MA135 Vectors and Matrices Samir Siksek
... to argue along the following lines. Let, say, α = 1 + 2i and β = 3 + 4i. Then we check that αβ = −5 + 10i = βα. “So the multiplication of complex numbers is commutative”. Is this an acceptable proof? No, it is not. This is merely an example. It shows that multiplication is commutative for this parti ...
... to argue along the following lines. Let, say, α = 1 + 2i and β = 3 + 4i. Then we check that αβ = −5 + 10i = βα. “So the multiplication of complex numbers is commutative”. Is this an acceptable proof? No, it is not. This is merely an example. It shows that multiplication is commutative for this parti ...
lecture notes 5
... it remains fixed under conjugation by a set of generators of G. Example 1.7. Let’s compute all the subgroups of the symmetric group Σ3 , and let’s see which ones are conjugate to which other ones, and which ones are normal. Using the presentation hσ, τ | σ2 = id, τ3 = id, στ = τ2 σi for Σ3 , we look ...
... it remains fixed under conjugation by a set of generators of G. Example 1.7. Let’s compute all the subgroups of the symmetric group Σ3 , and let’s see which ones are conjugate to which other ones, and which ones are normal. Using the presentation hσ, τ | σ2 = id, τ3 = id, στ = τ2 σi for Σ3 , we look ...
Triangularizability of Polynomially Compact Operators
... An operator T on an infinite-dimensional complex Banach space X is called polynomially compact if there exists a non-zero complex polynomial p such that the operator p(T ) is compact. If T k is compact for some k we say that T is a power compact operator. Trivial examples of polynomially compact ope ...
... An operator T on an infinite-dimensional complex Banach space X is called polynomially compact if there exists a non-zero complex polynomial p such that the operator p(T ) is compact. If T k is compact for some k we say that T is a power compact operator. Trivial examples of polynomially compact ope ...
GROUP ALGEBRAS. We will associate a certain algebra to a
... Proposition 0.3. Let φ : V −→ W be a F [G]-homomorphism. Then ker(φ) and im(φ) are F [G]-submodules of V and W respectively. Definition 0.3. Let G be a finite group, F a field and V a finite dimensional vector space over F . A representation ρ of G on V is a group homomorphism ρ : G −→ GL(V ) A repr ...
... Proposition 0.3. Let φ : V −→ W be a F [G]-homomorphism. Then ker(φ) and im(φ) are F [G]-submodules of V and W respectively. Definition 0.3. Let G be a finite group, F a field and V a finite dimensional vector space over F . A representation ρ of G on V is a group homomorphism ρ : G −→ GL(V ) A repr ...
Parallel numerical linear algebra
... These issues are locality and regularity of computation. Locality refers to the proximity of the arithmetic and storage components of computers. Computers store data in memories, which are physically separated from the computational units that perform useful arithmetic or logical functions. The time ...
... These issues are locality and regularity of computation. Locality refers to the proximity of the arithmetic and storage components of computers. Computers store data in memories, which are physically separated from the computational units that perform useful arithmetic or logical functions. The time ...
Tensors and hypermatrices
... multilinear functional. The next sections discuss the various generalizations of well-known linear algebraic and matrix theoretic notions, such as rank, norm, and determinant, to tensors and hypermatrices. The realization that these notions may be defined for order-d hypermatrices where d > 2 and th ...
... multilinear functional. The next sections discuss the various generalizations of well-known linear algebraic and matrix theoretic notions, such as rank, norm, and determinant, to tensors and hypermatrices. The realization that these notions may be defined for order-d hypermatrices where d > 2 and th ...
Linear Algebra and Differential Equations
... The present text consists of 130 pages of lecture notes, including numerous pictures and exercises, for a one-semester course in Linear Algebra and Differential Equations. The notes are reasonably self-contained. In particular, prior knowledge of Multivariable Calculus is not required. Calculators a ...
... The present text consists of 130 pages of lecture notes, including numerous pictures and exercises, for a one-semester course in Linear Algebra and Differential Equations. The notes are reasonably self-contained. In particular, prior knowledge of Multivariable Calculus is not required. Calculators a ...
![arXiv:math/0607084v3 [math.NT] 26 Sep 2008](http://s1.studyres.com/store/data/014838781_1-63e86760b3063903e159a8155f16fef1-300x300.png)
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.