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:

    Every normal person, in fact, is only normal on the average. His ego approximates to that of the psychotic in some part or other and to a greater or lesser extent.
    Sigmund Freud (1856–1939)

    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)