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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.
Proof: Let A be any normal upper triangular matrix. Since
(A*A)ii = (AA*)ii,
using subscript notation, one can write the equivalent expression using instead the ith unit vector () to select the ith row and ith column:
is equivalent, and so is
which shows that the ith row must have the same norm as the ith column.
Consider i = 1. The first entry of row 1 and column 1 are the same (because of normality), and the rest of column 1 is zero (because of triangularity). This implies the first row must be zero for entries 2 through n. Continuing this argument for row-column pairs 2 through n shows A is diagonal.◻
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 A, and the columns of U are the eigenvectors of A. The matching eigenvalues in ? come in the same order as the eigenvectors are ordered as columns of U.
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 spectral theorem for normal matrices is a special case of the more general Schur decomposition which holds for all square matrices. Let A be a square matrix. Then by Schur decomposition it is unitary similar to an upper-triangular matrix, say, B. If A is normal, so is B. But then B 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 is self-adjoint if and only if its spectrum is contained in . In other words: A normal matrix is Hermitian if and only if all its eigenvalues are real.
In general, the sum or product of two normal matrices need not be normal. However, the following holds:
Proposition. If A and B are normal with AB = BA, then both AB and A + B are also normal. Furthermore there exists a unitary matrix U such that UAU* and UBU* are diagonal matrices. In other words A and B are simultaneously diagonalizable.
In this special case, the columns of U* are eigenvectors of both A and B and form an orthonormal basis in Cn. 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.
It is possible to give a fairly long list of equivalent definitions of a normal matrix. Let A be a n × n complex matrix. Then the following are equivalent: