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

Famous quotes containing the words normal and/or matrix:

    Freedom is poetry, taking liberties with words, breaking the rules of normal speech, violating common sense. Freedom is violence.
    Norman O. Brown (b. 1913)

    In all cultures, the family imprints its members with selfhood. Human experience of identity has two elements; a sense of belonging and a sense of being separate. The laboratory in which these ingredients are mixed and dispensed is the family, the matrix of identity.
    Salvador Minuchin (20th century)