Normal Matrix - Consequences

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

where

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.

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 can be seen as a special case of the more general result which holds for all square matrices: Schur decomposition. In fact, 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, a normal matrix is unitary if and only if its spectrum is contained in the unit circle of the complex plane. Also, a normal matrix is self-adjoint if and only if its spectrum consists of reals.

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. 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.