Normal Matrix

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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].



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 AA 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.