Home»Documentation»Vector and Matrix Library User's Guide»Matrix Decompositions»The Symmetric Indefinite Decomposition

## The Symmetric Indefinite Decomposition | Extreme Optimization Numerical Libraries for .NET Professional |

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.

The symmetric indefinite decomposition is implemented by the
SymmetricIndefiniteDecomposition

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

Copyright Â© 2004-2023,
Extreme Optimization. All rights reserved.

*Extreme Optimization,* *Complexity made simple*, *M#*, and *M
Sharp* are trademarks of ExoAnalytics Inc.

*Microsoft*, *Visual C#, Visual Basic, Visual Studio*, *Visual
Studio.NET*, and the *Optimized for Visual Studio* logo

are
registered trademarks of Microsoft Corporation.