direct sum of Hermitian and skew-Hermitian matrices
In this example, we show that any square matrix with complex entries can uniquely be decomposed into the sum of one Hermitian matrix and one skew-Hermitian matrix. A fancy way to say this is that complex square matrices is the direct sum of Hermitian and skew-Hermitian matrices.
Since and are vector subspaces of , it is clear that is a vector subspace of . Conversely, suppose . We can then define
Here , and is the complex conjugate of , and is the transpose of . It follows that is Hermitian and is anti-Hermitian. Since , any element in can be written as the sum of one element in and one element in . Let us check that this decomposition is unique. If , then , so . We have established equation 1.
|Title||direct sum of Hermitian and skew-Hermitian matrices|
|Date of creation||2013-03-22 13:36:30|
|Last modified on||2013-03-22 13:36:30|
|Last modified by||mathcam (2727)|