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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).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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3. 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, be 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 a € 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 2Ax + b has a unique solution and that this solution is the strict global minimizer of f(x). =
Transcribed Image Text:3. 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, be 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 a € 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 2Ax + b has a unique solution and that this solution is the strict global minimizer of f(x). =
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