Question: 7.4 Let A be a symmetric positive definite n n matrix and let b be an n-vector. Consider the minimization problem minimize xTAx such

7.4 Let A be a symmetric positive definite n × n matrix and let b be an n-vector.

Consider the minimization problem minimize xTAx such that bTx = 1, over x ∈ ℝn. Show that the solution is given by x = A−1b∕(bTA−1b).

Hint: Using a Lagrange multiplier ????, minimize the unconstrained objective function xTAx + 2????(1 − bTx) over x ∈ ℝn to get x = ????A−1b, and show that the constraint is satisfied if ???? = 1∕(bTA−1b).

Step by Step Solution

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock blur-text-image

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Students Have Also Explored These Related Regression Analysis Questions!

Show that f(x, y) = x 2 ye x2y2 has maximum values at (1, 1/2) and minimum values at (1, 1/2). Show also that f has infinitely many other critical points and D = 0 at each of them. Which of them give...

Macroeconomics problems solve... Question 3 Emma is a first-year Ph.D. student of economics. She is very lucky because she got a place to stay in the Aggie Village, the nicest part of Davis. She...

B2 For n E N, consider the regression model of the form Y = arte, E~ N(0, E), where the covariate r E R" with x 0 and the positive definite sym- metric matrix E e R"x" are fixed and known whereas the...

Problem #1: Part a: Consider the function (z) =% , where z R"and Qis nxn symmetric positive definite matrix. Minimize f(z)subject to the linear constraint Az = b where A isan mx n matrix (assume m 0...

1. Let 1 and 2 be two symmetric, positive definite matrices. Let A be a diagonal matrix containing the eigenvalues of 2. Let A contain the corresponding eigenvectors of 2 as its rows in the same...

3. |Consider the following problem: min 23:? + 33:13:? + 4:1:+ 23:2 2.172.123 + 5331+ 3:122 47:3. Use the method described in Problem 2 or an}r other method, generate three conjugate directions....

Let A E Rnx" be a symmetric matrix in Rnx. Let A1, ..., An be n distinct eigenvalues of A satisfying 1 > 12 > . . . > An > 0 and u1, u2, . .., Un be the corresponding eigenvectors. Let's consider the...

1. [25 marks| Consider minimizing the strictly convex quadratic function glx) = -;-:'Gx+d"x+c where G is an (n X n) symmetric positive definite constant matrix, d is a constant n x 1 vector and c is...

MATHEMATICS FOR MACHINE LEARNING Marc Peter Deisenroth A. Aldo Faisal Cheng Soon Ong Contents Foreword 1 Part I Mathematical Foundations 9 1 Introduction and Motivation 11 1.1 Finding Words for...

2. (Do not hand in) We consider quadratic forms. Let A be an n x n matrix and let X = Consider the function of n variables defined by q(11, 12, . . ., In) = XTAX. . . . In This is called a quadratic...

Least Squares Approximation Hector D. Ceniceros 1 Least Squares Approximation Let f be a continuous function on [a, b]. We would like to find the best approximation to f by a polynomial of degree at...

A swimmer doing a flip turn moves from the pool wall to the left because of lower extremity forces Choose one: a. applied to the left b. not enough information to answer c. having little effect d....

The probability of scoring a penalty against a goalkeeper is 80%, If 10 penalties are taken, find the probability of: a. Hit 4 Penalties b. Miss all Penalties c. Hit a Penalty. Hit between 3 and 5...

Which of the following statements regarding nominal exchange rate forecasts is FALSE? Forward exchange rates perform better than spot exchange rates as the forecasting horizon is extended beyond one...

The total cost to produce x large jewelry-making kits and y small ones is given by C(x, y) = 2x 2 + 6y 2 + 4xy + 10. If a total of ten kits must be made, how should production be allocated so that...

Suppose that we have fit the straight-line regression model $\hat{y}=\hat{\beta}_{0}+\hat{\beta}_{1} x_{1}$ but the response is affected by a second variable $x_{2}$ such that the true regression...

Consider the simple linear regression model $y=\beta_{0}+\beta_{1} x+\varepsilon$, with $E(\varepsilon)=0$, $\operatorname{Var}(\varepsilon)=\sigma^{2}$, and $\varepsilon$ uncorrelated. a. Show that...

Dion Company reports the absorption costing income statement below for May. The company began the month with no finished goods inventory Dion produced 22.000 units, and 3,000 units remain in ending...

Elon Musk is having financial difficulty and therefore has asked Jeff Bezzo to restructure its $1 million note outstanding. The present note has 2 years remaining and pays a current rate of interest...

4. Morley Company pays $10,000 for land that remains unused and pays $5,000 to install an underground sprinkler system to maintain the plants on its property. As a result, Morley's A) plant assets...