Hta11 2. Prove that for the (Hilbert) matrix is positive definite. i+j-1 i.j-1 Hnts: (Proceed fro...
5. Recall that a symmetric matrix A is positive definite (SPD for short) if and only if T Ar > O for every nonzero vector 2. 5a. Find a 2-by-2 matrix A that (1) is symmetric, (2) is not singular, and (3) has all its elements greater than zero, but (1) is not SPD. Show a nonzero vector such that zAx < 0. 5b. Let B be a nonsingular matrix, of any size, not necessarily symmetric. Prove that the matrix...
(a) Let S be a symmetric positive definite matrix and define a function | on R" by 1/2 xx Sx . Prove that this function defines a vector norm. Hint: Use the Cholesky decomposition. (b) Find an example of square matrices A an This shows that ρ(A) is not a norm. Note: there are very simple examples. d B such that ρ(A+B)>ρ(A) + ρ(8)
(a) Let S be a symmetric positive definite matrix and define a function | on R"...
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2. (a) Use Octave as a Calculator to answer this question. Suppose that A and B are two 8 x 9 matrices. The (i.j)-entry of the matrix B is given by i *j -1. The (i. j)-entry of the matrix A equals 0 if i + j is divisible by 5 and equals the (i,j)-entry of the matrix B otherwise. i. What are the rank and nullity of matrices A and B? ii. Is vector u 9,64,-71,...
Homework 06b For...Next Statements The elements of a Hilbert matrix are given by the formula bu=1/(i+j-1), where į and j are the row and column indices of the matrix. For example, the 5x5 Hilbert matrix is: [ 1 172 1/3 1/4 175 172 173 174 175 176 173 174 175 176 177 174 175 176 177 178 175 176 177 178 179 Write a Sub that writes the elements of a mxm Hilbert matrix on an Excel spreadsheet, where m...
Problem 5 (a) Let A be an n × m matrix, and suppose that there exists a m × n matrix B such that BA = 1- (i) Let b є Rn be such that the system of equations Ax b has at least one solution. Prove that this solution must be unique. (ii) Must it be the case that the system of equations Ax = b has a solution for every b? Prove or provide a counterexample. (b) Let...
(j) ſte tanº e de Hint: Etan2 e. Bonus Questions 1. (2 points) Evaluate the definite integral 2-2 dx as a limit of Riemann sums. Hint: take the sample points x* = Xi-1Xi, i = 1, 2, ..., n. The idea behind partial fractions might also prove helpful. 2. (3 points) Develop a technique of integration using the Quotient Rule (in the same way that the Chair Rule was used to develop the Substitution Rule and the Product Rule was...
Problem 1. Please note a correction: A matrix is positive definite if all its lead- ing principal minors are positive and negative definite if the principal minor:s alternate in sign, starting with negative. I got confused and wrote the other way in class on Wednesday. It was a mistake Provide formulas/rules for each of the following: State the necessary and sufficient conditions for maximizing a function with more than two variables. State the necessary and sufficient conditions for minimizing a...
2 (5 points) Recalled that null space of a matrix A € Mnxn is defined as N(A) = {r € R” : Ar =0}. Now, the eigenspace of A corresponding to the eigenvalue 1 (denoted by Ex(A)) is defined as the nullspace of A-XI, that is, EX(A) = N(A – XI) = {v ER”: (A – XI)v = 0}. You should have three distinct eigenvalues in Problem 1 above. Let say there are li, 12, and 13. (i) Find the...
(MATLAB): Suppose that you are given a positive definite symmetric matrix A, a vector b, and a real number c. Write MATLAB code which finds the minimum of the function f() r A bc subject to the constraint rT =1 for some vector r and real number . Note: This is a Lagrange Multi pliers problem It turns out that the Lagrange multiplier algebra is simply matrix algebra, which you can easily do in MATLAB. It may be a In...
In this exercise, you will work with a QR factorization of an mxn matrix. We will proceed in the way that is chosen by MATLAB, which is different from the textbook presentation. An mxn matrix A can be presented as a product of a unitary (or orthogonal) mxm matrix Q and an upper-triangular m × n matrix R, that is, A = Q * R . Theory: a square mxm matrix Q is called unitary (or orthogona) if -,or equivalently,...