Complex Linear Algebra

The following table lists the complex equivalent of the real vector classes:

The complex dot product uses the conjugate of the first argument, so the dot product of a complex vector with itself is always a real number.

Most of the classes for real matrices have an equivalent complex matrix class. Complex symmetric matrices are relatively uncommon. Much more common are hermitian matrices, whose components above the diagonal are the conjugate of the corresponding element below the diagonal. For this reason, there is no ComplexSymmetricMatrix class, but there is a new ComplexHermitianMatrix class that represents hermitian matrices.

Complex matrices have four additional methods. GetRealPart()()() returns a real matrix with the real components of each matrix element. GetImaginaryPart()()() returns a real matrix with the imaginary components of each matrix element. Conjugate()()() returns a complex matrix whose components are the conjugate of the corresponding component in the original matrix. ConjugateTranspose()()() returns a complex matrix whose components are the conjugate of the corresponding component in the transpose of the original matrix.

The following table lists the complex equivalent of the real matrix classes:

There are some additional differences, mainly related to the ComplexHermitianMatrix class. You can only extract an upper triangular matrix from a Hermitian matrix if the Hermitian matrix' data is stored in the upper triangular part. Likewise, you can only extract a lower triangular matrix if the Hermitian matrix' data is stored in the lower triangular part.

For the same reason, the form of the complex Cholesky decomposition depends on how the data is stored. The TriangularFactor property of the ComplexCholeskyDecomposition class returns a ComplexTriangularMatrix that is upper triangular if the Hermitian matrix' data is stored in the upper triangular part, and lower triangular if the Hermitian matrix' data is stored in the lower triangular part.