By definition, a matrix [D] (as well as its inverse [D]-1) is positive definite if the determinants of all submatrices of the series:
 | (12–6) |
including the determinant of the full matrix [D], are positive. The series could have started out at any other diagonal term and then had row and column sets added in any order. Thus, two necessary (but not sufficient) conditions for a symmetric matrix to be positive definite are given here for convenience:
 | (12–7) |
 | (12–8) |
If any of the above determinants are zero (and the rest positive), the matrix is said to be positive semidefinite. If all of the above determinants are negative, the matrix is said to be negative definite.
In virtually all circumstances, matrices representing the complete structure with the appropriate boundary conditions must be positive definite. If they are not, the message “NEGATIVE PIVOT . . .” appears. This usually means that insufficient boundary conditions were specified. An exception is a piezoelectric analysis, which works with negative definite matrices, but does not generate any error messages.