fact, the pair of

principal vectors corresponding to the angle 1 with cos(1) = 1/p1 + a2 , and u2 and w2 are the principal vectors

corresponding to the angle 2 with cos(2) = 1/p1 + b2

To construct a pair of subspaces with any given set of k angles 1; : : : ; k in a 2k (or larger) dimensional Euclideanspace, take a subspace U with an orthonormal basis (e1; : : : ; ek) and complete it to an orthonormal basis (e1; : : : ; en)of the Euclidean space, where n 2k . Then, an orthonormal basis of the other subspaceW is, e.g.,

(cos(1)e1 + sin(1)ek+1; : : : ; cos(k)ek + sin(k)e2k):

5.3.3 Basic propertiesIf the largest angle is zero, one subspace is a subset of the other.If the smallest angle is zero, the subspaces intersect at least in a line.The number of angles equal to zero is the dimension of the space where the two subspaces intersect.

5.4 References[1] Jordan, C. (1875). Essai sur la gomtrie n dimensions. Bull. Soc. Math. France 3: 103.

[2] Afriat, S. N. (1957). Orthogonal and oblique projectors and the characterisation of pairs of vector spaces. Math. Proc.Cambridge Philos. Soc. 53: 800. doi:10.1017/S0305004100032916.

[3] Bjrck, .; Golub, G. H. (1973). Numerical Methods for Computing Angles Between Linear Subspaces. Math. Comp.27: 579. doi:10.2307/2005662.

[4] Galntai, A.; Hegeds, Cs. J. (2006). Jordans principal angles in complex vector spaces. Numer. Linear Algebra Appl.13: 589. doi:10.1002/nla.491.

Chapter 6

Antilinear map

In mathematics, a mapping f : V ! W from a complex vector space to another is said to be antilinear (orconjugate-linear or semilinear, though the latter term is more general) if

f(ax+ by) = af(x) + bf(y)

for all a; b 2 C and all x; y 2 V , where a andb are the complex conjugates of a and b respectively. The compositionof two antilinear maps is complex-linear.An antilinear map f : V !W may be equivalently described in terms of the linear map f : V ! W from V to thecomplex conjugate vector space W .Antilinear maps occur in quantummechanics in the study of time reversal and in spinor calculus, where it is customaryto replace the bars over the basis vectors and the components of geometric objects by dots put above the indices.

6.1 References Horn and Johnson, Matrix Analysis, Cambridge University Press, 1985. ISBN 0-521-38632-2. (antilinearmaps are discussed in section 4.6).

Budinich, P. and Trautman, A. The Spinorial Chessboard. Spinger-Verlag, 1988. ISBN 0-387-19078-3. (an-tilinear maps are discussed in section 3.3).

6.2 See also Linear map Complex conjugate Sesquilinear form Matrix consimilarity Time reversal

20

Chapter 7

Antiunitary operator

In mathematics, an antiunitary transformation, is a bijective antilinear map

U : H1 ! H2between two complex Hilbert spaces such that

hUx;Uyi = hx; yi

for all x and y in H1 , where the horizontal bar represents the complex conjugate. If additionally one has H1 = H2then U is called an antiunitary operator.Antiunitary operators are important in Quantum Theory because they are used to represent certain symmetries, suchas time-reversal symmetry. Their fundamental importance in quantum physics is further demonstrated by WignersTheorem.

7.1 Invariance transformationsIn Quantum mechanics, the invariance transformations of complex Hilbert spaceH leave the absolute value of scalarproduct invariant:

jhTx; Tyij = jhx; yij

for all x and y inH . Due to Wigners Theorem these transformations fall into two categories, they can be unitary orantiunitary.

7.1.1 Geometric InterpretationCongruences of the plane form two distinct classes. The rst conserves the orientation and is generated by translationsand rotations. The second does not conserve the orientation and is obtained from the rst class by applying a reection.On the complex plane these two classes corresponds (up to translation) to unitaries and antiunitaries, respectively.

7.2 Properties hUx;Uyi = hx; yi = hy; xi holds for all elements x; y of the Hilbert space and an antiunitary U . When U is antiunitary then U2 is unitary. This follows from

21

22 CHAPTER 7. ANTIUNITARY OPERATOR

hU2x;U2yi = hUx;Uyi = hx; yi:

For unitary operator V the operator V K , whereK is complex conjugate operator, is antiunitary. The reverseis also true, for antiunitary U the operator UK is unitary.

For antiunitary U the denition of the adjoint operator U is changed into

hUx; yi = hx;Uyi

The adjoint of an antiunitary U is also antiunitary and

UU = UU = 1: (This is not to be confused with the denition of unitary operators, as U is notcomplex linear.)

7.3 Examples The complex conjugate operatorK;Kz = z; is an antiunitary operator on the complex plane. The operator

U = yK =

0 ii 0

K;

where y is the second Pauli matrix andK is the complex conjugate operator, is antiunitary. It satises U2 = 1 .

7.4 Decomposition of an antiunitary operator into a direct sum of elemen-tary Wigner antiunitaries

An antiunitary operator on a nite-dimensional space may be decomposed as a direct sum of elementary WignerantiunitariesW , 0 . The operatorW0 : C ! C is just simple complex conjugation on C

W0(z) = z

For 0 < , the operationW acts on two-dimensional complex Hilbert space. It is dened by

W((z1; z2)) = (ei/2z2; e

i/2z1):

Note that for 0 <

W(W((z1; z2))) = (eiz1; e

iz2);

so suchW may not be further decomposed intoW0 's, which square to the identity map.Note that the above decomposition of antiunitary operators contrasts with the spectral decomposition of unitaryoperators. In particular, a unitary operator on a complex Hilbert space may be decomposed into a direct sum ofunitaries acting on 1-dimensional complex spaces (eigenspaces), but an antiunitary operator may only be decomposedinto a direct sum of elementary operators on 1- and 2-dimensional complex spaces.

7.5 References Wigner, E. Normal Form of Antiunitary Operators, Journal of Mathematical Physics Vol 1, no 5, 1960, pp.409412

Wigner, E. Phenomenological Distinction between Unitary and Antiunitary Symmetry Operators, Journal ofMathematical Physics Vol1, no5, 1960, pp.414416

7.6. SEE ALSO 23

7.6 See also Unitary operator Wigners Theorem Particle physics and representation theory

Chapter 8

Asymmetric norm

In mathematics, an asymmetric norm on a vector space is a generalization of the concept of a norm.

8.1 DenitionLet X be a real vector space. Then an asymmetric norm on X is a function p : X R satisfying the followingproperties:

non-negativity: for all x X, p(x) 0; deniteness: for x X, x = 0 if and only if p(x) = p(x) = 0; homogeneity: for all x X and all 0, p(x) = p(x); the triangle inequality: for all x, y X, p(x + y) p(x) + p(y).

8.2 Examples On the real line R, the function p given by

p(x) =

(jxj; x 0;2jxj; x 0;

is an asymmetric norm but not a norm.

More generally, given a strictly positive function g : Sn1 R dened on the unit sphere Sn1 in Rn (withrespect to the usual Euclidean norm ||, say), the function p given by

p(x) = g(x/jxj)jxj

is an asymmetric norm on Rn but not necessarily a norm.

8.3 References Cobza, S. (2006). Compact operators on spaces with asymmetric norm. Stud. Univ. Babe-Bolyai Math.51 (4): 6987. ISSN 0252-1938. MR 2314639.

24

8.4. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 25

8.4 Text and image sources, contributors, and licenses8.4.1 Text

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8.4.3 Content license Creative Commons Attribution-Share Alike 3.0

Absolutely convex setProperties Absolutely convex hull ReferencesSee also

Adjugate matrixDefinition Examples 2 2 generic matrix 3 3 generic matrix 3 3 numeric matrix

Properties InversesCharacteristic polynomialJacobis formulaCayleyHamilton formula

See alsoReferencesExternal links

Affine spaceInformal descriptionsDefinitionSubtraction and Weyls axiomsAffine combinations

ExamplesAffine subspaces Affine combinations and affine dependenceGeometric objects as points and vectorsAxiomsRelation to projective spaces See alsoNotesReferences

AmitsurLevitzki theoremStatementProofsReferences

Angles between flatsJordans definition*[1]Angles between subspacesVariational characterisation*[3]DefinitionExamplesBasic properties

References

Antilinear mapReferencesSee also

Antiunitary operatorInvariance transformationsGeometric Interpretation

PropertiesExamplesDecomposition of an antiunitary operator into a direct sum of elementary Wigner antiunitariesReferencesSee also

Asymmetric normDefinitionExamplesReferencesText and image sources, contributors, and licensesTextImagesContent license

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