diff --git a/src/rulesets/LinearAlgebra/norm.jl b/src/rulesets/LinearAlgebra/norm.jl
index 910e37e1f..abd422387 100644
--- a/src/rulesets/LinearAlgebra/norm.jl
+++ b/src/rulesets/LinearAlgebra/norm.jl
@@ -17,7 +17,7 @@ function frule((_, Δx), ::typeof(norm), x::Number, p::Real)
     return y, ∂y
 end
 
-function rrule(::typeof(norm), x::AbstractArray, p::Real)
+function rrule(::typeof(norm), x::AbstractArray{<:Number}, p::Real)
     y = LinearAlgebra.norm(x, p)
     function norm_pullback_p(Δy)
         ∂x = if isempty(x) || p == 0
@@ -48,7 +48,7 @@ function rrule(::typeof(norm), x::AbstractArray, p::Real)
     norm_pullback_p(::Zero) = (NO_FIELDS, Zero(), Zero())
     return y, norm_pullback_p
 end
-function rrule(::typeof(norm), x::AbstractArray)
+function rrule(::typeof(norm), x::AbstractArray{<:Number})
     y = LinearAlgebra.norm(x)
     function norm_pullback_2(Δy)
         ∂x = if isempty(x)
@@ -64,7 +64,7 @@ function rrule(::typeof(norm), x::AbstractArray)
     norm_pullback_2(::Zero) = (NO_FIELDS, Zero())
     return y, norm_pullback_2
 end
-function rrule(::typeof(norm), x::Union{LinearAlgebra.AdjOrTransAbsVec}, p::Real)
+function rrule(::typeof(norm), x::LinearAlgebra.AdjOrTransAbsVec{<:Number}, p::Real)
     y, inner_pullback = rrule(norm, parent(x), p)
     function norm_pullback(Δy)
         (∂self, ∂x′, ∂p) = inner_pullback(Δy)
@@ -93,7 +93,7 @@ end
 ##### `normp`
 #####
 
-function rrule(::typeof(LinearAlgebra.normp), x::AbstractArray, p)
+function rrule(::typeof(LinearAlgebra.normp), x::AbstractArray{<:Number}, p)
     y = LinearAlgebra.normp(x, p)
     function normp_pullback(Δy)
         ∂x = @thunk _normp_back_x(x, p, y, Δy)
@@ -138,14 +138,14 @@ end
 ##### `normMinusInf`/`normInf`
 #####
 
-function rrule(::typeof(LinearAlgebra.normMinusInf), x::AbstractArray)
+function rrule(::typeof(LinearAlgebra.normMinusInf), x::AbstractArray{<:Number})
     y = LinearAlgebra.normMinusInf(x)
     normMinusInf_pullback(Δy) = (NO_FIELDS, _normInf_back(x, y, Δy))
     normMinusInf_pullback(::Zero) = (NO_FIELDS, Zero())
     return y, normMinusInf_pullback
 end
 
-function rrule(::typeof(LinearAlgebra.normInf), x::AbstractArray)
+function rrule(::typeof(LinearAlgebra.normInf), x::AbstractArray{<:Number})
     y = LinearAlgebra.normInf(x)
     normInf_pullback(Δy) = (NO_FIELDS, _normInf_back(x, y, Δy))
     normInf_pullback(::Zero) = (NO_FIELDS, Zero())
@@ -169,7 +169,7 @@ end
 ##### `norm1`
 #####
 
-function rrule(::typeof(LinearAlgebra.norm1), x::AbstractArray)
+function rrule(::typeof(LinearAlgebra.norm1), x::AbstractArray{<:Number})
     y = LinearAlgebra.norm1(x)
     norm1_pullback(Δy) = (NO_FIELDS, InplaceableThunk(
         @thunk(_norm1_back(x, y, Δy)),
@@ -201,7 +201,7 @@ function frule((_, Δx), ::typeof(LinearAlgebra.norm2), x)
     return y, _norm2_forward(x, Δx, y)
 end
 
-function rrule(::typeof(LinearAlgebra.norm2), x::AbstractArray)
+function rrule(::typeof(LinearAlgebra.norm2), x::AbstractArray{<:Number})
     y = LinearAlgebra.norm2(x)
     norm2_pullback(Δy) = (NO_FIELDS, InplaceableThunk(
         @thunk(_norm2_back(x, y, Δy)),
@@ -233,7 +233,7 @@ end
 ##### `normalize`
 #####
 
-function rrule(::typeof(normalize), x::AbstractVector, p::Real)
+function rrule(::typeof(normalize), x::AbstractVector{<:Number}, p::Real)
     nrm, inner_pullback = rrule(norm, x, p)
     Ty = typeof(first(x) / nrm)
     y = copyto!(similar(x, Ty), x)
@@ -248,7 +248,7 @@ function rrule(::typeof(normalize), x::AbstractVector, p::Real)
     normalize_pullback(::Zero) = (NO_FIELDS, Zero(), Zero())
     return y, normalize_pullback
 end
-function rrule(::typeof(normalize), x::AbstractVector)
+function rrule(::typeof(normalize), x::AbstractVector{<:Number})
     nrm = LinearAlgebra.norm2(x)
     Ty = typeof(first(x) / nrm)
     y = copyto!(similar(x, Ty), x)