The Symmetric Indefinite Decomposition

The symmetric indefinite decomposition or Bunch-Kaufman decomposition is defined for symmetric matrices that may not be positive definite. It expresses a matrix as the product of a lower triangular matrix, a block diagonal matrix, and the transpose of the triangular matrix.

The symmetric indefinite decomposition algorithm exploits the special structure of symmetric matrices. As a result, it is about twice as fast as the LU decomposition. Moreover, the decomposition is fairly stable, so it is suitable for use in a broader range of problems.

Working with symmetric indefinite decompositions

The symmetric indefinite decomposition is implemented by the SymmetricIndefiniteDecomposition<T> class. It has no constructors. Instead, it is created by calling the GetSymmetricIndefiniteDecomposition method on the matrix. This method has two overloads. The first overload has no arguments. The second overload takes a Boolean value that specifies whether the contents of the matrix may be overwritten by the decomposition. The default is false.

var A = Matrix.CreateSymmetric(4, new double[] {
    4.16,-3.12, 0.56,-0.10,
    0   ,-5.03,-0.83, 1.18,
    0   , 0   , 0.76, 0.34,
    0   , 0   , 0   , 1.18
}, MatrixTriangle.Upper, MatrixElementOrder.RowMajor);
var ldlt = A.GetSymmetricIndefiniteDecomposition();

The Decompose method performs the actual decomposition. This method copies the matrix if necessary. It then calls the appropriate LAPACK routine to perform the actual decomposition. This method is called by other methods as needed. You will rarely need to call it explicitly.

Once the decomposition is computed, a number of operations can be performed in much less time. You can repeatedly solve a system of linear equations with different right-hand sides. You can also calculate the determinant and the inverse of the base matrix:

var b = Matrix.Create(4, 2, new double[] {
    8.70,-13.35, 1.89,-4.14,
    8.30,  2.13, 1.61, 5.00
}, MatrixElementOrder.ColumnMajor);
var x = ldlt.Solve(b);
var invA = ldlt.GetInverse();
var detA = ldlt.GetDeterminant();

The LowerTriangularFactor property returns a TriangularMatrix<T> containing the lower triangular matrix, G, of the decomposition. There is no property to obtain the block diagonal matrix.

var L = ldlt.LowerTriangularFactor;
var LT = ldlt.UpperTriangularFactor;