In mathematics, a complex square matrix A is normal if
where A* is the conjugate transpose of A. That is, a matrix is normal if it commutes with its conjugate transpose.
A matrix A with real entries satisfies A*=AT, and is therefore normal if ATA = AAT.
Normality is a convenient test for diagonalizability: a matrix is normal if and only if it is unitarily similar to a diagonal matrix, and therefore any matrix A satisfying the equation A*A=AA* is diagonalizable.
The concept of normal matrices can be extended to normal operators on infinite dimensional Hilbert 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.
Other articles related to "matrix, normal, normal matrix":
... The adjoint M* of a complex matrix M is the transpose of the conjugate of M M * = M T ... A square matrix A is called normal if it commutes with its adjoint A*A = AA* ... All hermitian matrices are normal ...
... of the relationships of different kinds of normal matrices as analogous to the relationships between different kinds of complex numbers Invertible matrices are analogous to non-ze ...
Famous quotes containing the words matrix and/or normal:
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