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analysis.inner_product_space.rayleigh

The Rayleigh quotient #

The Rayleigh quotient of a self-adjoint operator T on an inner product space E is the function λ x, ⟪T x, x⟫ / ∥x∥ ^ 2.

The main results of this file are is_self_adjoint.has_eigenvector_of_is_max_on and is_self_adjoint.has_eigenvector_of_is_min_on, which state that if E is complete, and if the Rayleigh quotient attains its global maximum/minimum over some sphere at the point x₀, then x₀ is an eigenvector of T, and the supr/infi of λ x, ⟪T x, x⟫ / ∥x∥ ^ 2 is the corresponding eigenvalue.

The corollaries is_self_adjoint.has_eigenvalue_supr_of_finite_dimensional and is_self_adjoint.has_eigenvalue_supr_of_finite_dimensional state that if E is finite-dimensional and nontrivial, then T has some (nonzero) eigenvectors with eigenvalue the supr/infi of λ x, ⟪T x, x⟫ / ∥x∥ ^ 2.

TODO #

A slightly more elaborate corollary is that if E is complete and T is a compact operator, then T has some (nonzero) eigenvector with eigenvalue either ⨆ x, ⟪T x, x⟫ / ∥x∥ ^ 2 or ⨅ x, ⟪T x, x⟫ / ∥x∥ ^ 2 (not necessarily both).

theorem continuous_linear_map.rayleigh_smul {𝕜 : Type u_1} [is_R_or_C 𝕜] {E : Type u_2} [inner_product_space 𝕜 E] (T : E →L[𝕜] E) (x : E) {c : 𝕜} (hc : c 0) :
(λ (x : E), T.re_apply_inner_self x / x ^ 2) (c x) = (λ (x : E), T.re_apply_inner_self x / x ^ 2) x
theorem continuous_linear_map.image_rayleigh_eq_image_rayleigh_sphere {𝕜 : Type u_1} [is_R_or_C 𝕜] {E : Type u_2} [inner_product_space 𝕜 E] (T : E →L[𝕜] E) {r : } (hr : 0 < r) :
(λ (x : E), T.re_apply_inner_self x / x ^ 2) '' {0} = (λ (x : E), T.re_apply_inner_self x / x ^ 2) '' metric.sphere 0 r
theorem continuous_linear_map.supr_rayleigh_eq_supr_rayleigh_sphere {𝕜 : Type u_1} [is_R_or_C 𝕜] {E : Type u_2} [inner_product_space 𝕜 E] (T : E →L[𝕜] E) {r : } (hr : 0 < r) :
(⨆ (x : {x // x 0}), (λ (x : E), T.re_apply_inner_self x / x ^ 2) x) = ⨆ (x : (metric.sphere 0 r)), (λ (x : E), T.re_apply_inner_self x / x ^ 2) x
theorem continuous_linear_map.infi_rayleigh_eq_infi_rayleigh_sphere {𝕜 : Type u_1} [is_R_or_C 𝕜] {E : Type u_2} [inner_product_space 𝕜 E] (T : E →L[𝕜] E) {r : } (hr : 0 < r) :
(⨅ (x : {x // x 0}), (λ (x : E), T.re_apply_inner_self x / x ^ 2) x) = ⨅ (x : (metric.sphere 0 r)), (λ (x : E), T.re_apply_inner_self x / x ^ 2) x
theorem inner_product_space.is_self_adjoint.eq_smul_self_of_is_local_extr_on {𝕜 : Type u_1} [is_R_or_C 𝕜] {E : Type u_2} [inner_product_space 𝕜 E] [complete_space E] {T : E →L[𝕜] E} (hT : inner_product_space.is_self_adjoint T) {x₀ : E} (hextr : is_local_extr_on T.re_apply_inner_self (metric.sphere 0 x₀) x₀) :
T x₀ = ((λ (x : E), T.re_apply_inner_self x / x ^ 2) x₀) x₀
theorem inner_product_space.is_self_adjoint.has_eigenvector_of_is_local_extr_on {𝕜 : Type u_1} [is_R_or_C 𝕜] {E : Type u_2} [inner_product_space 𝕜 E] [complete_space E] {T : E →L[𝕜] E} (hT : inner_product_space.is_self_adjoint T) {x₀ : E} (hx₀ : x₀ 0) (hextr : is_local_extr_on T.re_apply_inner_self (metric.sphere 0 x₀) x₀) :

For a self-adjoint operator T, a local extremum of the Rayleigh quotient of T on a sphere centred at the origin is an eigenvector of T.

theorem inner_product_space.is_self_adjoint.has_eigenvector_of_is_max_on {𝕜 : Type u_1} [is_R_or_C 𝕜] {E : Type u_2} [inner_product_space 𝕜 E] [complete_space E] {T : E →L[𝕜] E} (hT : inner_product_space.is_self_adjoint T) {x₀ : E} (hx₀ : x₀ 0) (hextr : is_max_on T.re_apply_inner_self (metric.sphere 0 x₀) x₀) :
module.End.has_eigenvector T (⨆ (x : {x // x 0}), (λ (x : E), T.re_apply_inner_self x / x ^ 2) x) x₀

For a self-adjoint operator T, a maximum of the Rayleigh quotient of T on a sphere centred at the origin is an eigenvector of T, with eigenvalue the global supremum of the Rayleigh quotient.

theorem inner_product_space.is_self_adjoint.has_eigenvector_of_is_min_on {𝕜 : Type u_1} [is_R_or_C 𝕜] {E : Type u_2} [inner_product_space 𝕜 E] [complete_space E] {T : E →L[𝕜] E} (hT : inner_product_space.is_self_adjoint T) {x₀ : E} (hx₀ : x₀ 0) (hextr : is_min_on T.re_apply_inner_self (metric.sphere 0 x₀) x₀) :
module.End.has_eigenvector T (⨅ (x : {x // x 0}), (λ (x : E), T.re_apply_inner_self x / x ^ 2) x) x₀

For a self-adjoint operator T, a minimum of the Rayleigh quotient of T on a sphere centred at the origin is an eigenvector of T, with eigenvalue the global infimum of the Rayleigh quotient.

The supremum of the Rayleigh quotient of a self-adjoint operator T on a nontrivial finite-dimensional vector space is an eigenvalue for that operator.

The infimum of the Rayleigh quotient of a self-adjoint operator T on a nontrivial finite-dimensional vector space is an eigenvalue for that operator.