Linear Algebra using Dirac Notation: Pt. 2 PHYS 476Q - Southern Illinois University February 6, 2018 PHYS 476Q - Southern Illinois University Linear Algebra using Dirac Notation: Pt. 2 February 6, 2018 1 / 15
Adjoint Operators For every operator R L(H 1, H 2 ), there exists a unique operator R L(H 2, H 1 ) called the adjoint of R that is defined by the condition that β R α = α R β α H 1, β H 2. Bra-ket rule: ket: R β bra: β R. In terms of its matrix representation, we have j R i = i R j = R ij. The matrix for R is the conjugate transpose of the matrix for R. PHYS 476Q - Southern Illinois University Linear Algebra using Dirac Notation: Pt. 2 February 6, 2018 2 / 15
Adjoint Operators Example Let R = 0 + + 1, where ± = 1/2( 0 ± i 1 ). What is R? PHYS 476Q - Southern Illinois University Linear Algebra using Dirac Notation: Pt. 2 February 6, 2018 3 / 15
Eigenvalues and Eigenvectors The eigenvalue of a linear operator R is a complex number λ such that R ψ = λ ψ for some vector ψ. For a fixed eigenvalue λ, the set V λ = { ϕ : R ϕ = λ ϕ } forms a vector space. This vector space V λ is called the eigenspace associated with λ (or the λ-eigenspace), and any vector from this space is an eigenvector of R with eigenvalue λ. The kernel (or null space) of an operator R is the eigenspace associated with the eigenvalue 0, and it is denoted by ker(r). The rank of an operator R : H 1 H 2, denoted rk(r) is the number given rk(r) = dim(h 1 ) dim(ker(r)). PHYS 476Q - Southern Illinois University Linear Algebra using Dirac Notation: Pt. 2 February 6, 2018 4 / 15
Important Classes of Operators The Picture: PHYS 476Q - Southern Illinois University Linear Algebra using Dirac Notation: Pt. 2 February 6, 2018 5 / 15
Projections A projection on a d-dimensional space H is any operator of the form P V = s ɛ i ɛ i, i=1 where the { ɛ i } s i=1 1 s d. is an orthonormal set of vectors for some integer The { ɛ i } s i=1 form a basis for some s-dimensional vector subspace V H, and P V is called the subspace projector onto V : P V ψ V for any ψ H. The identity I is the trivial projector onto all of H. PHYS 476Q - Southern Illinois University Linear Algebra using Dirac Notation: Pt. 2 February 6, 2018 6 / 15
Projections Example: For any normalized vector ψ, the operator ψ ψ is a rank-one projector. Example: Prove that every projector satisfies P 2 = P. Example: Prove that 0 and 1 are the only possible eigenvalues for a projector. PHYS 476Q - Southern Illinois University Linear Algebra using Dirac Notation: Pt. 2 February 6, 2018 7 / 15
Normal Operators An operator N L(H) is called normal if N N = NN. Lemma: If N L(H) is normal with distinct eigenvalues λ 1 and λ 2, then the associated eigenspaces V λ1 and V λ2 are orthogonal. Furthermore, H = λ i V λi, where the union is taken over all distinct eigenvalues of N. The eigenspace decomposition picture: PHYS 476Q - Southern Illinois University Linear Algebra using Dirac Notation: Pt. 2 February 6, 2018 8 / 15
Normal Operators: The Spectral Decomposition Spectral Decomposition Theorem: Every normal operator N L(H) can be uniquely written as N = n λ i P λi, i=1 where the {λ i } n i=1 are the distinct eigenvalues of N and the {P λ i } n i=1 are the corresponding eigenspace projectors. We can further decompose the eigenspace projectors and write: N = dim(h) k=1 Here the λ k are not necessarily distinct. λ k λ k λ k. PHYS 476Q - Southern Illinois University Linear Algebra using Dirac Notation: Pt. 2 February 6, 2018 9 / 15
Unitary Operators An operator U L(H) is called unitary if U U = UU = I. Example For arbitrary α, β, γ R the operator U(α, β, γ) on C 2 given by is unitary. U(α, β, γ). = ( e i(α+γ)/2 cos β/2 e i(α γ)/2 ) sin β/2 e i( α+γ)/2 sin β/2 e i(α+γ)/2 cos β/2 PHYS 476Q - Southern Illinois University Linear Algebra using Dirac Notation: Pt. 2 February 6, 2018 10 / 15
Unitary Operators Unitary operators preserve inner products: For vectors U ψ and U φ, their inner product is ψ U U φ = ψ I φ = ψ φ. If { ɛ i } d i=1 and { δ i } d i=1 are orthonormal bases for H, then U = d i=1 ɛ i δ i is a unitary operator transforming one basis to another U U = d d δ i ɛ i ɛ j δ j = i=1 j=1 d δ i δ i = I. i=1 The spectral decomposition can be written as N = UΛU, where Λ is a diagonal matrix in the computational basis with the eigenvalues of N along the diagonal. PHYS 476Q - Southern Illinois University Linear Algebra using Dirac Notation: Pt. 2 February 6, 2018 11 / 15
Hermitian and Positive Operators An operator A is called hermitian (or self-adjoint) if A = A. Fact: Every hermitian operator has real eigenvalues. Proof: A positive operator is any normal operator A with non-negative eigenvalues. Equivalently, A is positive iff ψ A ψ 0 for all ψ H. For any operator R L(H 1, H 2 ), the operators R R and RR are positive. PHYS 476Q - Southern Illinois University Linear Algebra using Dirac Notation: Pt. 2 February 6, 2018 12 / 15
Functions of Normal Operators The spectral decomposition allows us to define functions of operators. Notice that N = n λ k P λk N m = k=1 n λ m k P λ k for all m = 1,. Then for any complex function f (z) with Taylor expansion f (z) = k=0 α kz k, we define the operator function ˆf (N) := k=0 α kn k. Consequently, for a function f : X C, if N has eigenvalues lying in X, then we can define a new operator k=1 ˆf (N) = n f (λ i )P λi. i=1 PHYS 476Q - Southern Illinois University Linear Algebra using Dirac Notation: Pt. 2 February 6, 2018 13 / 15
Functions of Normal Operators Example: For the function f (z) = e z, we have ˆf (N) = e N = n e λ i P λi. i=1 For functions f (z) not defined at z = 0, define ˆf (N) = λ i 0 f (λ i)p λi. Example: For the multiplicative inverse f (z) = z 1 = 1 z, define N 1 := λ i 0 λ 1 i P λi. Note that N 1 N = NN 1 = λ i P λi λ 1 j P λj = P λi = P supp(n). λ i 0 λ j 0 λ i 0 where supp(n) := λi 0V λi is the support of N. PHYS 476Q - Southern Illinois University Linear Algebra using Dirac Notation: Pt. 2 February 6, 2018 14 / 15
Singular Value Decomposition Singular Value Decomposition Theorem: Every operator R L(H) can be written as R = n σ k UP σk, k=1 where {σ k } n k=1 are the distinct singular values of R, U is a unitary operator, and the {P σk } n k=1 are projections on the eigenspaces of R R. Equivalently, R can be written as R = V Λ σ W, where V and W are unitaries, and Λ σ is a diagonal matrix in the computational basis with diagonal elements being the singular values of R. PHYS 476Q - Southern Illinois University Linear Algebra using Dirac Notation: Pt. 2 February 6, 2018 15 / 15