In mathematics, acomplexsquare matrixA isnormal if itcommutes with itsconjugate transposeA*:
The concept of normal matrices can be extended tonormal operators oninfinite-dimensionalnormed spaces and to normal elements inC*-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.
Thespectral theorem states that a matrix is normal if and only if it isunitarily similar to adiagonal matrix, and therefore any matrixA satisfying the equationA*A =AA* is diagonalizable. (The converse does not hold because diagonalizable matrices may have non-orthogonal eigenspaces.) Thus andwhere is a diagonal matrix whose diagonal values are in general complex.
The left and right singular vectors in thesingular value decomposition of a normal matrix differ only in complex phase from each other and from the corresponding eigenvectors, since the phase must be factored out of the eigenvalues to form singular values.
Among complex matrices, allunitary,Hermitian, andskew-Hermitian matrices are normal, with all eigenvalues being unit modulus, real, and imaginary, respectively. Likewise, among real matrices, allorthogonal,symmetric, andskew-symmetric matrices are normal, with all eigenvalues being complex conjugate pairs on the unit circle, real, and imaginary, respectively. However, it isnot the case that all normal matrices are either unitary or (skew-)Hermitian, as their eigenvalues can be any complex number, in general. For example,is neither unitary, Hermitian, nor skew-Hermitian, because its eigenvalues are; yet it is normal because
Proposition—A normaltriangular matrix isdiagonal.
LetA be any normal upper triangular matrix. Sinceusing subscript notation, one can write the equivalent expression using instead theith unit vector () to select theith row andith column:The expressionis equivalent, and so is
which shows that theith row must have the same norm as theith column.Consideri = 1. The first entry of row 1 and column 1 are the same, and the rest of column 1 is zero (because of triangularity). This implies the first row must be zero for entries 2 throughn. Continuing this argument for row–column pairs 2 throughn showsA is diagonal.Q.E.D.
The concept of normality is important because normal matrices are precisely those to which thespectral theorem applies:
Proposition—A matrixA is normal if and only if there exist adiagonal matrixΛ and aunitary matrixU such thatA =UΛU*.
The diagonal entries ofΛ are theeigenvalues ofA, and the columns ofU are theeigenvectors ofA. The matching eigenvalues inΛ come in the same order as the eigenvectors are ordered as columns ofU.
Another way of stating thespectral theorem is to say that normal matrices are precisely those matrices that can be represented by a diagonal matrix with respect to a properly chosenorthonormal basis ofCn. Phrased differently: a matrix is normal if and only if itseigenspaces spanCn and are pairwiseorthogonal with respect to the standard inner product ofCn.
The spectral theorem for normal matrices is a special case of the more generalSchur decomposition which holds for all square matrices. LetA be a square matrix. Then by Schur decomposition it is unitary similar to an upper-triangular matrix, say,B. IfA is normal, so isB. But thenB 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:
Proposition—A normal matrix is unitary if and only if all of its eigenvalues (its spectrum) lie on the unit circle of the complex plane.
Proposition—A normal matrix isself-adjoint if and only if its spectrum is contained in. In other words: A normal matrix isHermitian if and only if all its eigenvalues arereal.
In general, the sum or product of two normal matrices need not be normal. However, the following holds:
Proposition—IfA andB are normal withAB =BA, then bothAB andA +B are also normal. Furthermore there exists a unitary matrixU such thatUAU* andUBU* are diagonal matrices. In other wordsA andB aresimultaneously diagonalizable.
In this special case, the columns ofU* are eigenvectors of bothA andB and form an orthonormal basis inCn. This follows by combining the theorems that, over an algebraically closed field,commuting matrices aresimultaneously triangularizable and a normal matrix is diagonalizable – the added result is that these can both be done simultaneously.
It is possible to give a fairly long list of equivalent definitions of a normal matrix. LetA be an ×n complex matrix. Then the following are equivalent:
Some but not all of the above generalize to normal operators on infinite-dimensional Hilbert spaces. For example, a bounded operator satisfying (9) is onlyquasinormal.
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It is occasionally useful (but sometimes misleading) to think of the relationships of special kinds of normal matrices as analogous to the relationships of the corresponding type of complex numbers of which their eigenvalues are composed. This is because any function of a non-defective matrix acts directly on each of its eigenvalues, and the conjugate transpose of its spectral decomposition is, where is the diagonal matrix of eigenvalues. Likewise, if two normal matrices commute and are therefore simultaneously diagonalizable, any operation between these matrices also acts on each corresponding pair of eigenvalues.
As a special case, the complex numbers may be embedded in the normal 2×2 real matrices by the mappingwhich preserves addition and multiplication. It is easy to check that this embedding respects all of the above analogies.
