According to the second-order condition for saddle points, what must be true about the eigenvalues of the Hessian matrix at a saddle point?
A. At least one eigenvalue is positive and one is negative
B. All eigenvalues are positive
C. All eigenvalues are negative
D. All eigenvalues are zero
Answer (2)
In mathematics, the concept of a saddle point appears in the study of optimization and is particularly significant when analyzing functions of multiple variables. At a saddle point, the behavior of the function mimics that of a saddle surface — it can appear to be minimal from one perspective and maximal from another.
To delve deeper into the second-order condition for saddle points, it's important to consider the role of the Hessian matrix. The Hessian matrix is a square matrix of second-order partial derivatives of a scalar-valued function.
Understanding Eigenvalues:
The eigenvalues of the Hessian matrix inform us about the curvature of the function at a particular point.
Saddle Points and Eigenvalues:
For a point to be classified as a saddle point, the Hessian matrix at that point must have both positive and negative eigenvalues. This indicates that the function curves differently in different directions: concave in some directions (negative curvature) and convex in others (positive curvature).
Second-Order Condition:
If you examine the eigenvalues of the Hessian at a critical point, at least one must be positive and one must be negative for the point to be a saddle point.
Given the self-contained nature of this condition, the correct choice among the options provided is:
(A) At least one eigenvalue is positive and one is negative
This condition captures the essence of the mixed curvature at a saddle point in a multi-dimensional space.
At a saddle point, the Hessian matrix must have at least one positive and one negative eigenvalue, indicating mixed curvature. Therefore, the correct answer is (A) At least one eigenvalue is positive and one is negative.
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