Normal matrix

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A complex square matrix A is a normal matrix if

A*A=AA*

where A* is the conjugate transpose of A. That is, a matrix is normal if it commutes with its conjugate transpose.

If A is a real matrix, then A*=AT; it is normal if ATA = AAT.

Normality is a convenient test for diagonizability: every normal matrix can be converted to a diagonal matrix by a unitary transform, and every matrix which can be made diagonal by a unitary transform is also normal, but finding the desired transform requires much more work than simply testing to see whether the matrix is normal.

[edit] Special cases

Among complex matrices, all unitary, Hermitian, and skew-Hermitian matrices are normal. If A is unitary, then A*A = AA* = I. If A is Hermitian, then A* = A and so AA* = AA = A*A. Likewise, among real matrices, all orthogonal, symmetric, and skew-symmetric matrices are normal.

If A is both a triangular matrix and a normal matrix, then A is diagonal. This can be seen by looking at the diagonal entries of A*A and AA*, where A is a normal, triangular matrix.

However, it is not the case that all normal matrices are either unitary or (skew-)Hermitian. As an example, the matrix

A = \begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{pmatrix}

is normal because

AA^* = \begin{pmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{pmatrix} = A^*A.

The matrix A is neither unitary, Hermitian, nor skew-Hermitian.

The sum or product of two normal matrices is usually not normal.

[edit] Consequences

The concept of normality is important because normal matrices are precisely those to which the spectral theorem applies: a matrix A is normal if and only if it can be represented by a diagonal matrix Λ and a unitary matrix U by the formula

\mathbf{A} = \mathbf{U} \mathbf{\Lambda} \mathbf{U}^*

where

\mathbf{\Lambda} = diag(\lambda_1, \lambda_2, \dots)
\mathbf{U}^*\mathbf{U} = \mathbf{U} \mathbf{U}^* = \mathbf{I}.

The entries λ of diagonal matrix Λ are the eigenvalues of A, and the columns of U are the eigenvectors of A. The matching eigenvalues in Λ come in the same order as the eigenvectors are ordered as columns of U.

In general, the sum or product of two normal matrices need not be normal. However, there is a special case: if A and B are normal with AB = BA, then both AB and A + B are also normal. Furthermore the two are simultaneously diagonalizable, that is: both A and B are made diagonal by the same unitary matrix U. Both UAU* and UBU* are diagonal matrices. In this special case, the columns of U* are eigenvectors of both A and B and form an orthonormal basis in Cn.

Any square matrix A has polar decomposition A = UP where U is unitary and P is some positive semidefinite matrix. If A is invertible, then U and P are unique. If A is normal, then UP = PU. (The converse is true only in the finite dimensional case.)

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 Cn. Phrased differently: a matrix is normal if and only if its eigenspaces span Cn and are pairwise orthogonal with respect to the standard inner product of Cn.

The concept of normal matrices can be extended to normal operators on infinite dimensional Hilbert 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.

[edit] Analogy

It is often useful to think of the relationships of different kinds of normal matrices as analogous to the relationships between different kinds of complex numbers: