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... Observation 4.4. Let A be an adjacency matrix of a labeled graph G, with n vertices. Then the entry aij of Ak , where k ≥ 1, is the number of walks of length k from vertex j to vertex i. Proof by Induction. The statement holds for k = 1 by the definition of an adjacency matrix. Assume the statement ...
... Observation 4.4. Let A be an adjacency matrix of a labeled graph G, with n vertices. Then the entry aij of Ak , where k ≥ 1, is the number of walks of length k from vertex j to vertex i. Proof by Induction. The statement holds for k = 1 by the definition of an adjacency matrix. Assume the statement ...
Galois Field Computations A Galois field is an algebraic field that
... to manipulate ordinary MATLAB arrays of real numbers, making the GF(2 m) capabilities of the toolbox easy to learn and use. The topics in this section are ...
... to manipulate ordinary MATLAB arrays of real numbers, making the GF(2 m) capabilities of the toolbox easy to learn and use. The topics in this section are ...
Lecture 1: Lattice ideals and lattice basis ideals
... free, there exists an embedding Zn /L ⊂ Zd for some d. Let e1 , . . . , en be the canonical basis of Zn . Then for i = 1, . . . , n, d e i + L is mapped to ai ∈ Z via this embedding. It follows that P n t i =1 bi ai = 0 if and only if b = (b1 , . . . , bn ) ∈ L. In other words, b ∈ L if and only if ...
... free, there exists an embedding Zn /L ⊂ Zd for some d. Let e1 , . . . , en be the canonical basis of Zn . Then for i = 1, . . . , n, d e i + L is mapped to ai ∈ Z via this embedding. It follows that P n t i =1 bi ai = 0 if and only if b = (b1 , . . . , bn ) ∈ L. In other words, b ∈ L if and only if ...
power-associative rings - American Mathematical Society
... simple algebras. The final line of investigation we shall present here is a complete determination of those algebras 21 such that 2I(+) is a simple Jordan algebra. We are first led to attach to any algebra 33 over a field % an algebra 33(X) defined for every X of %. This algebra is the same vector s ...
... simple algebras. The final line of investigation we shall present here is a complete determination of those algebras 21 such that 2I(+) is a simple Jordan algebra. We are first led to attach to any algebra 33 over a field % an algebra 33(X) defined for every X of %. This algebra is the same vector s ...
Hankel Matrices: From Words to Graphs
... General Hankel Matrices: I Let C be a class possibly labeled graphs, hyper-graphs or τ -structures. Let 2 be a binary operation define on C. Let Gi be an enumeration of all (labeled) finite graphs Let f be graph parameter. The (full) Hankel matrix M (f, 2) is defined by M (f, 2)i,j = f (Gi 2Gj ) and ...
... General Hankel Matrices: I Let C be a class possibly labeled graphs, hyper-graphs or τ -structures. Let 2 be a binary operation define on C. Let Gi be an enumeration of all (labeled) finite graphs Let f be graph parameter. The (full) Hankel matrix M (f, 2) is defined by M (f, 2)i,j = f (Gi 2Gj ) and ...
lecture13_densela_1_.. - People @ EECS at UC Berkeley
... New lower bound for all “direct” linear algebra Let M = “fast” memory size per processor = cache size (sequential case) or n2/p (parallel case) #words_moved by at least one processor = (#flops / M1/2 ) #messages_sent by at least one processor = (#flops / M3/2 ) • Holds for • BLAS, LU, QR, eig, S ...
... New lower bound for all “direct” linear algebra Let M = “fast” memory size per processor = cache size (sequential case) or n2/p (parallel case) #words_moved by at least one processor = (#flops / M1/2 ) #messages_sent by at least one processor = (#flops / M3/2 ) • Holds for • BLAS, LU, QR, eig, S ...
lecture notes - TU Darmstadt/Mathematik
... Lorenz, and Krishnan Narayanan gave valuable hints for a better exposition or informed me about typos or omissions in the text. There are certainly still many errors in this text. If you find one it would be nice if you write me an email. Any other feedback is also welcome. ...
... Lorenz, and Krishnan Narayanan gave valuable hints for a better exposition or informed me about typos or omissions in the text. There are certainly still many errors in this text. If you find one it would be nice if you write me an email. Any other feedback is also welcome. ...
Implementing a Toolkit for Ring
... Next, the notion of lattices and the decoding and discretization algorithms are presented in Section 1.3. Finally, Section 1.4 deals with algebraic number theory and cyclotomic number fields. This includes some lesser-known concepts, like tensorial decomposition in prime power cyclotomics and the du ...
... Next, the notion of lattices and the decoding and discretization algorithms are presented in Section 1.3. Finally, Section 1.4 deals with algebraic number theory and cyclotomic number fields. This includes some lesser-known concepts, like tensorial decomposition in prime power cyclotomics and the du ...
Representation Theory of Finite Groups
... Definition 2.13 (Equivalence of Representations). Let (ρ, V ) and (ρ′ , V ′ ) be two representations of G. The representations (ρ, V ) and (ρ′ , V ′ ) are called G-equivalent (or equivalent) if there exists a linear isomorphism T : V → V ′ such that ρ′ (g) = T ρ(g)T −1 for all g ∈ G. Let ρ be a repr ...
... Definition 2.13 (Equivalence of Representations). Let (ρ, V ) and (ρ′ , V ′ ) be two representations of G. The representations (ρ, V ) and (ρ′ , V ′ ) are called G-equivalent (or equivalent) if there exists a linear isomorphism T : V → V ′ such that ρ′ (g) = T ρ(g)T −1 for all g ∈ G. Let ρ be a repr ...
Jordan normal form
In linear algebra, a Jordan normal form (often called Jordan canonical form)of a linear operator on a finite-dimensional vector space is an upper triangular matrix of a particular form called a Jordan matrix, representing the operator with respect to some basis. Such matrix has each non-zero off-diagonal entry equal to 1, immediately above the main diagonal (on the superdiagonal), and with identical diagonal entries to the left and below them. If the vector space is over a field K, then a basis with respect to which the matrix has the required form exists if and only if all eigenvalues of the matrix lie in K, or equivalently if the characteristic polynomial of the operator splits into linear factors over K. This condition is always satisfied if K is the field of complex numbers. The diagonal entries of the normal form are the eigenvalues of the operator, with the number of times each one occurs being given by its algebraic multiplicity.If the operator is originally given by a square matrix M, then its Jordan normal form is also called the Jordan normal form of M. Any square matrix has a Jordan normal form if the field of coefficients is extended to one containing all the eigenvalues of the matrix. In spite of its name, the normal form for a given M is not entirely unique, as it is a block diagonal matrix formed of Jordan blocks, the order of which is not fixed; it is conventional to group blocks for the same eigenvalue together, but no ordering is imposed among the eigenvalues, nor among the blocks for a given eigenvalue, although the latter could for instance be ordered by weakly decreasing size.The Jordan–Chevalley decomposition is particularly simple with respect to a basis for which the operator takes its Jordan normal form. The diagonal form for diagonalizable matrices, for instance normal matrices, is a special case of the Jordan normal form.The Jordan normal form is named after Camille Jordan.