Google search-bar gives the definition of Hermitian as:
Hermitian: denoting or relating to a matrix in which those pairs of elements that are symmetrically placed with respect to the principal diagonal are complex conjugates
I have thought that Hermitian was synonymous with "real", meaning, if the matrix (A, for example) is Hermitian then that means there are no complex values in the matrix. I also believe it means the complex conjugate of the matrix is equal to the matrix like so:
$$A = A^\dagger.$$
However, there also exist Hermitian functions (which are complex?!) and the Hermitian operator (does not have to be real). Could someone please tell me what does the word "Hermitian" mean and what are the differences between the three: Hermitian matrix, Hermitian function, and Hermitian Operator? I am confused.
(PS: Please feel free to correct me if I have tagged this question incorrectly.)
Best Answer
A Hermitian matrix is a matrix that is equal to its conjugate transpose. This generalizes the concept of a "symmetric matrix", since every real symmetric matrix is Hermitian. However, there are certainly complex matrices that are Hermitian, such as
$$ \begin{bmatrix} 0 & i \\ -i & 0 \end{bmatrix} $$
By the Spectral theorem, every Hermitian matrix is unitarily diagonalizable with all real eigenvalues. Therefore, while a Hermitian matrix can have complex entries, in an appropriate basis it behaves like a real matrix.
When we describe a linear operator (or linear function) as being Hermitian, we are using a basis-free approach to Hermitian-ness. A linear operator $T: V \rightarrow V$ on an inner product space $V$ is said to be Hermitian if $T = T^*$, where $T^*$ is the unique operator such that $\langle Tx, y \rangle = \langle x, T^* y \rangle$ for every $x,y \in V$.
The connection between the two concepts is that an $n \times n$ Hermitian matrix is the matrix representation of a Hermitian operator on $\mathbb{C}^n$ equipped with the standard dot product.