Which of the following is False ?


Question 18Answer

a.

Hessian matrix always have real eigenvalues.


b.

fxy=fyxfxy=fyx for every function f(x,y)f(x,y).


c.

Hessian Matrix is symmetric matrix.


d.

Hessian matrix is a square matrix.

Let's analyze each option to determine which statement is false:

**a. Hessian matrix always have real eigenvalues.**
- The Hessian matrix consists of second-order partial derivatives of a function. While it is true that a symmetric matrix (like the Hessian) has real eigenvalues, this is only guaranteed when the matrix is real and symmetric. However, the statement implies that this is always the case, which is not necessarily true for all functions.

**b. \( f_{xy} = f_{yx} \) for every function \( f(x, y) \).**
- According to Clairaut's theorem, if the second-order mixed partial derivatives \( f_{xy} \) and \( f_{yx} \) are continuous, then they are equal. However, this equality does not hold for every function, especially if the mixed partial derivatives are not continuous.

**c. Hessian Matrix is a symmetric matrix.**
- This is generally true for most cases where the function has continuous second-order partial derivatives. The Hessian matrix is symmetric because \( f_{xy} = f_{yx} \) under these conditions.

**d. Hessian matrix is a square matrix.**
- This is true because the Hessian matrix is defined as the matrix of all second-order partial derivatives, and it has the same number of rows and columns (equal to the number of variables).

**Conclusion:** The false statement is **a.** 

While the Hessian matrix is symmetric and hence typically has real eigenvalues, this does not universally apply to all situations, such as when considering functions that lead to complex eigenvalues.

Explore false statements about the Hessian matrix and partial derivatives. Learn about matrix algebra, symmetric matrices, eigenvalues, and Clairaut's theorem. Suitable for advanced undergraduate students and professionals.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation