Normal Matrix
A Normal Matrix is a Complex Square Matrix, [math]\displaystyle{ A }[/math] where [math]\displaystyle{ A^*A=AA^* }[/math] where [math]\displaystyle{ A^* }[/math] is the conjugate transpose of [math]\displaystyle{ A^* }[/math].
- See: C*-Algebra, Complex Number, Matrix_(Mathematics)#Square_matrices, Matrix (Mathematics), Conjugate Transpose, Commute (Mathematics), Real Number, Diagonalizable, Similar Matrix, Diagonal Matrix, Normal Operator, Hilbert Space.
References
2015
- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/normal_matrix Retrieved:2015-2-16.
- In mathematics, a complex square matrix is normal if : [math]\displaystyle{ A^*A=AA^* }[/math] where A∗ is the conjugate transpose of A. That is, a matrix is normal if it commutes with its conjugate transpose.
That is, a matrix is normal if it commutes with its conjugate transpose.
A real square matrix 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 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.
- In mathematics, a complex square matrix is normal if : [math]\displaystyle{ A^*A=AA^* }[/math] where A∗ is the conjugate transpose of A. That is, a matrix is normal if it commutes with its conjugate transpose.