A Hermitian matrix is a square matrix that is equal to its own conjugate transpose. This means that if we take the complex conjugate of each entry in the matrix and transpose it, we will get the original matrix. In simpler terms, a Hermitian matrix is symmetric about its main diagonal, with the values on the diagonal being real numbers.
A skew Hermitian matrix is also a square matrix that is equal to its own conjugate transpose, but with the opposite sign. In other words, if we take the complex conjugate of each entry in the matrix and transpose it, we will get the negative of the original matrix. Similar to a Hermitian matrix, a skew Hermitian matrix is also symmetric about its main diagonal, but with the values on the diagonal being purely imaginary numbers.
The characteristic roots of a Hermitian matrix are the eigenvalues of the matrix. These are the values that, when multiplied by the corresponding eigenvectors, give back the original matrix. In the case of a Hermitian matrix, the characteristic roots are always real numbers.
No, a Hermitian matrix can only have real eigenvalues. This is because the characteristic polynomial of a Hermitian matrix has only real coefficients, which means that the solutions (eigenvalues) must also be real.
The characteristic roots of a Hermitian matrix and its skew Hermitian counterpart are complex conjugates of each other. This means that if a Hermitian matrix has eigenvalues a+bi (where a and b are real numbers), then its skew Hermitian counterpart will have eigenvalues -a+bi. Additionally, the eigenvectors of a Hermitian matrix and its skew Hermitian counterpart are also complex conjugates of each other.