normal matrix (original) (raw)

A complex matrix A∈ℂn×n is said to be normal if A∗⁢A=A⁢A∗ where ∗ denotes the conjugate transpose.
Similarly for a real matrix A∈ℝn×n is said to be normal if AT⁢A=A⁢AT where T denotes the transpose.

properties:

• •
  Equivalently a complex matrix A∈ℂn×n is said to be normal if it satisfies [A,A∗]=0 where [,] is the commutator bracket.
• •
  Equivalently a real matrix A∈ℝn×n is said to be normal if it satisfies [A,AT]=0 where [,] is the commutator bracket.
• •
  Let A be a square complex matrix of order n. It follows from Schur’s inequality that if A is a normal matrix then ∑i=1n|λi|2=trace⁡A∗⁢A where ∗ is the conjugate transpose and λi are the eigenvalues of A.
• •
  A complex square matrix is diagonal if and only if it is normal, triangular.(see theorem for normal triangular matrices).

examples:

• •
  (ab-ba) where a,b∈ℝ
• •
  (1i-i1)

see also:

• •
  Wikipedia, http://www.wikipedia.org/wiki/Normal\_matrixnormal matrix