From aa59d8fc16e8d07f142621b7818b6c05d072e184 Mon Sep 17 00:00:00 2001
From: hyrodium <hyrodium@gmail.com>
Date: Sat, 23 Oct 2021 23:44:08 +0900
Subject: [PATCH 01/11] add support for eigen values and eigen vectors

---
 src/Rotations.jl |  1 +
 src/eigen.jl     | 14 ++++++++++++++
 test/eigen.jl    | 15 +++++++++++++++
 test/runtests.jl |  1 +
 4 files changed, 31 insertions(+)
 create mode 100644 src/eigen.jl
 create mode 100644 test/eigen.jl

diff --git a/src/Rotations.jl b/src/Rotations.jl
index 8455832c..3d19b373 100644
--- a/src/Rotations.jl
+++ b/src/Rotations.jl
@@ -22,6 +22,7 @@ include("principal_value.jl")
 include("rodrigues_params.jl")
 include("error_maps.jl")
 include("rotation_error.jl")
+include("eigen.jl")
 include("deprecated.jl")
 
 export
diff --git a/src/eigen.jl b/src/eigen.jl
new file mode 100644
index 00000000..828df82d
--- /dev/null
+++ b/src/eigen.jl
@@ -0,0 +1,14 @@
+function LinearAlgebra.eigvals(R::Rotation{3})
+    θ = rotation_angle(R)
+    return SVector(exp(θ*im), exp(-θ*im), 1)
+end
+
+function LinearAlgebra.eigvecs(R::Rotation{3})
+    n3 = normalize(rotation_axis(R))
+    n1 = normalize(perpendicular_vector(n3))
+    n2 = normalize(n3×n1)
+    v1 = n1*im + n2
+    v2 = n1 + n2*im
+    v3 = n3
+    return hcat(v1,v2,v3)
+end
diff --git a/test/eigen.jl b/test/eigen.jl
new file mode 100644
index 00000000..fafcd2f4
--- /dev/null
+++ b/test/eigen.jl
@@ -0,0 +1,15 @@
+@testset "Eigen" begin
+    all_types = (RotMatrix{3}, AngleAxis, RotationVec,
+                 UnitQuaternion, RodriguesParam, MRP,
+                 RotXYZ, RotYZX, RotZXY, RotXZY, RotYXZ, RotZYX,
+                 RotXYX, RotYZY, RotZXZ, RotXZX, RotYXY, RotZYZ,
+                 RotX, RotY, RotZ,
+                 RotXY, RotYZ, RotZX, RotXZ, RotYX, RotZY)
+
+    for type in all_types
+        R = rand(type)
+        λs = eigvals(R)
+        vs = eigvecs(R)
+        @test R * vs ≈ transpose(λs) .* vs
+    end
+end
diff --git a/test/runtests.jl b/test/runtests.jl
index 967e52fd..33904e2f 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -28,6 +28,7 @@ include("rodrigues_params.jl")
 include("quatmaps.jl")
 include("rotation_error.jl")
 include("distribution_tests.jl")
+include("eigen.jl")
 include("deprecated.jl")
 
 include(joinpath(@__DIR__, "..", "perf", "runbenchmarks.jl"))

From b3cc7f09b70a2596c3bd0a2e0290948523902584 Mon Sep 17 00:00:00 2001
From: hyrodium <hyrodium@gmail.com>
Date: Sat, 23 Oct 2021 23:56:47 +0900
Subject: [PATCH 02/11] update methods for eigen

---
 src/eigen.jl | 10 ++++++++--
 1 file changed, 8 insertions(+), 2 deletions(-)

diff --git a/src/eigen.jl b/src/eigen.jl
index 828df82d..695912c9 100644
--- a/src/eigen.jl
+++ b/src/eigen.jl
@@ -7,8 +7,14 @@ function LinearAlgebra.eigvecs(R::Rotation{3})
     n3 = normalize(rotation_axis(R))
     n1 = normalize(perpendicular_vector(n3))
     n2 = normalize(n3×n1)
-    v1 = n1*im + n2
-    v2 = n1 + n2*im
+    v1 = normalize(n1*im + n2)
+    v2 = normalize(n1 + n2*im)
     v3 = n3
     return hcat(v1,v2,v3)
 end
+
+function LinearAlgebra.eigen(R::Rotation{3})
+    λs = eigvals(R)
+    vs = eigvecs(R)
+    return Eigen(λs, vs)
+end

From 6d5bbf43dc622bfa4af3ac99db2efbe127dda555 Mon Sep 17 00:00:00 2001
From: hyrodium <hyrodium@gmail.com>
Date: Sat, 23 Oct 2021 23:57:13 +0900
Subject: [PATCH 03/11] update tests for eigen

---
 test/eigen.jl | 13 +++++++++++--
 1 file changed, 11 insertions(+), 2 deletions(-)

diff --git a/test/eigen.jl b/test/eigen.jl
index fafcd2f4..521181b7 100644
--- a/test/eigen.jl
+++ b/test/eigen.jl
@@ -6,10 +6,19 @@
                  RotX, RotY, RotZ,
                  RotXY, RotYZ, RotZX, RotXZ, RotYX, RotZY)
 
-    for type in all_types
-        R = rand(type)
+    for T in all_types, F in (one, rand)
+        R = F(T)
         λs = eigvals(R)
         vs = eigvecs(R)
+        E = eigen(R)
+        v1 = vs[:,1]
+        v2 = vs[:,2]
+        v3 = vs[:,3]
         @test R * vs ≈ transpose(λs) .* vs
+        @test norm(v1) ≈ 1
+        @test norm(v2) ≈ 1
+        @test norm(v3) ≈ 1
+        @test E.values == λs
+        @test E.vectors == vs
     end
 end

From b12554cb1046c2c9f599a48b182c8ff382902684 Mon Sep 17 00:00:00 2001
From: hyrodium <hyrodium@gmail.com>
Date: Mon, 25 Oct 2021 23:51:06 +0900
Subject: [PATCH 04/11] update 3d eigen

---
 src/eigen.jl | 7 ++++---
 1 file changed, 4 insertions(+), 3 deletions(-)

diff --git a/src/eigen.jl b/src/eigen.jl
index 695912c9..64525c91 100644
--- a/src/eigen.jl
+++ b/src/eigen.jl
@@ -1,14 +1,15 @@
+# 3D
 function LinearAlgebra.eigvals(R::Rotation{3})
     θ = rotation_angle(R)
-    return SVector(exp(θ*im), exp(-θ*im), 1)
+    return SVector(exp(-θ*im), exp(θ*im), 1)
 end
 
 function LinearAlgebra.eigvecs(R::Rotation{3})
     n3 = normalize(rotation_axis(R))
     n1 = normalize(perpendicular_vector(n3))
     n2 = normalize(n3×n1)
-    v1 = normalize(n1*im + n2)
-    v2 = normalize(n1 + n2*im)
+    v1 = normalize(n1 + n2*im)
+    v2 = normalize(n1*im + n2)
     v3 = n3
     return hcat(v1,v2,v3)
 end

From 3b42712a1353c47005c1357a092af225cecb033c Mon Sep 17 00:00:00 2001
From: hyrodium <hyrodium@gmail.com>
Date: Mon, 25 Oct 2021 23:51:47 +0900
Subject: [PATCH 05/11] add eigen methods for 2d

---
 src/eigen.jl | 17 +++++++++++++++++
 1 file changed, 17 insertions(+)

diff --git a/src/eigen.jl b/src/eigen.jl
index 64525c91..4e4e27c0 100644
--- a/src/eigen.jl
+++ b/src/eigen.jl
@@ -19,3 +19,20 @@ function LinearAlgebra.eigen(R::Rotation{3})
     vs = eigvecs(R)
     return Eigen(λs, vs)
 end
+
+
+# 2D
+function LinearAlgebra.eigvals(R::Rotation{2})
+    θ = rotation_angle(R)
+    return SVector(exp(-θ*im), exp(θ*im))
+end
+
+function LinearAlgebra.eigvecs(R::Rotation{2})
+    return @SMatrix [1/√2 1/√2;im/√2 -im/√2]
+end
+
+function LinearAlgebra.eigen(R::Rotation{2})
+    λs = eigvals(R)
+    vs = eigvecs(R)
+    return Eigen(λs, vs)
+end

From 8f6bc9388920f335879e76d8a4bf68023c38ff3b Mon Sep 17 00:00:00 2001
From: hyrodium <hyrodium@gmail.com>
Date: Mon, 25 Oct 2021 23:52:16 +0900
Subject: [PATCH 06/11] update tests for eigen

---
 test/eigen.jl | 29 +++++++++++++++++++++++++++--
 1 file changed, 27 insertions(+), 2 deletions(-)

diff --git a/test/eigen.jl b/test/eigen.jl
index 521181b7..d84ca09e 100644
--- a/test/eigen.jl
+++ b/test/eigen.jl
@@ -1,12 +1,13 @@
-@testset "Eigen" begin
+@testset "Eigen_3D" begin
     all_types = (RotMatrix{3}, AngleAxis, RotationVec,
                  UnitQuaternion, RodriguesParam, MRP,
                  RotXYZ, RotYZX, RotZXY, RotXZY, RotYXZ, RotZYX,
                  RotXYX, RotYZY, RotZXZ, RotXZX, RotYXY, RotZYZ,
                  RotX, RotY, RotZ,
                  RotXY, RotYZ, RotZX, RotXZ, RotYX, RotZY)
+    oneaxis_types = (RotX, RotY, RotZ)
 
-    for T in all_types, F in (one, rand)
+    @testset "$(T)" for T in all_types, F in (one, rand)
         R = F(T)
         λs = eigvals(R)
         vs = eigvecs(R)
@@ -20,5 +21,29 @@
         @test norm(v3) ≈ 1
         @test E.values == λs
         @test E.vectors == vs
+        if !(T in oneaxis_types)
+            # If the rotation angle is in [0°, 180°], then the eigvals will be equal.
+            # Note that the randomized RotX (and etc.) have rotation angle in [0°, 360°].
+            @test eigvals(R) ≈ eigvals(collect(R))
+        end
+    end
+end
+
+@testset "Eigen_2D" begin
+    all_types = (RotMatrix{2}, Angle2d)
+
+    @testset "$(T)" for T in all_types, F in (one, rand)
+        R = F(T)
+        λs = eigvals(R)
+        vs = eigvecs(R)
+        E = eigen(R)
+        v1 = vs[:,1]
+        v2 = vs[:,2]
+        @test R * vs ≈ transpose(λs) .* vs
+        @test norm(v1) ≈ 1
+        @test norm(v2) ≈ 1
+        @test E.values == λs
+        @test E.vectors == vs
+        @test eigvals(R) ≈ eigvals(collect(R))
     end
 end

From 2ff7b115a262a70e1e05f2c72e540abd00f3d717 Mon Sep 17 00:00:00 2001
From: hyrodium <hyrodium@gmail.com>
Date: Tue, 26 Oct 2021 00:05:41 +0900
Subject: [PATCH 07/11] update tests for Julia 1.0

---
 test/eigen.jl | 4 +++-
 1 file changed, 3 insertions(+), 1 deletion(-)

diff --git a/test/eigen.jl b/test/eigen.jl
index d84ca09e..ebbd3e26 100644
--- a/test/eigen.jl
+++ b/test/eigen.jl
@@ -21,9 +21,11 @@
         @test norm(v3) ≈ 1
         @test E.values == λs
         @test E.vectors == vs
-        if !(T in oneaxis_types)
+        if !(T in oneaxis_types) && VERSION ≥ v"1.2"
             # If the rotation angle is in [0°, 180°], then the eigvals will be equal.
             # Note that the randomized RotX (and etc.) have rotation angle in [0°, 360°].
+            # This needs Julia(≥1.2) to get sorted eigenvalues in a canonical order
+            # See https://github.com/JuliaLang/julia/pull/21598
             @test eigvals(R) ≈ eigvals(collect(R))
         end
     end

From e0f04c4faaad05415a80e8b8ae9e378b56fc12c5 Mon Sep 17 00:00:00 2001
From: hyrodium <hyrodium@gmail.com>
Date: Tue, 26 Oct 2021 00:14:45 +0900
Subject: [PATCH 08/11] update tests for Julia 1.0

---
 test/eigen.jl | 6 +++++-
 1 file changed, 5 insertions(+), 1 deletion(-)

diff --git a/test/eigen.jl b/test/eigen.jl
index ebbd3e26..69121286 100644
--- a/test/eigen.jl
+++ b/test/eigen.jl
@@ -46,6 +46,10 @@ end
         @test norm(v2) ≈ 1
         @test E.values == λs
         @test E.vectors == vs
-        @test eigvals(R) ≈ eigvals(collect(R))
+        if !(T in oneaxis_types) && VERSION ≥ v"1.2"
+            # This needs Julia(≥1.2) to get sorted eigenvalues in a canonical order
+            # See https://github.com/JuliaLang/julia/pull/21598
+            @test eigvals(R) ≈ eigvals(collect(R))
+        end
     end
 end

From ca3d8708bfd1e206abbfa422a7f3ed9aabf35be2 Mon Sep 17 00:00:00 2001
From: hyrodium <hyrodium@gmail.com>
Date: Tue, 26 Oct 2021 00:21:27 +0900
Subject: [PATCH 09/11] fix bug in eigen test

---
 test/eigen.jl | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/test/eigen.jl b/test/eigen.jl
index 69121286..d9c3ffcb 100644
--- a/test/eigen.jl
+++ b/test/eigen.jl
@@ -46,7 +46,7 @@ end
         @test norm(v2) ≈ 1
         @test E.values == λs
         @test E.vectors == vs
-        if !(T in oneaxis_types) && VERSION ≥ v"1.2"
+        if VERSION ≥ v"1.2"
             # This needs Julia(≥1.2) to get sorted eigenvalues in a canonical order
             # See https://github.com/JuliaLang/julia/pull/21598
             @test eigvals(R) ≈ eigvals(collect(R))

From 4014fc8108ef0627405dc3bcdc5f5622d705adc1 Mon Sep 17 00:00:00 2001
From: hyrodium <hyrodium@gmail.com>
Date: Tue, 26 Oct 2021 00:38:57 +0900
Subject: [PATCH 10/11] fix bug in eigen test

---
 test/eigen.jl | 6 ++++--
 1 file changed, 4 insertions(+), 2 deletions(-)

diff --git a/test/eigen.jl b/test/eigen.jl
index d9c3ffcb..2caa7fe2 100644
--- a/test/eigen.jl
+++ b/test/eigen.jl
@@ -34,8 +34,8 @@ end
 @testset "Eigen_2D" begin
     all_types = (RotMatrix{2}, Angle2d)
 
-    @testset "$(T)" for T in all_types, F in (one, rand)
-        R = F(T)
+    @testset "$(T)" for T in all_types, θ in 0.0:0.1:π
+        R = T(Angle2d(θ))
         λs = eigvals(R)
         vs = eigvecs(R)
         E = eigen(R)
@@ -47,6 +47,8 @@ end
         @test E.values == λs
         @test E.vectors == vs
         if VERSION ≥ v"1.2"
+            # If the rotation angle θ is in [0°, 180°], then the eigvals will be equal.
+            # Note that the randomized RotX (and etc.) have rotation angle in [0°, 360°].
             # This needs Julia(≥1.2) to get sorted eigenvalues in a canonical order
             # See https://github.com/JuliaLang/julia/pull/21598
             @test eigvals(R) ≈ eigvals(collect(R))

From 143f9aa8786429bfc7fe70daae9cce32b70c7bda Mon Sep 17 00:00:00 2001
From: hyrodium <hyrodium@gmail.com>
Date: Tue, 26 Oct 2021 13:01:12 +0900
Subject: [PATCH 11/11] update performance

---
 src/eigen.jl | 4 ++++
 1 file changed, 4 insertions(+)

diff --git a/src/eigen.jl b/src/eigen.jl
index 4e4e27c0..56622162 100644
--- a/src/eigen.jl
+++ b/src/eigen.jl
@@ -15,6 +15,10 @@ function LinearAlgebra.eigvecs(R::Rotation{3})
 end
 
 function LinearAlgebra.eigen(R::Rotation{3})
+    return eigen(AngleAxis(R))
+end
+
+function LinearAlgebra.eigen(R::AngleAxis)
     λs = eigvals(R)
     vs = eigvecs(R)
     return Eigen(λs, vs)