Homework 2. Solutions 1 a) Show that (x, y) = x1y1 + x2y2 + x3y3
... i.e. rotation on the angle ϕ is a composition of two reflections. 7† Prove the Cauchy–Bunyakovsky–Schwarz inequality (x, y)2 ≤ (x, x)(y, y) , where x, y are arbitrary two vectors and ( , ) is a scalar product in Euclidean space. Hint: For any two given vectors x, y consider the quadratic polynomial ...
... i.e. rotation on the angle ϕ is a composition of two reflections. 7† Prove the Cauchy–Bunyakovsky–Schwarz inequality (x, y)2 ≤ (x, x)(y, y) , where x, y are arbitrary two vectors and ( , ) is a scalar product in Euclidean space. Hint: For any two given vectors x, y consider the quadratic polynomial ...
Physics 70007, Fall 2009 Answers to HW set #2
... where H11 , H22 , and H12 are real numbers with the dimension of energy, and |1i and |2i are eigenkets of some observable (6= H ). Find the energy eigenkets and corresponding energy eigenvalues. Make sure that your answer makes good sense for H12 = 0. The eigenvalues are found in the usual manner: ...
... where H11 , H22 , and H12 are real numbers with the dimension of energy, and |1i and |2i are eigenkets of some observable (6= H ). Find the energy eigenkets and corresponding energy eigenvalues. Make sure that your answer makes good sense for H12 = 0. The eigenvalues are found in the usual manner: ...
Matrix Arithmetic
... rigor of linear transformations may find this useful. Please be sure to cover the text’s Chapter One first. This material is Free, including the LaTeX source, under the Creative Commons AttributionShareAlike 2.5 License. The latest version should be the one on this site. Any feedback on the note is ...
... rigor of linear transformations may find this useful. Please be sure to cover the text’s Chapter One first. This material is Free, including the LaTeX source, under the Creative Commons AttributionShareAlike 2.5 License. The latest version should be the one on this site. Any feedback on the note is ...
Classical groups and their real forms
... First of all we want to get familiar with classical groups. Therefor we will define them and consider both coordinate and coordinate-free representations. Definition 1. Let V be a vector space over E. A classical group is either GL(V ) or a subgroup that preserves a non-degenerate sesquilinear form be ...
... First of all we want to get familiar with classical groups. Therefor we will define them and consider both coordinate and coordinate-free representations. Definition 1. Let V be a vector space over E. A classical group is either GL(V ) or a subgroup that preserves a non-degenerate sesquilinear form be ...
Linear transformations and matrices Math 130 Linear Algebra
... Aha! You see it! If you take the elements in the ith row from A and multiply them in order with elements in the column of v, then add those n products together, you get the ith element of T (v). With this as our definition of multiplication of an m × n matrix by a n × 1 column vector, we have Av = T ...
... Aha! You see it! If you take the elements in the ith row from A and multiply them in order with elements in the column of v, then add those n products together, you get the ith element of T (v). With this as our definition of multiplication of an m × n matrix by a n × 1 column vector, we have Av = T ...
Application: directed graphs
... Definition 1. Let G be a graph with m edges and n nodes. The edge-node incident matrix of G is the m × n matrix A with ...
... Definition 1. Let G be a graph with m edges and n nodes. The edge-node incident matrix of G is the m × n matrix A with ...
Semester 2 Program
... Topic 4: Applications of Trigonometry, Topic 5: Linear and Exponential Functions and their Graphs, and Topic 6: Matrices and Networks Lesson 1 – Single Lesson Term Two Week 9 ...
... Topic 4: Applications of Trigonometry, Topic 5: Linear and Exponential Functions and their Graphs, and Topic 6: Matrices and Networks Lesson 1 – Single Lesson Term Two Week 9 ...
cg-type algorithms to solve symmetric matrix equations
... There are alternative strategies for solving (1) by iterative methods. In [6], the block CG (Bl-CG) method has been presented when A is an SPD matrix. Another method which is based on Krylov subspace methods has been proposed in [9] for linear system of equations with general coefficient matrices. R ...
... There are alternative strategies for solving (1) by iterative methods. In [6], the block CG (Bl-CG) method has been presented when A is an SPD matrix. Another method which is based on Krylov subspace methods has been proposed in [9] for linear system of equations with general coefficient matrices. R ...
Working with Your Data (Chapter 2 in the Little
... Is the sum of products of the elements of the ith row of A each multiplied by the corresponding element of x. From this definition and from the example it is easily seen that Ax is defined only when the number of rows in A are equal to the number of elements in the rows of A (i.e. number of columns) ...
... Is the sum of products of the elements of the ith row of A each multiplied by the corresponding element of x. From this definition and from the example it is easily seen that Ax is defined only when the number of rows in A are equal to the number of elements in the rows of A (i.e. number of columns) ...
Stochastic Matrices in a Finite Field Introduction Literature review
... matrices will have solutions to their characteristic equations that are in their respective finite fields and as a consequence, the trace and determinant of any stochastic matrix with a finite field equal the sum and product of its eigenvalues respectively. We have also proven that we have only one ...
... matrices will have solutions to their characteristic equations that are in their respective finite fields and as a consequence, the trace and determinant of any stochastic matrix with a finite field equal the sum and product of its eigenvalues respectively. We have also proven that we have only one ...
Matrices with a strictly dominant eigenvalue
... is called regular if A satisfies condition (R). Now we have the following well-known theorem (for another proof of this theorem cf. e.g. [6]): Theorem 3.1 The state vectors of a regular Markov chain converge to the unique right eigenvector of the corresponding transition matrix with component sum 1 ...
... is called regular if A satisfies condition (R). Now we have the following well-known theorem (for another proof of this theorem cf. e.g. [6]): Theorem 3.1 The state vectors of a regular Markov chain converge to the unique right eigenvector of the corresponding transition matrix with component sum 1 ...
MATH 310, REVIEW SHEET 1 These notes are a very short
... is that we’re now thinking of the equation in terms of linear combinations. • Do basic operations on vectors • Convert a system of linear equations into a vector equation, and vice versa • Find the general solution of a system of vector equations • Determine if a vector is a linear combination of ot ...
... is that we’re now thinking of the equation in terms of linear combinations. • Do basic operations on vectors • Convert a system of linear equations into a vector equation, and vice versa • Find the general solution of a system of vector equations • Determine if a vector is a linear combination of ot ...
ON PSEUDOSPECTRA AND POWER GROWTH 1. Introduction and
... 1.2. What about diagonalizable matrices? The matrices A, B in the Greenbaum–Trefethen example are nilpotent, as are those constructed in the proof of Theorem 1.1 above. Obviously, these are rather special. What happens if, instead, we consider more generic matrices, for example diagonalizable matric ...
... 1.2. What about diagonalizable matrices? The matrices A, B in the Greenbaum–Trefethen example are nilpotent, as are those constructed in the proof of Theorem 1.1 above. Obviously, these are rather special. What happens if, instead, we consider more generic matrices, for example diagonalizable matric ...
Linear Algebra, Section 1.9 First, some vocabulary: A function is a
... If we don’t want to specify that a function is onto its codomain, we will say that f maps x into the codomain. • A function y = f (x) is said to be 1 − 1 if f (x1 ) = f (x2 ) ⇒ x1 = x2 or (this is logically equivalent to the equation above): x1 6= x2 ⇒ f (x1 ) 6= f (x2 ) In words, this means that ea ...
... If we don’t want to specify that a function is onto its codomain, we will say that f maps x into the codomain. • A function y = f (x) is said to be 1 − 1 if f (x1 ) = f (x2 ) ⇒ x1 = x2 or (this is logically equivalent to the equation above): x1 6= x2 ⇒ f (x1 ) 6= f (x2 ) In words, this means that ea ...
