The ket } } can be computed by normal matrix multiplication.
2.
A symmetric matrix is necessarily a normal matrix.
3.
The spectral theorem says that every normal matrix is unitarily equivalent to some diagonal matrix.
4.
A normal matrix is self-adjoint if and only if its spectrum is contained in.
5.
For example, the spectral theorem for normal matrices states every normal matrix is unitarily diagonalizable.
6.
In other words : A normal matrix is Hermitian if and only if all its eigenvalues are real.
7.
In the latter case, the Gelfand Naimark representation theorem is one avenue in the development of spectral theory for normal operators, and generalizes the notion of diagonalizing a normal matrix.
8.
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.
9.
A Hermitian matrix ( or the special case of a real symmetric matrix ) or a unitary matrix is never defective; more generally, a normal matrix ( which includes Hermitian and unitary as special cases ) is never defective.
10.
In the special case that is a normal matrix, which by definition must be square, the positive semi-definite, the decomposition "'UDU "' " } } is also a singular value decomposition.
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