12.3 Hermitian matrices again

Recall that in section 5.4.1, we defined a square n-by-n complex matrix A on a vector space V to be Hermitian if A = A^†: A is equal to its conjugate transpose and is self-adjoint. For the inner product ⟨,⟩,

for all vectors v₁ and v₂ in V. We now need some additional properties of Hermitian matrices for the consideration of NISQ algorithms. matrix$Hermitian matrix$self-adjoint

It follows from the definition of a Hermitian matrix that:

• The diagonal elements of A are real.
• If c is a real number, cA is Hermitian.
• The eigenvalues λ_j of A are real. Some eigenvalues may appear more than once.
• The eigenvectors corresponding to distinct eigenvalues are orthogonal.
• By the Spectral Theorem in section 5.10, we can find {u₁, …, u_n} orthonormal eigenvectors of V with corresponding eigenvalues {λ₁, …, λ_n}. Many proofs regarding...