Yes, all real symmetric matrices are Hermitian.
Understanding Hermitian Matrices
A Hermitian matrix is a square complex matrix that is equal to its conjugate transpose. In simpler terms, a Hermitian matrix remains the same if you swap its rows and columns, and then take the complex conjugate of each element.
Real Symmetric Matrices
A real symmetric matrix is a square matrix with real entries where each element is equal to its corresponding element across the main diagonal.
Relationship
Since a real symmetric matrix has real entries, its complex conjugate is itself. Therefore, its conjugate transpose is identical to its original transpose, which is the matrix itself due to its symmetric property. This means that a real symmetric matrix satisfies the definition of a Hermitian matrix.
Examples
- The matrix A = [[1, 2], [2, 3]] is a real symmetric matrix and is also Hermitian.
- The matrix B = [[1, i], [-i, 1]] is a Hermitian matrix but not a real symmetric matrix because it contains complex elements.
Conclusion
In summary, all real symmetric matrices are Hermitian because they satisfy the condition of being equal to their conjugate transpose. However, not all Hermitian matrices are real symmetric matrices as they can contain complex elements.
