improve eigvals & co docstrings (JuliaLang#43904)

stdlib/LinearAlgebra/src/eigen.jl

@@ -140,7 +140,8 @@ end
 sorteig!(λ::AbstractVector, sortby::Union{Function,Nothing}=eigsortby) = sortby === nothing ? λ : sort!(λ, by=sortby)
 
 """
-    eigen!(A, [B]; permute, scale, sortby)
+    eigen!(A; permute, scale, sortby)
+    eigen!(A, B; sortby)
 
 Same as [`eigen`](@ref), but saves space by overwriting the input `A` (and
 `B`), instead of creating a copy.
@@ -179,7 +180,7 @@ end
 """
     eigen(A; permute::Bool=true, scale::Bool=true, sortby) -> Eigen
 
-Computes the eigenvalue decomposition of `A`, returning an [`Eigen`](@ref) factorization object `F`
+Compute the eigenvalue decomposition of `A`, returning an [`Eigen`](@ref) factorization object `F`
 which contains the eigenvalues in `F.values` and the eigenvectors in the columns of the
 matrix `F.vectors`. (The `k`th eigenvector can be obtained from the slice `F.vectors[:, k]`.)
 
@@ -316,7 +317,7 @@ Return the eigenvalues of `A`.
 
 For general non-symmetric matrices it is possible to specify how the matrix is balanced
 before the eigenvalue calculation. The `permute`, `scale`, and `sortby` keywords are
-the same as for [`eigen!`](@ref).
+the same as for [`eigen`](@ref).
 
 # Examples
 ```jldoctest
@@ -462,17 +463,18 @@ function eigen!(A::StridedMatrix{T}, B::StridedMatrix{T}; sortby::Union{Function
 end
 
 """
-    eigen(A, B) -> GeneralizedEigen
+    eigen(A, B; sortby) -> GeneralizedEigen
 
-Computes the generalized eigenvalue decomposition of `A` and `B`, returning a
+Compute the generalized eigenvalue decomposition of `A` and `B`, returning a
 [`GeneralizedEigen`](@ref) factorization object `F` which contains the generalized eigenvalues in
 `F.values` and the generalized eigenvectors in the columns of the matrix `F.vectors`.
 (The `k`th generalized eigenvector can be obtained from the slice `F.vectors[:, k]`.)
 
 Iterating the decomposition produces the components `F.values` and `F.vectors`.
 
-Any keyword arguments passed to `eigen` are passed through to the lower-level
-[`eigen!`](@ref) function.
+By default, the eigenvalues and vectors are sorted lexicographically by `(real(λ),imag(λ))`.
+A different comparison function `by(λ)` can be passed to `sortby`, or you can pass
+`sortby=nothing` to leave the eigenvalues in an arbitrary order.
 
 # Examples
 ```jldoctest
@@ -563,7 +565,7 @@ end
 """
     eigvals(A, B) -> values
 
-Computes the generalized eigenvalues of `A` and `B`.
+Compute the generalized eigenvalues of `A` and `B`.
 
 # Examples
 ```jldoctest

stdlib/LinearAlgebra/src/symmetriceigen.jl

@@ -16,7 +16,7 @@ eigen!(A::RealHermSymComplexHerm{<:BlasReal,<:StridedMatrix}, irange::UnitRange)
 """
     eigen(A::Union{SymTridiagonal, Hermitian, Symmetric}, irange::UnitRange) -> Eigen
 
-Computes the eigenvalue decomposition of `A`, returning an [`Eigen`](@ref) factorization object `F`
+Compute the eigenvalue decomposition of `A`, returning an [`Eigen`](@ref) factorization object `F`
 which contains the eigenvalues in `F.values` and the eigenvectors in the columns of the
 matrix `F.vectors`. (The `k`th eigenvector can be obtained from the slice `F.vectors[:, k]`.)
 
@@ -42,7 +42,7 @@ eigen!(A::RealHermSymComplexHerm{T,<:StridedMatrix}, vl::Real, vh::Real) where {
 """
     eigen(A::Union{SymTridiagonal, Hermitian, Symmetric}, vl::Real, vu::Real) -> Eigen
 
-Computes the eigenvalue decomposition of `A`, returning an [`Eigen`](@ref) factorization object `F`
+Compute the eigenvalue decomposition of `A`, returning an [`Eigen`](@ref) factorization object `F`
 which contains the eigenvalues in `F.values` and the eigenvectors in the columns of the
 matrix `F.vectors`. (The `k`th eigenvector can be obtained from the slice `F.vectors[:, k]`.)
 
@@ -86,7 +86,7 @@ eigvals!(A::RealHermSymComplexHerm{<:BlasReal,<:StridedMatrix}, irange::UnitRang
 """
     eigvals(A::Union{SymTridiagonal, Hermitian, Symmetric}, irange::UnitRange) -> values
 
-Returns the eigenvalues of `A`. It is possible to calculate only a subset of the
+Return the eigenvalues of `A`. It is possible to calculate only a subset of the
 eigenvalues by specifying a [`UnitRange`](@ref) `irange` covering indices of the sorted eigenvalues,
 e.g. the 2nd to 8th eigenvalues.
 
@@ -127,7 +127,7 @@ eigvals!(A::RealHermSymComplexHerm{T,<:StridedMatrix}, vl::Real, vh::Real) where
 """
     eigvals(A::Union{SymTridiagonal, Hermitian, Symmetric}, vl::Real, vu::Real) -> values
 
-Returns the eigenvalues of `A`. It is possible to calculate only a subset of the eigenvalues
+Return the eigenvalues of `A`. It is possible to calculate only a subset of the eigenvalues
 by specifying a pair `vl` and `vu` for the lower and upper boundaries of the eigenvalues.
 
 # Examples