103B - Homework 1 Solutions - Roman Kitsela Exercise 1. Q6 Proof
... Proof. We need to determine whether the n × n real matrices with determinant 2 form a subgroup of GL(n, R). The group operation is matrix multiplication (inherited from GL(n, R)) and the main thing that can go wrong in subgroup problems is the closure property fails (if I have two elements both in t ...
... Proof. We need to determine whether the n × n real matrices with determinant 2 form a subgroup of GL(n, R). The group operation is matrix multiplication (inherited from GL(n, R)) and the main thing that can go wrong in subgroup problems is the closure property fails (if I have two elements both in t ...
AN INTRODUCTION TO REPRESENTATION THEORY. 2. Lecture 2
... Let us discuss the case A = k[x]. Since this algebra is commutative, the irreducible representations of A are always 1-dimensional representations ρ(x) = λ ∈ k. The classification of indecomposable representations of k[x] is more interesting. Recall that any linear operator on a finite dimensional v ...
... Let us discuss the case A = k[x]. Since this algebra is commutative, the irreducible representations of A are always 1-dimensional representations ρ(x) = λ ∈ k. The classification of indecomposable representations of k[x] is more interesting. Recall that any linear operator on a finite dimensional v ...
Constructing Lie Algebras of First Order Differential Operators
... n variables; this is the goal of the paper at hand. We will restrict our attention to a particular type of Lie algebras of differential operators, which can be formally described as follows. Let K be a field of characteristic 0, let g be a Lie algebra over K, and let k be a subalgebra of g of finite ...
... n variables; this is the goal of the paper at hand. We will restrict our attention to a particular type of Lie algebras of differential operators, which can be formally described as follows. Let K be a field of characteristic 0, let g be a Lie algebra over K, and let k be a subalgebra of g of finite ...
Non-commutative arithmetic circuits with division
... division (equivalently, inverse) gates. Such a circuit computes a ”non-commutative rational function” – a far more complicated object than its commutative counterpart. Traditionally, arithmetic circuit complexity focuses on the computation of polynomials, with rational functions receiving minor atte ...
... division (equivalently, inverse) gates. Such a circuit computes a ”non-commutative rational function” – a far more complicated object than its commutative counterpart. Traditionally, arithmetic circuit complexity focuses on the computation of polynomials, with rational functions receiving minor atte ...
Compact quantum groups
... finite-dimensional representations (the fundamental representation u, its conjugate and their tensor products). This fact essentially simplifies the theory. In the present paper we have chosen the second more ambitious approach. We start with a unital C∗ -algebra A. Elements of the algebra are inter ...
... finite-dimensional representations (the fundamental representation u, its conjugate and their tensor products). This fact essentially simplifies the theory. In the present paper we have chosen the second more ambitious approach. We start with a unital C∗ -algebra A. Elements of the algebra are inter ...
PDF version of lecture with all slides
... In MATLAB, in order to properly calculate the dot product of two vectors use >>sum(a.*b) element by element mul>plica>on (.*) sum the results A . prior to the * or / indicates that matlab shou ...
... In MATLAB, in order to properly calculate the dot product of two vectors use >>sum(a.*b) element by element mul>plica>on (.*) sum the results A . prior to the * or / indicates that matlab shou ...
Semidefinite and Second Order Cone Programming Seminar Fall 2001 Lecture 9
... that U is a subalgebra of V if it is isomorphic to a subalgebra of V. Thus we have shown that all associative algebras are essentially homomorphic to some some algebra of square matrices. Let us see some examples of subalgebras and isomorphisms within the algebra of matrices. Example 3 (Some subalge ...
... that U is a subalgebra of V if it is isomorphic to a subalgebra of V. Thus we have shown that all associative algebras are essentially homomorphic to some some algebra of square matrices. Let us see some examples of subalgebras and isomorphisms within the algebra of matrices. Example 3 (Some subalge ...
Chapter 4: Polynomials A polynomial is an expression of the form p
... p(X), that is, p(a) = 0. The above discussion tells that p(X) = (X − a)q(X) for some polynomial q(X). Now we ask the question: is a also a root of q(X)? The answer can be found by computing q(a), the value of q(X) at a, to see if it is zero. If q(a) = 0, the answer is No and in that case we say tha ...
... p(X), that is, p(a) = 0. The above discussion tells that p(X) = (X − a)q(X) for some polynomial q(X). Now we ask the question: is a also a root of q(X)? The answer can be found by computing q(a), the value of q(X) at a, to see if it is zero. If q(a) = 0, the answer is No and in that case we say tha ...
5.6 UNITARY AND ORTHOGONAL MATRICES
... Notice that because U∗ U = I ⇐⇒ UU∗ = I, the columns of U are orthonormal if and only if the rows of U are orthonormal, and this is why the definitions of unitary and orthogonal matrices can be stated either in terms of orthonormal columns or orthonormal rows. Another nice feature is that multiplicat ...
... Notice that because U∗ U = I ⇐⇒ UU∗ = I, the columns of U are orthonormal if and only if the rows of U are orthonormal, and this is why the definitions of unitary and orthogonal matrices can be stated either in terms of orthonormal columns or orthonormal rows. Another nice feature is that multiplicat ...
