# Definition A Hermitian (or self-adjoint) operator is one defined by: $ H^{\dagger} = H $ where $\dagger$ is the conjugate transpose or Hermitian adjoint. # Properties * Hermitian operators have real eigenvalues. * [[Eigentate|Eigenstates]] of Hermitian operators associated with different [[eigenvalue|eigenvalues]] are orthogonal. # Hermitian Matrices [[Matrix|Matrices]] representing Hermitian operators are Hermitian, i.e., they also obey $H^\dagger = H$. In component form: $ h_{ij} = h_{ji}^* $ Hermitian matrices are [[Diagonalizable Matrix|diagonalizable]] via suitable [[Similarity Transformation|unitary transformations]]. Any Hermitian matrix can be decomposed into a real [[symmetric matrix|symmetric]] piece and an imaginary [[skew-symmetric matrix|antisymmetric]] piece as follows: $ H = \frac{1}{2}(H + H^T) - \frac{i}{2}(H - H^T). $ ## Vector Space $n\times n$ Hermitian matrices with complex entries form a [[vector space]] denoted $H_n(\mathbb{C})$. It is a [[vector subspace]] of $M_n(\mathbb{C})$ (see [[Matrix#Vector Space|here]]). Example elements of this vector space include the [[Pauli matrices]]. Note that $H_n(\mathbb{C})$ is *not* a complex vector space, in fact, it is a real one. This is due to the fact that if you multiply a Hermitian matrix by $i$, you get an anti-Hermitian matrix, violating closure. ## $H_2(\mathbb{C})$ A [[basis]] for this space is $\mathcal{B} = \{\sigma_i | i = 0, 3\}$ where $\sigma_{1,2,3} = \sigma_{x,y,z}$ are the [[Pauli Matrices]] and $\sigma_0$ is the identity matrix. This fact is very often used in quantum mechanics. # Applications in Physics Observals in quantum mechanics are represented by Hermitian operators, which, on finite dimensional [[Ket Space|ket spaces]], can be represented as Hermitian matrices.