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A complex square matrix A is a normal matrix iff
- A * A = AA *
where A* is the conjugate transpose of
A (if A is a real matrix, this is the same as the transpose
of A).
Examples of normal matrices are unitary matrices, hermitian matrices and positive definite matrices.
It is useful to think of normal matrices in analogy to complex numbers, invertible normal matrices in analogy to non-zero
complex numbers, the conjugate transpose in analogy to the complex
conjugate, unitary matrices in analogy to complex numbers of absolute value 1, hermitian matrices in analogy to real numbers
and positive definite matrices in analogy to positive real numbers.
The concept of normality is mainly important because normal matrices are precisely the ones to which the spectral theorem applies; in other words, 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.
In general, the sum or product of two normal matrices need not be normal. However, if A and B are normal
with AB = BA, then both AB and A + B are also normal and furthermore we can
simultaneously diagonalize A and B in the following sense: there exists a unitary matrix U such
UAU* and UBU* are both diagonal matrices. In this case, the columns of
U* are eigenvectors of both A and B and form an orthonormal basis of
Cn.
If A is an invertible normal matrix, then there exists a unitary matrix U and a positive definite matrix
R such that A = RU = UR. The matrices R and U are uniquely determined by
A. This statement can be seen as an analog (and generalization) of the polar representation of non-zero complex
numbers.
The concept of normal matrices can be generalized to normal
operators on Hilbert spaces and to normal elements in C-star algebras.
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