Final answer:
To prove a quadratic form is positive definite in a real vector space, it is necessary that all diagonal elements of its matrix in any basis are positive, but this is not sufficient, as positive diagonal elements alone do not ensure positive definiteness due to potential influence of off-diagonal elements.
Step-by-step explanation:
Positive Definiteness of a Quadratic Form
For a quadratic form to be positive definite, it is necessary but not sufficient that all diagonal elements of its matrix in a chosen basis are positive. This condition is necessary because if any diagonal element were negative, it would contribute a term that allows the quadratic form to take on negative values for some vectors, thus violating the definition of positive definiteness. However, this is not a sufficient condition because the off-diagonal elements also affect the definiteness of the quadratic form. These elements can introduce terms that make the quadratic form negative for some non-zero vectors, even when all diagonal elements are positive.
For example, consider the quadratic form Q(x,y) = x² - 2xy + y², represented in matrix form as A = [1, -1; -1, 1] with respect to the standard basis. While the diagonal elements are positive, the quadratic form is not positive definite because it evaluates to zero for x = y, indicating it's at best semi-definite. Therefore, the simple presence of positive diagonal elements does not guarantee positive definiteness.
To ensure positive definiteness, one must verify that the quadratic form satisfies the more stringent conditions, such as all the leading principal minors of the matrix being positive (Sylvester's criterion), or that the quadratic form evaluates to a positive number for all non-zero vectors in the space.