Fall 2010
Matrix Analysis and Computations
Course numbers
ECE 210A, CHE 211A, CMPSC 211A, ME 210A, MATH 206A
Course description
This course gives an introduction to matrix analysis and linear algebra at the graduate
level for students in engineering and a
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Al-gebra and Co-gebra
are brother and sister
Zbigniew Oziewicz
Seht Ihr den Mond dort stehen
er ist nur halb zu sehen
und ist doch rund und scho n
so sind gar manche Sachen
die wir getrost belachen
weil unsr
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be removed for a fruitful usage, e.g. in projective geometry. This is done by introducing cross
ratios. The group which maps two linearly ordered bases onto another is gl n and sln for the
mapping of unimodula
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which is called integral, see [130]. Physicists traditionally chose 5 for this element.
We allow extensors to be inserted into a bracket according to the following rule
A = a1 , . . . , ar ,
B = b 1 , . . . , bs ,
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We will shortly recall the second definition of the regressive product as given in the A2 by
Gramann. First of all we have to define the Erganzung of an extensor A
denoted by a vertical bar |A. Let A be an ext
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This renders the present definition of the meet as computational inefficient. Moreover, it is
unsatisfactory that the meet is a derived product and not directly given as the join or wedge.
The rule of the double
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A is a permutation p SstepA+1 of A such that every block Bs remains to be reduced. In other
words, the blocks Bs consist of ordered subsequences of letters from the word representing A.
The meet of k factors
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as basic objects and constructs lines and planes as joins of points. Recently projective duality
was studied in terms of Clifford algebras [36, 37, 38]. Clifford algebras have been employed for
projective geometr
Chapter 2
Basics on Clifford algebras
2.1
Algebras recalled
In this section we recall some definitions and facts from module and ring theory. In the sense we
use the terms algebra and ring, they are synonyms. We want to address the structure of the
scalar
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We are able to use the bracket to write for the action of a vector x on a reciprocal vector u
x u = xiuk [ei , ek ]
= xiuk [ei , |ek ] = xi uk k
= xiui k.
(1-39)
V
V
This mechanism can be generalized to an ac
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1.4.1
The bracket
While we follow Rota et al. in their mathematical treatment, we separate explicitely from the
comments about co-vectors and Hopf algebras in their writing in the above cited references. It is
les
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where Aij = Aji . It should however be remarked, that this symplectic Clifford algebras are
not related to classical groups in a such direct manner as the orthogonal Clifford algebras. The
point is, that sympl
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equation. Furthermore, we can learn from the polarization process that this type of algebra is
related to anticommutation relations:
X i j
Q(x) =
x x ei ej i
X i j
p (x, y) =
2B
x y (eiej + ej ei )
(1-10)
i,j
whic
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laboration with him which took place in Konstanz in Summer 1999, major problems had been
solved which led to the formation of the BIGEBRA package [3] in December 1999. The package
proved to be calculationable s
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related to C -algebraic methods without assuming positivity.
However, this method was not widely used in spite of reasonable and unique achievements,
most likely due to its lengthy and cumbersome calculations. Whe
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and computational ease.
It turned out to be extremely useful to have geometrical ideas at hand which can be transformed into the QF theoretical framework. As a general rule, it is true that sane geometric
con
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We use Hopf algebraic methods to derive the basic formulas of Clifford algebra theory
(classical and QCA). One of them will be called Pieri-formula of Clifford algebra.
We discuss the Rota-Stein cliffordiza
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[51, 54, 5, 53] where this topic is addressed. A detailed explanation why quantum has been
used as prefix in QCA can be found in [57]. Geometry is reduced to algebra, which is a pity. A
broader treatment, e.g. C
Chapter 1
Peano Space and Gramann-Cayley
Algebra
In this section we will turn our attention to the various possibilities which arise if additional
structures are added to a linear space (k-module or k-vector space). It will turn out that a second
structur
2
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A Treatise on Quantum Clifford Algebras
Normed space normed algebra
Given only a linear space we own very few rules to manipulate its elements. Usually one is
interested in a reasonable extension, e.g. by a distance or length function acting on elem
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Hilbert space, quadratic space classical Clifford algebra
A slightly more general concept is to concentrate in the first place on an inner product. Let
< . | . > : V V k
< x | y > = < y | x >
(1-7)
be a sy
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A morphism of (unital) rings is a mapping f : (R, +, ) (S, +, ) satisfying
f (a + b) = f (a) + f (b)
f (a b) = f (a) f (b)
f (eR ) = eS
if eR , eS do exist.
The kernel of a ring homomorphism f : R S is an ideal
I
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regular if it exists an element b R with aba = a.
left (right) invertible if R is unital and it exists an element b R such that ab = e (ba = e).
invertible if it is left and right invertible.
central if f
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are the Schubert varieties, which are extremely useful in algebraic geometry. Gramannians, flag
manifolds and cohomological aspects can be treated along this route.
Given the variety of approaches to Quantum
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This operation is quite sensible to the chosen basis, as we will exemplify once more on the
element u = u0Id + ui ei + uij ei ej .
GB
u = u0Id + ui ei + uij ei ej
u = u0Id + ui ei uij ei ej
CB
u = u0Id + ui e
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u = u0 Id + uiei + uij ei ej and compute
GB
u = u0 Id + ui ei + uij ei ej
CB
(u) = u0
u = u0 Id + ui ei + uij ei ej
= u0 Id + ui ei + uij (ei ej + Bij )
(u) = u0 + uij Bij
u = u0 Id + ui ei + uij ei ej
dGB
= u0 I
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where t is chosen so that n = (t 2 + t2 ), and (a, b) = 1 when a = b and 0 otherwise. This
state model is a link covariant called the Kauffman bracket [75], which is essentially the Jones
polynomial up to normali
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The components of v can be obtained by applying the canonical co-vectors
relation
i
which fixes #Ilinear forms
i
i
on v, assuming the
(ej ) = i ,
(3-5)
V . Obviously we have
i
(v) =
i
(v j ej ) = v j i = v i,
Chapter 3
Graphical calculi
3.1
3.1.1
The Kuperberg graphical method
Origin of the method
In 1991, Kuperberg introduced a graphical method to visualize tensorial equations [84]. His
method received some recognition, e.g. [76, 77], since he derived a valua
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called co-variant. This notion reflects the fact
coefficients transform in an inverse (contra that under a way as the basis itself. Co-vector inagainst) (linear) change of the basis the vector
=
dices transform c