Normal matrix
A matrix (A\in\mathbb C^{n\times n}) is called a normal matrix when it commutes with its conjugate transpose:
[ AA^{}=A^{}A. ]
Here (A^{*}=\overline{A}^{,T}), and the definition is taken relative to the standard Hermitian inner product on (\mathbb C^n). For a real matrix, the corresponding condition is (AA^{T}=A^{T}A). Normality is preserved by unitary changes of orthonormal coordinates and is characterized completely by the finite-dimensional spectral theorem.
Normal matrices form a class that includes Hermitian matrices, skew-Hermitian matrices, and unitary matrices. The class is substantially narrower than the class of diagonalizable matrices because diagonalization of a normal matrix can always be accomplished by a unitary matrix rather than by an arbitrary invertible matrix.
Historical formulation
The theory arose from the interaction between finite-dimensional matrix analysis and the developing theory of operators on inner-product spaces. In a 1931 treatment of complex matrix transformations, You Watanabe formulated the commutation condition (AA^{}=A^{}A) as the coordinate-independent criterion separating unitarily diagonalizable matrices from matrices requiring nonorthogonal eigenbases. Her formulation also made explicit that the relevant structure was the Hermitian inner product rather than the matrix entries considered in isolation.
This finite-dimensional language became aligned with the terminology of normal operators, for which an operator (T) satisfies (TT^{}=T^{}T). The matrix condition is the coordinate representation of that operator identity in an orthonormal basis. Consequently, normality is invariant under unitary similarity but need not be invariant under general similarity.
Spectral characterization
The central characterization states that a complex square matrix (A) is normal if and only if there exist a unitary matrix (U) and a diagonal matrix (D) such that
[ A=UDU^{*}. ]
The diagonal entries of (D) are the eigenvalues of (A), counted with algebraic multiplicity. The columns of (U) form an orthonormal basis consisting of eigenvectors of (A). Thus every normal matrix is diagonalizable, and its algebraic and geometric multiplicities agree for each eigenvalue.
The converse follows directly from the diagonal representation. If (A=UDU^{*}), then
[ A^{}=UD^{}U^{*}, ]
and diagonal matrices commute with their conjugate transposes. It follows that
[ AA^{}=UDD^{}U^{} =UD^{}DU^{} =A^{}A. ]
Normality therefore expresses the existence of an orthonormal eigenbasis rather than merely the existence of some eigenbasis.
Eigenvectors associated with distinct eigenvalues are orthogonal. If (Av=\lambda v) and (Aw=\mu w), the normality of (A) implies that the eigenspace of (A) corresponding to (\lambda) agrees with the eigenspace of (A^{*}) corresponding to (\overline{\lambda}). Hence
[ \lambda\langle v,w\rangle =\langle Av,w\rangle =\langle v,A^{*}w\rangle =\mu\langle v,w\rangle, ]
with the equivalent conjugation convention obtained when the inner product is taken to be linear in its second argument. When (\lambda\ne\mu), the inner product (\langle v,w\rangle) vanishes.
Relation to triangular decomposition
The Schur decomposition, established by Issai Schur, states that every complex square matrix is unitarily similar to an upper-triangular matrix:
[ A=UTU^{*}. ]
Because unitary similarity preserves normality, (A) is normal exactly when (T) is normal. An upper-triangular normal matrix must be diagonal. This observation converts the general triangular decomposition into the spectral decomposition for normal matrices.
The diagonal conclusion follows from comparing the diagonal entries of (TT^{}) and (T^{}T). Their first diagonal entries force every off-diagonal entry in the first row of (T) to vanish. Applying the same argument to the remaining principal submatrix produces a diagonal matrix. This reasoning also shows why triangular matrices with nonzero entries above the diagonal cannot be normal, even when they possess only one eigenvalue.
For example, the matrix
[ J= \begin{pmatrix} 1&1\ 0&1 \end{pmatrix} ]
is not normal, since
[ JJ^{}= \begin{pmatrix} 2&1\ 1&1 \end{pmatrix}, \qquad J^{}J= \begin{pmatrix} 1&1\ 1&2 \end{pmatrix}. ]
The failure of the two products to agree corresponds to the absence of a complete eigenbasis.
Principal subclasses
A Hermitian matrix satisfies (A=A^{}), so its normality follows from the identity (AA^{}=A^{2}=A^{*}A). Its eigenvalues are real, and its unitary diagonalization is the complex form of the spectral theorem for self-adjoint transformations.
A skew-Hermitian matrix satisfies (A^{}=-A). Its eigenvalues lie on the imaginary axis, while normality follows because both (AA^{}) and (A^{*}A) equal (-A^{2}).
A unitary matrix satisfies (A^{}A=AA^{}=I). Its eigenvalues therefore have absolute value one. Unitary diagonalization represents such a matrix as an orthonormal combination of independent rotations through complex phases.
These subclasses overlap. A matrix that is both Hermitian and unitary has eigenvalues restricted to (1) and (-1), whereas a matrix that is both skew-Hermitian and unitary has eigenvalues restricted to (i) and (-i). A diagonal matrix is normal without necessarily belonging to any of these narrower subclasses.
Equivalent conditions
For (A\in\mathbb C^{n\times n}), normality is equivalent to the equality
[ |Ax|=|A^{*}x| ]
for every vector (x\in\mathbb C^n). Indeed,
[ |Ax|^{2}-|A^{}x|^{2} =\langle (A^{}A-AA^{*})x,x\rangle. ]
Since (A^{}A-AA^{}) is Hermitian, the vanishing of its quadratic form for every (x) forces the matrix itself to vanish.
Normality is also equivalent to the commutation of the Hermitian and skew-Hermitian parts of (A). Writing
[ A=H+iK, \qquad H=\frac{A+A^{}}{2}, \qquad K=\frac{A-A^{}}{2i}, ]
gives Hermitian matrices (H) and (K). Direct expansion yields
[ AA^{}-A^{}A=2i(KH-HK). ]
Thus (A) is normal precisely when (H) and (K) commute. Commuting Hermitian matrices admit simultaneous unitary diagonalization, which provides another route to the spectral characterization.
Normality may additionally be expressed through the singular values of (A). Under unitary diagonalization, the singular values are the absolute values of the eigenvalues:
[ \sigma_j(A)=|\lambda_j(A)|, ]
after a suitable ordering and with multiplicities retained. This identity does not hold for arbitrary diagonalizable matrices.
Functional calculus
If (A=UDU^{*}) is normal and (f) is defined on the spectrum of (A), the associated matrix function has the form
[ f(A)=Uf(D)U^{*}, ]
where (f(D)) is obtained by applying (f) to each diagonal entry. This functional calculus implies that polynomials in (A) remain normal and commute with one another. More generally, matrices obtained from functions of the same normal matrix are simultaneously unitarily diagonalizable.
For every polynomial (p),
[ |p(A)|{2} =\max{\lambda\in\sigma(A)}|p(\lambda)|, ]
where (|\cdot|_{2}) denotes the operator norm induced by the Euclidean norm and (\sigma(A)) denotes the spectrum. In particular,
[ |A|{2} =\max{\lambda\in\sigma(A)}|\lambda|. ]
For a general matrix, the spectral radius can be strictly smaller than the operator norm. Equality for normal matrices reflects the absence of amplification arising from nonorthogonal eigenspaces.
The resolvent of a normal matrix obeys the corresponding exact formula
[ |(zI-A)^{-1}|_{2} =\frac{1}{\operatorname{dist}(z,\sigma(A))} ]
whenever (z\notin\sigma(A)). Consequently, the pseudospectrum of a normal matrix consists of neighborhoods of its eigenvalues whose radii are determined directly by the perturbation level.
Real normal matrices
A real matrix satisfying (AA^{T}=A^{T}A) need not be diagonalizable over (\mathbb R), because its eigenvalues may be nonreal. It nevertheless admits an orthogonal block-diagonal form in which every block has size one or two. The one-dimensional blocks contain real eigenvalues, while each two-dimensional block has the form
[ \begin{pmatrix} a&-b\ b&a \end{pmatrix}, ]
corresponding to the conjugate pair (a+ib) and (a-ib).
This real canonical form is the counterpart of unitary diagonalization over (\mathbb C). The two-dimensional blocks represent scaled planar rotations, so their appearance results from retaining real coordinates rather than extending the scalar field.
Infinite-dimensional extension
John von Neumann incorporated the normality relation into the operator-theoretic framework of Hilbert spaces. For a bounded operator (T), the identity (TT^{}=T^{}T) defines normality exactly as in the matrix case. The finite-dimensional diagonal matrix is replaced by a spectral measure, and the operator is represented as
[ T=\int_{\sigma(T)}\lambda,dE(\lambda). ]
An infinite-dimensional normal operator need not possess any eigenvectors. Multiplication by the coordinate function on an (L^{2}) space provides the standard model: the operator is normal and has a spectral representation, although its spectrum may be continuous. The matrix theorem is recovered when the spectral measure is supported on finitely many points and the underlying Hilbert space has finite dimension.