Question 21: In the context of quadratic forms, what does it mean for a quadratic form to be positive definite? (A) The quadratic form is convex (B) The quadratic form is concave (C) The quadratic form is non-convex (D) The quadratic form is indefinite

In the context of quadratic forms, understanding whether a quadratic form is positive definite is crucial for discussing their properties and applications.
A quadratic form in several variables is an expression of the form:
Q ( x ) = x T A x
where x is a column vector, x T is the transpose of x , and A is a symmetric matrix.
A quadratic form Q ( x ) is said to be positive definite if it satisfies the following condition:
0 \quad \text{for all non-zero vectors } x."> Q ( x ) > 0 for all non-zero vectors x .
This means that the quadratic form will produce positive values for any input vector that is not the zero vector. In geometric terms, this implies that the graph of the quadratic form (as part of a function f ( x ) = Q ( x ) ) would describe a paraboloid that curves upward, indicating that it's convex in nature.
Therefore, in reference to the multiple-choice options given:

(A) The quadratic form is convex
(B) The quadratic form is concave
(C) The quadratic form is non-convex
(D) The quadratic form is indefinite

The correct answer is (A) because a positive definite quadratic form is convex. Convexity implies that the graph of the function opens upwards, which complements the nature of a positive definite attempt at ensuring all values produced (except at the origin) are positive.
So, the correct choice is (A) The quadratic form is convex .

Answered by IsabellaRoseDavis | 2025-07-22