Remark

• Non-negative definiteness of the matrix implies that all its eigen values are nonnegative. All the leading minors at the top-left corner of a non-negative definite matrix are non-negative. This property can be used to check the non-negative definiteness of a matrix in the simple case.



• The covariance matrix represents second-order relationship between each pair of the random variables and plays an important role in applications of random variables.
  Example 2 The symmetric matrix is non-negative definite because
  and det =4.
  On the other hand, the symmetric matrix is not non-negative definite because