more doc updates and small GKL bug fix

docs/src/man/implementation.md

@@ -29,11 +29,18 @@ whereas the inverse calculation is obtained as
 KrylovKit.unproject!
 ```
 
-Finally, an orthonormal basis can be transformed using a rank-1 update using
+An orthonormal basis can be transformed using a rank-1 update using
 ```@docs
 KrylovKit.rank1update!
 ```
 
+Note that this changes the subspace. A mere rotation of the basis, which does not change
+the subspace spanned by it, can be computed using
+
+```@docs
+KrylovKit.basistransform!
+```
+
 ## Dense linear algebra
 
 KrylovKit relies on Julia's `LinearAlgebra` module from the standard library for most of its

src/factorizations/gkl.jl

@@ -229,7 +229,7 @@ function initialize!(iter::GKLIterator, state::GKLFactorization; verbosity::Int
     αs = empty!(state.αs)
     βs = empty!(state.βs)
 
-    u = mul!(V[1], iter.u₀, 1 / norm(iter.u₀))
+    u = mul!(U[1], iter.u₀, 1 / norm(iter.u₀))
     v = iter.operator(u, true)
     α = norm(v)
     rmul!(v, 1 / α)

src/orthonormal.jl

@@ -272,6 +272,20 @@ end
     return b
 end
 
+"""
+    basistransform!(b::OrthonormalBasis, U::AbstractMatrix)
+
+Transform the orthonormal basis `b` by the matrix `U`. For `b` an orthonormal basis,
+the matrix `U` should be real orthogonal or complex unitary; it is up to the user to ensure
+this condition is satisfied. The new basis vectors are given by
+
+```
+    b[j] ← b[i] * U[i,j]
+```
+
+and are stored in `b`, so the old basis vectors are thrown away. Note that, by definition,
+the subspace spanned by these basis vectors is exactly the same.
+"""
 function basistransform!(b::OrthonormalBasis{T}, U::AbstractMatrix) where {T} # U should be unitary or isometric
     if T <: AbstractArray && IndexStyle(T) isa IndexLinear && Threads.nthreads() > 1
         return basistransform_linear_multithreaded!(b, U)