Properties
- The eigenvalues of a skew-Hermitian matrix are all purely imaginary. Furthermore, skew-Hermitian matrices are normal. Hence they are diagonalizable and their eigenvectors for distinct eigenvalues must be orthogonal.
- All entries on the main diagonal of a skew-Hermitian matrix have to be pure imaginary, i.e., on the imaginary axis (the number zero is also considered purely imaginary).
- If A, B are skew-Hermitian, then aA + bB is skew-Hermitian for all real scalars a and b.
- If A is skew-Hermitian, then both iA and -iA are Hermitian.
- If A is skew-Hermitian, then Ak is Hermitian if k is an even integer and skew-Hermitian if k is an odd integer.
- An arbitrary (square) matrix C can uniquely be written as the sum of a Hermitian matrix A and a skew-Hermitian matrix B:
- If A is skew-Hermitian, then eA is unitary.
- The space of skew-Hermitian matrices forms the Lie algebra u(n) of the Lie group U(n).
Read more about this topic: Skew-Hermitian Matrix
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)