Positivity

The positivity (of a matrix) can be defined only for square, symmetric matrices; a matrix A is positive-definite if File:Hepa img840.gif for all non-zero vectors x. A necessary and sufficient condition for this is that all the eigenvalues of A be strictly positive. An analogous definition exists for negative-definite.

If all the eigenvalues of a symmetric matrix are non-negative, the matrix is said to be positive semi-definite. If a matrix has both positive and negative eigenvalues, it is indefinite.

When the elements of the matrix are subject to experimental errors or to rounding errors, which is nearly always the case in real calculations, one must be careful in recognizing a zero eigenvalue. The important quantity is then not the value of the smallest eigenvalue, but the ratio of the smallest to the largest eigenvalue. When this ratio is smaller than the relative accuracy inherent in the calculation, the smallest eigenvalue must be considered to be compatible with zero.