3. A quadratic function in n variables is any function defined on R" which can beexpressed in the formf(x) = a + b·x + x• Ax,where a € R, be R", and A is an n × n-symmetric matrix.(a) Show that the function f(x) defined on R² byf(x₁, x₂) = (x₁ - x₂)² + (x₁ + 2x₂ + 1)² − 8x₁x₂is a quadratic function of two variables by finding the appropriate a € R, b = R²,and the 2 x 2-symmetric matrix A.(b) Compute the gradient Vf(x) and the Hessian Hf(x) of the quadratic functionin (a) and express these quantities in terms of the a € R, b = R², and the2 x 2-symmetric matrix A computed in (a).(c) Show that a quadratic function f(x) of n variables is convex if and only if thecorresponding n × n-symmetric matrix A is positive semidefinite, and is strictlyconvex if A is positive definite.(d) If f(x) is a quadratic function of n variables such that the corresponding matrixA is positive definite, show that 0 2Ax + b has a unique solution and thatthis solution is the strict global minimizer of f(x).=

Answered: Image /qna-images/answer/c3b0bf89-60b5-4fd5-9631-6607009d84ba.jpg

A quadratic function in n variables is any function defined on R" which can be expressed in the form f(x) = a + b x + x. Ax, where a € R, b = R", and A is an n × n-symmetric matrix. (a) Show that the function f(x) defined on R² by f(x₁, x₂) = (x₁ - x₂)² + (x₁ + 2x₂ + 1)² − 8x₁x₂ is a quadratic function of two variables by finding the appropriate a € R, b = R², and the 2 x 2-symmetric matrix A. (b) Compute the gradient Vf(x) and the Hessian Hf(x) of the quadratic function in (a) and express these quantities in terms of the ae R, b = R², and the 2 x 2-symmetric matrix A computed in (a). (c) Show that a quadratic function f(x) of n variables is convex if and only if the corresponding n × n-symmetric matrix A is positive semidefinite, and is strictly convex if A is positive definite. (d) If f(x) is a quadratic function of n variables such that the corresponding matrix A is positive definite, show that 0 = 24x + b has a unique solution and that this solution is the strict global minimizer of f(x).

A quadratic function in n variables is any function defined on R" which can be expressed in the form f(x) = a + b x + x. Ax, where a € R, b = R", and A is an n × n-symmetric matrix. (a) Show that the function f(x) defined on R² by f(x₁, x₂) = (x₁ - x₂)² + (x₁ + 2x₂ + 1)² − 8x₁x₂ is a quadratic function of two variables by finding the appropriate a € R, b = R², and the 2 x 2-symmetric matrix A. (b) Compute the gradient Vf(x) and the Hessian Hf(x) of the quadratic function in (a) and express these quantities in terms of the ae R, b = R², and the 2 x 2-symmetric matrix A computed in (a). (c) Show that a quadratic function f(x) of n variables is convex if and only if the corresponding n × n-symmetric matrix A is positive semidefinite, and is strictly convex if A is positive definite. (d) If f(x) is a quadratic function of n variables such that the corresponding matrix A is positive definite, show that 0 = 24x + b has a unique solution and that this solution is the strict global minimizer of f(x).

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.6: Applications And The Perron-frobenius Theorem

Problem 70EQ

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