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MATH 310, REVIEW SHEET 1 These notes are a very short
... The other thing we talked about was network flow. Here’s the set up: you’re given a network, which consists of a bunch of nodes, and a bunch of branches connecting the nodes. You can think of it as a network of streets, where the nodes are intersections, and the branches are the roads between the in ...
... The other thing we talked about was network flow. Here’s the set up: you’re given a network, which consists of a bunch of nodes, and a bunch of branches connecting the nodes. You can think of it as a network of streets, where the nodes are intersections, and the branches are the roads between the in ...
Coloring Random 3-Colorable Graphs with Non
... eigenvalue of A. We define the variance of the distribution G(A) to be σ 2 = maxu,v (Auv − A2uv ) (i. e., the maximal variance of individual entries). The variance is bounded by 1/4 and goes to zero if all entries of A go either to zero or to one. For instance, for the distribution G(r, p, 3), if p = ...
... eigenvalue of A. We define the variance of the distribution G(A) to be σ 2 = maxu,v (Auv − A2uv ) (i. e., the maximal variance of individual entries). The variance is bounded by 1/4 and goes to zero if all entries of A go either to zero or to one. For instance, for the distribution G(r, p, 3), if p = ...
Word
... Read Chapter 3.4 – 3.8 of the MATLAB book before coming to class. This preparation material is provided to supplement this reading. Students will learn a basic understanding of array operations in MATLAB. This material contains the following: 1. Matrix Arithmetic Overview a. Matrix Math (beyond this ...
... Read Chapter 3.4 – 3.8 of the MATLAB book before coming to class. This preparation material is provided to supplement this reading. Students will learn a basic understanding of array operations in MATLAB. This material contains the following: 1. Matrix Arithmetic Overview a. Matrix Math (beyond this ...
Numerical analysis of a quadratic matrix equation
... eigenvalues are real and nonpositive and there is a gap between the n largest (the primary eigenvalues) and the n smallest (the secondary eigenvalues); there are n linearly independent eigenvectors associated with the primary eigenvalues and likewise for the secondary eigenvalues; and Q(X ) has at l ...
... eigenvalues are real and nonpositive and there is a gap between the n largest (the primary eigenvalues) and the n smallest (the secondary eigenvalues); there are n linearly independent eigenvectors associated with the primary eigenvalues and likewise for the secondary eigenvalues; and Q(X ) has at l ...
MATH 2030: MATRICES Introduction to Linear Transformations We
... The matrix in the proof of the last theorem is called the standard matrix of the linear transformation T. Example 0.7. Q: Show that a rotation about the origin through an angle θ defines a linear transformation from R2 to R2 and find its standard matrix. A: Let Rθ be the rotation, we will prove this ...
... The matrix in the proof of the last theorem is called the standard matrix of the linear transformation T. Example 0.7. Q: Show that a rotation about the origin through an angle θ defines a linear transformation from R2 to R2 and find its standard matrix. A: Let Rθ be the rotation, we will prove this ...
Question Sheet 1 1. Let u = (−1,1,2) v = (2,0,3) w = (1,3,12
... to another one, and the area of the image will be det(A) times the area of the original figure. A similar statement is true in higher dimensions: for example in R3 ; but one should replace “area” by “volume ” in this case. One easy case is when the matrix is a 3 by 3 scalar, say 5I3 . Then T will st ...
... to another one, and the area of the image will be det(A) times the area of the original figure. A similar statement is true in higher dimensions: for example in R3 ; but one should replace “area” by “volume ” in this case. One easy case is when the matrix is a 3 by 3 scalar, say 5I3 . Then T will st ...
Chapter 9 Linear transformations
... with ad − bc = 0 but a, b, c, d not all zero? Suppose for example that none of a, b, c, d is zero. Then ad − bc = 0 implies that a/c = b/d, so the second column of A is a scalar multiple of the first column. Writing u for first column of A and λu for the second column of A we see that ...
... with ad − bc = 0 but a, b, c, d not all zero? Suppose for example that none of a, b, c, d is zero. Then ad − bc = 0 implies that a/c = b/d, so the second column of A is a scalar multiple of the first column. Writing u for first column of A and λu for the second column of A we see that ...
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.