Hermitian Matrix:
A Hermitian matrix is a complex square matrix that is equal to its own conjugate transpose. In other words, for a matrix H to be Hermitian, it must satisfy the condition H = H^H, where H^H denotes the conjugate transpose of H.

Given Matrix H:
In this case, the given matrix H = [1 1+i; 1-i 4] is a Hermitian matrix because it is equal to its conjugate transpose.

Diagonalization:
To find a non-singular matrix P such that P^(-1)HP is diagonal, we can use the following steps:
1. Find the eigenvalues and eigenvectors of the Hermitian matrix H.
2. Construct the matrix P using the eigenvectors as columns.
3. Verify that P^(-1)HP is a diagonal matrix.

Calculation:
1. First, we find the eigenvalues of H by solving the characteristic equation det(H - λI) = 0, where I is the identity matrix.
det([1-λ 1+i; 1-i 4-λ]) = 0
(1-λ)(4-λ) - (1+i)(1-i) = 0
Solving this equation, we get the eigenvalues λ1 = 3+2i and λ2 = 2-2i.
2. Next, we find the eigenvectors corresponding to these eigenvalues by solving the equation (H - λI)v = 0.
For λ1 = 3+2i, we find the eigenvector v1 = [1; i].
For λ2 = 2-2i, we find the eigenvector v2 = [1; -i].
3. Construct the matrix P using the eigenvectors as columns: P = [1 1; i -i].
4. Finally, verify that P^(-1)HP is a diagonal matrix. Calculate P^(-1) = P^H/(det(P)) and then compute P^(-1)HP to obtain the diagonal matrix.

Conclusion:
In conclusion, by following the above steps, we can find a non-singular matrix P such that P^(-1)HP is diagonal for the given Hermitian matrix H. This process of diagonalization is a common technique used in linear algebra to simplify the analysis of complex matrices.