diff --git a/stdlib/LinearAlgebra/src/diagonal.jl b/stdlib/LinearAlgebra/src/diagonal.jl
index 1a5eca3c0ce05..06494130acbd0 100644
--- a/stdlib/LinearAlgebra/src/diagonal.jl
+++ b/stdlib/LinearAlgebra/src/diagonal.jl
@@ -116,13 +116,15 @@ ishermitian(D::Diagonal{<:Number}) = isreal(D.diag)
 ishermitian(D::Diagonal) = all(ishermitian, D.diag)
 issymmetric(D::Diagonal{<:Number}) = true
 issymmetric(D::Diagonal) = all(issymmetric, D.diag)
-isposdef(D::Diagonal) = all(x -> x > 0, D.diag)
+isposdef(D::Diagonal) = all(isposdef, D.diag)
 
 factorize(D::Diagonal) = D
 
 real(D::Diagonal) = Diagonal(real(D.diag))
 imag(D::Diagonal) = Diagonal(imag(D.diag))
 
+isdiag(D::Diagonal) = all(isdiag, D.diag)
+isdiag(D::Diagonal{<:Number}) = true
 istriu(D::Diagonal) = true
 istril(D::Diagonal) = true
 function triu!(D::Diagonal,k::Integer=0)
diff --git a/stdlib/LinearAlgebra/src/uniformscaling.jl b/stdlib/LinearAlgebra/src/uniformscaling.jl
index be1379e9034f3..5d0190bd43f4c 100644
--- a/stdlib/LinearAlgebra/src/uniformscaling.jl
+++ b/stdlib/LinearAlgebra/src/uniformscaling.jl
@@ -74,10 +74,12 @@ oneunit(J::UniformScaling{T}) where {T} = oneunit(UniformScaling{T})
 zero(::Type{UniformScaling{T}}) where {T} = UniformScaling(zero(T))
 zero(J::UniformScaling{T}) where {T} = zero(UniformScaling{T})
 
+isdiag(::UniformScaling) = true
 istriu(::UniformScaling) = true
 istril(::UniformScaling) = true
 issymmetric(::UniformScaling) = true
 ishermitian(J::UniformScaling) = isreal(J.λ)
+isposdef(J::UniformScaling) = isposdef(J.λ)
 
 (+)(J::UniformScaling, x::Number) = J.λ + x
 (+)(x::Number, J::UniformScaling) = x + J.λ
diff --git a/stdlib/LinearAlgebra/test/diagonal.jl b/stdlib/LinearAlgebra/test/diagonal.jl
index ee11b5db9854b..95003b76c1cd9 100644
--- a/stdlib/LinearAlgebra/test/diagonal.jl
+++ b/stdlib/LinearAlgebra/test/diagonal.jl
@@ -50,6 +50,9 @@ Random.seed!(1)
         @test D[1,2] == 0
 
         @test issymmetric(D)
+        @test isdiag(D)
+        @test isdiag(Diagonal([[1 0; 0 1], [1 0; 0 1]]))
+        @test !isdiag(Diagonal([[1 0; 0 1], [1 0; 1 1]]))
         @test istriu(D)
         @test istril(D)
         if elty <: Real
@@ -298,6 +301,10 @@ end
 @testset "isposdef" begin
     @test isposdef(Diagonal(1.0 .+ rand(n)))
     @test !isposdef(Diagonal(-1.0 * rand(n)))
+    @test isposdef(Diagonal(complex(1.0, 0.0) .+ rand(n)))
+    @test !isposdef(Diagonal(complex(1.0, 1.0) .+ rand(n)))
+    @test isposdef(Diagonal([[1 0; 0 1], [1 0; 0 1]]))
+    @test !isposdef(Diagonal([[1 0; 0 1], [1 0; 1 1]]))
 end
 
 @testset "getindex" begin
diff --git a/stdlib/LinearAlgebra/test/uniformscaling.jl b/stdlib/LinearAlgebra/test/uniformscaling.jl
index c544736f9feac..6aea3fd6899f3 100644
--- a/stdlib/LinearAlgebra/test/uniformscaling.jl
+++ b/stdlib/LinearAlgebra/test/uniformscaling.jl
@@ -30,13 +30,18 @@ end
     @test conj(UniformScaling(1.0+1.0im))::UniformScaling{Complex{Float64}} == UniformScaling(1.0-1.0im)
 end
 
-@testset "istriu, istril, issymmetric, ishermitian, isapprox" begin
+@testset "isdiag, istriu, istril, issymmetric, ishermitian, isposdef, isapprox" begin
+    @test isdiag(I)
     @test istriu(I)
     @test istril(I)
     @test issymmetric(I)
     @test issymmetric(UniformScaling(complex(1.0,1.0)))
     @test ishermitian(I)
     @test !ishermitian(UniformScaling(complex(1.0,1.0)))
+    @test isposdef(I)
+    @test !isposdef(-I)
+    @test isposdef(UniformScaling(complex(1.0, 0.0)))
+    @test !isposdef(UniformScaling(complex(1.0, 1.0)))
     @test UniformScaling(4.00000000000001) ≈ UniformScaling(4.0)
     @test UniformScaling(4.32) ≈ UniformScaling(4.3) rtol=0.1 atol=0.01
     @test UniformScaling(4.32) ≈ 4.3 * [1 0; 0 1] rtol=0.1 atol=0.01