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
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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 si
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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 on
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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 foll
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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 e
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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 direc
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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 lette
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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]. Clif
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 s
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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
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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
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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
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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
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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 De
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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 lik
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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
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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 C
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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 t
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
2
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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 extensio
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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. Le
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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.
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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).
inver
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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
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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 +
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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)
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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],
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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
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 so
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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) (lin