# Direct Solution of Saddle-Point Problems

## Abstract

In this chapter we give a brief overview of direct techniques used for solution of saddle-point problems. They are all based on some factorization of the saddle-point matrix. Therefore, we first recall two main factorizations of general indefinite matrices that preserve the symmetry of the original system matrix. Then we discuss cases when the 2-by-2 block matrix in the form ( 1.2) admits an LDL^{T} factorization with the diagonal factor and gives its relation to the class of symmetric quasi-definite matrices. In particular, we look at the conditioning of factors in this Cholesky-like factorization in terms of the conditioning of the matrix ( 1.2) and its (1, 1)-block *A*. Then, we show that the two main direct approaches for solving the saddle-point problems via the Schur complement method or the null-space method can be seen not just a solution procedures but also as two different factorizations of the saddle-point matrix, namely, the block LDL^{T} factorization and the block QTQ^{T} factorization, respectively.

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