Normal matrix


In mathematics, a complex square matrix is normal if it commutes with its conjugate transpose :
The concept of normal matrices can be extended to normal operators on infinite dimensional normed spaces and to normal elements in C*-algebras. As in the matrix case, normality means commutativity is preserved, to the extent possible, in the noncommutative setting. This makes normal operators, and normal elements of C*-algebras, more amenable to analysis.
The spectral theorem states that a matrix is normal if and only if it is unitarily similar to a diagonal matrix, and therefore any matrix satisfying the equation is diagonalizable.

Special cases

Among complex matrices, all unitary, Hermitian, and skew-Hermitian matrices are normal. Likewise, among real matrices, all orthogonal, symmetric, and skew-symmetric matrices are normal. However, it is not the case that all normal matrices are either unitary or Hermitian. For example,
is neither unitary, Hermitian, nor skew-Hermitian, yet it is normal because

Consequences

The concept of normality is important because normal matrices are precisely those to which the spectral theorem applies:
The diagonal entries of are the eigenvalues of, and the columns of are the eigenvectors of. The matching eigenvalues in come in the same order as the eigenvectors are ordered as columns of.
Another way of stating the spectral theorem is to say that normal matrices are precisely those matrices that can be represented by a diagonal matrix with respect to a properly chosen orthonormal basis of. Phrased differently: a matrix is normal if and only if its eigenspaces span and are pairwise orthogonal with respect to the standard inner product of.
The spectral theorem for normal matrices is a special case of the more general Schur decomposition which holds for all square matrices. Let be a square matrix. Then by Schur decomposition it is unitary similar to an upper-triangular matrix, say,. If is normal, so is. But then must be diagonal, for, as noted above, a normal upper-triangular matrix is diagonal.
The spectral theorem permits the classification of normal matrices in terms of their spectra, for example:
In general, the sum or product of two normal matrices need not be normal. However, the following holds:
In this special case, the columns of are eigenvectors of both and and form an orthonormal basis in. This follows by combining the theorems that, over an algebraically closed field, commuting matrices are simultaneously triangularizable and a normal matrix is diagonalizable – the added result is that these can both be done simultaneously.

Equivalent definitions

It is possible to give a fairly long list of equivalent definitions of a normal matrix. Let be a complex matrix. Then the following are equivalent:
  1. is normal.
  2. is diagonalizable by a unitary matrix.
  3. There exists a set of eigenvectors of which forms an orthonormal basis for.
  4. for every.
  5. The Frobenius norm of can be computed by the eigenvalues of :.
  6. The Hermitian part and skew-Hermitian part of commute.
  7. is a polynomial in.
  8. for some unitary matrix.
  9. and commute, where we have the polar decomposition with a unitary matrix and some positive semidefinite matrix.
  10. commutes with some normal matrix with distinct eigenvalues.
  11. for all where has singular values and eigenvalues.
  12. The operator norm of a normal matrix equals the numerical and spectral radii of. Explicitly, this means:
  13. :
Some but not all of the above generalize to normal operators on infinite-dimensional Hilbert spaces. For example, a bounded operator satisfying is only quasinormal.

Analogy

It is occasionally useful to think of the relationships of different kinds of normal matrices as analogous to the relationships between different kinds of complex numbers:
As a special case, the complex numbers may be embedded in the normal 2×2 real matrices by the mapping
which preserves addition and multiplication. It is easy to check that this embedding respects all of the above analogies.