Skew hermitian matrix is defined as a Square matrix whose conjugate transpose is similar or equals to negative of matrix. That is when transpose of a matrix is equals to negative of matrix then it is said to be as skew hermitian matrix. Transpose of any matrix is defined as matrix in which rows and columns replace each other. That is rows become columns and columns becomes rows of a matrix. And a negative matrix is defined as matrix in which elements becomes negative that is for any matrix A its negative matrix will become –A thus forming each element of A to be negative.
Lets take an example for skew hermitian matrix. Consider a matrix having elements:
First element is ‘-і’ , second element of first row is 2 + і, first element of second row is -2 + і and second element of second row is 0. In this matrix we can see that the transpose of this matrix becomes as first element as –і second as -2+ і third as 2+ і and fourth element as 0. And the negative of a matrix is determined as first element to be –і second as -2+ і third as 2+ і and fourth element as 0. Thus, we can see that the transpose of a matrix is equals to the negative of a matrix. Hence it is said to be hermitian matrix.
A skew hermitian matrix is purely imaginary. These are normal matrices. Entries of diagonal elements in skew hermitian matrix must be purely imaginary in nature. That is its not mandatory that all elements must be imaginary but diagonal elements must be imaginary in nature. If a matrix 'M' is skew hermitian then its іM and –іM both wil be hermitian in nature.