# Normal Matrix

Normal Matrix

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":

Eigenvalue Algorithm - Eigenvalues and Eigenvectors - Normal, Hermitian, and Real-symmetric Matrices
... 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 ...
Normal Matrix - Analogy
... 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 ...

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As all historians know, the past is a great darkness, and filled with echoes. Voices may reach us from it; but what they say to us is imbued with the obscurity of the matrix out of which they come; and try as we may, we cannot always decipher them precisely in the clearer light of our day.
Margaret Atwood (b. 1939)

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Gilbert Keith Chesterton (1874–1936)