Normalmatrix

Normal matrix

In linear algebra, a square matrix A is called normal if it commutes with its conjugate transpose, that is A A = A A, where A denotes the conjugate transpose. Equivalently, A is normal if it commutes with its adjoint. This condition encompasses several important classes of matrices: Hermitian matrices (A = A), skew-Hermitian matrices (A = -A), and unitary matrices (A A = AA = I).

A central feature of normal matrices is their diagonalizability by a unitary transformation. Specifically, A is normal if and only if there exists a unitary matrix U such that U A U is diagonal. The diagonal entries are the eigenvalues of A, which may be complex even when A is real. For real normal matrices, one can obtain a real orthogonal similarity to a block diagonal form with 1×1 blocks for real eigenvalues and 2×2 blocks corresponding to complex conjugate eigenpairs; over the complex field, A is always unitarily diagonalizable.

An important consequence is that eigenvectors corresponding to distinct eigenvalues of a normal matrix are orthogonal. The spectral theorem for normal matrices underpins many applications, allowing functions to be applied to A via its diagonal form (functional calculus).

Normal matrices are widely used in various areas of mathematics and applied sciences, including quantum mechanics, signal processing, and numerical linear algebra. Examples include diagonal matrices, identity, Hermitian matrices such as symmetric matrices in real space, and unitary matrices. A non-example is a Jordan block with off-diagonal ones, which is not normal since it fails A A = A A.