Question

A quadratic form is positive semi - definite when ________

             a. all the eigen values are less than or equal to zero and atleast one eigen value is zero

             b. all the eigen values are greater than or equal to zero and atleast one eigen value is zero

             c. all the eigen values are greater than zero

             d. all the eigen values are greater than or equal to zero and atleast one eigen value is zero                                                                               ​

Answer 1

Given:

A quadratic form is positive semi-definite when ________

a. all the eigenvalues are less than or equal to zero and at least one eigenvalue is zero.

b. all the eigenvalues are greater than or equal to zero and at least one eigenvalue is zero

c. all the eigenvalues are greater than zero

d. all the eigenvalues are greater than or equal to zero and at least one eigenvalue is zero.

To Find:

When a quadratic form is positive semi-definite?

Solution:

Let A be a real symmetric matrix. If every eigenvalue of A is positive, then A is positive definite. So, option c is the correct one.

Answer: all the eigenvalues are greater than zero. (c)