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.

Read more about Normal Matrix:  Special Cases, Consequences, Equivalent Definitions, Analogy

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 ...

Famous quotes containing the words matrix and/or normal:

    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)

    Nothing is poetical if plain daylight is not poetical; and no monster should amaze us if the normal man does not amaze.
    Gilbert Keith Chesterton (1874–1936)