Skew-Hermitian matrix

Schiefhermitesch a matrix or matrix antihermitesch is a mathematical object from linear algebra. This special type of square matrices with complex coefficients is converted with respect to the complex Standardskalarproduktes with mirroring, the coefficients on the main diagonal in its adjoint matrix. Named these matrices are named after the mathematician Charles Hermite.

Definition

A matrix is called schiefhermitesch if it is equal to its adjoint negative, which means

Therefore applies to the entries of a matrix schiefhermiteschen

Examples

• The matrix

• The matrices

Properties

• Said matrix is square.
• The main diagonal elements are purely imaginary.
• The real part is skew-symmetric, the imaginary part is symmetric.
• Is schiefhermitesch, then is Hermitian.
• The eigenvalues ​​schiefhermitescher matrices are purely imaginary, the eigenvectors form an orthonormal system for the Hermitian standard form.
• Antihermitesche matrices can always be diagonalized.
• In the real case, the terms coincide schiefhermitesch and skew-symmetric. Real skew-symmetric matrices can be obtained by real change of basis to bring in block diagonal form with blocks

• If B schiefhermitesch, then Bk is Hermitian with straight k and k for odd schiefhermitesch
• If B schiefhermitesch, then eB is unitary.
• An arbitrary square matrix C can be clearly identified as the sum of a Hermitian matrix A and a matrix B schiefhermiteschen be written:

The Lie algebra of matrices schiefhermiteschen

The commutator schiefhermitescher matrices is schiefhermitesch again. So the schiefhermiteschen matrices form a Lie algebra, this is denoted by.

Is the Lie algebra of the Lie group of unitary matrices