Normal matrix

A regular array in a linear algebra matrix with property

Ie a matrix that commutes with its adjoint matrix. For a real matrix applies analogously

The spectral theorem states that a matrix is ​​then exactly normal when there is a unitary matrix, so that, with a diagonal matrix. So Normal matrices have the property that they are unitarily diagonalizable. So there exists an orthonormal basis of eigenvectors of A. The diagonal elements of are precisely the eigenvalues ​​of. In particular, any real symmetric matrix, and every complex Hermitian matrix are normal. In addition, any unitary matrix is normal.

Examples

The eigenvalues ​​can be complex even if the matrix is ​​real, and so are generally complex, as the example shows:

Only for the special case of a real symmetric matrix, the matrix and the eigenvalues ​​(that ) are always real.

Note that there are matrices diagonalisable but not normal though. In this case, there is no unitary diagonalizability, that is, it applies only where no is unitary, ie. An example of a non-normal but diagonalizable matrix

Normality and deviations from normality

The decomposition of the matrix in is called the Schur decomposition or Schur normal form. Basically:

A strict upper triangular matrix ( on the diagonal are therefore all zeros ) and the eigenvalues ​​of are. For normal matrices, the following applies:

Is not normal, it is referred as the deviation from normality. The standard refers to the Frobenius norm.

Normal matrices and normal operators

A normal operator is in two ways a generalization of the normal matrix:

The basic function of the concept of "normal" for a matrix is due to the definition of " adjoint " to the game: The adjoint matrix is to be defined by the following property:

This definition can also be based independent reading, but only when the vectors are coordinate vectors with respect to an orthonormal basis in this definition, the scalar product can be as a matrix product review (see also Matrix (mathematics) # vector spaces of matrices ), so that it follows for arbitrary matrices:

Only then the adjoint matrix to be always calculated by conjugation and transposition.