Skew-Hermitian matrix

From Free net encyclopedia

In linear algebra, a square matrix (or more generally, a linear transformation from a complex vector space with a sesquilinear norm to itself) A is said to be skew-Hermitian or antihermitian if its conjugate transpose A^* is also its negative. That is, if it satisfies the relation:

A^* = −A

or in component form, if A = (a_i,j):

<math>a_{i,j} = -\overline{a_{j,i}}</math>

for all i and j.

Examples

For example, the following matrix is skew-Hermitian:

<math>\begin{pmatrix}i & 2 + i \\ -2 + i & 3i \end{pmatrix}</math>

Properties

• All entries on the main diagonal of a skew-Hermitian matrix have to be pure imaginary, ie. on the imaginary axis. Hence the same is true for the eigenvalues of a skew-Hermitian matrix.
• If A is skew-Hermitian, then iA is Hermitian
• If A, B are skew-Hermitian, then aA + bB is skew-Hermitian for all real scalars a, b.
• All skew-Hermitian matrices are normal.
• If A is skew-Hermitian, then A² is Hermitian.
• If A is skew-Hermitian, then A raised to an odd power is skew-Hermitian.
• The difference of a matrix and its conjugate transpose (<math>C - C^*</math>) is skew-Hermitian.
• An arbitrary matrix C can be written as the sum of a Hermitian matrix A and a skew-Hermitian matrix B:

  <math>C = A+B \quad\mbox{with}\quad A = \frac{1}{2}(C + C^*) \quad\mbox{and}\quad B = \frac{1}{2}(C - C^*).</math>

  [edit]

  See also

  • skew-symmetric matrix
  • Hermitian matrix
  • normal matrix
  • unitary matrixit:Matrice antihermitiana
  fi:Vinohermiittinen matriisi