Question

Which of the following is not a condition for a given real nonsingular quadratic form, Q = X^TAX, to be a negative definite quadratic form?

(a) The number of positive square terms in the quadratic form is equal to zero

(b) The rank of the matrix A is equal to the number of variables in the quadratic form (index)

(c) All the eigen values of A are negative

(d) The rank and index are equal

I had been asked this question in an interview.

The query is from Canonical Form or Sum of Squares Form topic in portion Eigen Values and Eigen Vectors of Engineering Mathematics

Answer 1

Correct option is (d) The rank and index are equal

To explain: The quadratic form is said to be negative definitive if the rank is equal to index and the number of square terms is equal to zero or all the eigen values of the matrix are negative.