<正> The present paper proves the existence and uniqueness (in some given sense) ofthe LDL~T decomposition of real symmetric non-negative definite matrices, where Lis a unit lower triangular matrix with real elements and D is a diagonal matrixwith real elements. The proof is made in a constructive way. By taking advantageof this decomposition, a criterion for the consistency of the linear equation with sucha coefficient matrix and its whole solution set (or the least-squares solution if it isinconsistent) are obtained. Since it involves no row or column permutation, the pro-cess may be combined with any sparse technique on the computer, and hence is ofpractical importance in treating the large scale sparse matrices derived from suchproblems as the structure design by finite elements methods. Finally, the stability ofsuch a decomposition is discussed and a backward error analysis is given.