Give a square matrix its eigenvalues

Sources/AngouriMath/Core/Entity/Omni/Entity.Matrix.cs

@@ -376,6 +376,35 @@ public Entity AsScalar()
                 return polynomial ?? 0;
             }
 
+            /// <summary>
+            /// The eigenvalues of this square matrix: the roots of its
+            /// <see cref="CharacteristicPolynomial"/>, as a set, a repeated root once.
+            /// <see langword="null"/> where the matrix is not square.
+            /// </summary>
+            /// <remarks>
+            /// The characteristic polynomial divides by nothing, so its roots carry no condition an
+            /// elimination's pivots would bring: <c>[[0, J], [J, 0]]</c> has <c>sqrt(J^2)</c> and
+            /// <c>-sqrt(J^2)</c>, which hold at <c>J = 0</c> too. Up to the fourth degree the roots are
+            /// in radicals, and past it where the polynomial factors; otherwise the set is the one the
+            /// equation solver leaves.
+            /// <a href="https://github.com/asc-community/AngouriMath/issues/1676">#1676</a>
+            /// </remarks>
+            /// <example>
+            /// <code>
+            /// Console.WriteLine(MathS.Matrix(new Entity[,] { { 2, 0, 0 }, { 0, 3, 4 }, { 0, 4, 9 } }).Eigenvalues);
+            /// </code>
+            /// Prints
+            /// <code>
+            /// { 1, 2, 11 }
+            /// </code>
+            /// </example>
+            public Set? Eigenvalues => eigenvalues.GetValue(static @this =>
+            {
+                var lambda = Variable.CreateUnique(@this, "lambda");
+                return @this.CharacteristicPolynomial(lambda) is { } polynomial ? polynomial.SolveEquation(lambda) : null;
+            }, this);
+            private LazyPropertyA<Set?> eigenvalues;
+
             /// <summary>Returns an inverse matrix if it exists</summary>
             public Matrix? Inverse => inverse.GetValue(static @this =>
             {

Sources/MCP/AngouriMath.Mcp/Matrices.cs

@@ -136,8 +136,8 @@ public static (Matrix Value, List<string> Guards) Clean(Matrix m)
     }
 
     /// <summary>
-    /// Eigenvalues via the characteristic polynomial, which belongs in the library:
-    /// https://github.com/asc-community/AngouriMath/issues/1676
+    /// The characteristic polynomial, det(A - lambda I), and the library's eigenvalues,
+    /// `Matrix.Eigenvalues`. https://github.com/asc-community/AngouriMath/issues/1676
     ///
     /// The guard-dropping is not cosmetic. A determinant taken by dividing by pivots leaves a
     /// `provided` guard for each one, and on a symbolic matrix those are artefacts of the
@@ -163,12 +163,15 @@ public static Eigen Compute(Matrix a, Variable lambda)
         Entity? roots;
         try
         {
-            // The solver reintroduces pivot guards of its own, so the result is cleaned
-            // again — stripping only the characteristic polynomial leaves eigenvalues
-            // reading 'J provided not J = 0'.
-            var (cleaned, solverGuards) = Clean(poly.SolveEquation(lambda).Simplify());
-            roots = cleaned;
-            guards.AddRange(solverGuards);
+            // The library's, from a characteristic polynomial that divides by nothing; cleaned
+            // all the same, so that a guard any later route brings is reported rather than shown.
+            if (a.Eigenvalues is { } eigenvalues)
+            {
+                var (cleaned, solverGuards) = Clean(eigenvalues.Simplify());
+                roots = cleaned;
+                guards.AddRange(solverGuards);
+            }
+            else roots = null;
         }
         catch { roots = null; }
 

Sources/MCP/README.md

@@ -151,8 +151,5 @@ existing tool, then a prompt, and add a tool last. A tool's description is a rou
 documentation: say when to call it, and where a model would wrongly trust itself, say so.
 
 This is an adapter, not a second computer algebra system. Mathematics belongs in the library, where
-the terminal and the notebooks get it too, and three workarounds here wait for it:
-eigenvalues by the characteristic polynomial ([#1676](https://github.com/asc-community/AngouriMath/issues/1676)),
-a numeric comparison with a tolerance ([#1674](https://github.com/asc-community/AngouriMath/issues/1674)),
-and the numeric definite integral run twice to find its good digits
-([#1675](https://github.com/asc-community/AngouriMath/issues/1675)).
+the terminal and the notebooks get it too. Where the server has to work around the library, the
+workaround links the library's issue in a comment beside it, and the fix for that issue removes it.

Sources/Tests/UnitTests/Algebra/EigenvaluesTest.cs

@@ -0,0 +1,61 @@
+//
+// Copyright (c) 2019-2026 Angouri.
+// AngouriMath is licensed under MIT.
+// Details: https://github.com/asc-community/AngouriMath/blob/master/LICENSE.md.
+// Website: https://am.angouri.org.
+//
+
+using System.Linq;
+using AngouriMath.Extensions;
+using Xunit;
+using static AngouriMath.Entity;
+using static AngouriMath.Entity.Set;
+
+namespace AngouriMath.Tests.Algebra
+{
+    /// <summary>
+    /// <see cref="Matrix.Eigenvalues"/>: the roots of the characteristic polynomial, which divides by nothing,
+    /// so no pivot leaves a condition on them. Each one is checked by making <c>A - lambda I</c> singular.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/1676">#1676</a>
+    /// </summary>
+    [Trait("Area", "Algebra")]
+    public sealed class EigenvaluesTest
+    {
+        [Fact]
+        public void OfASymmetricMatrixAreItsRealRoots()
+            => Assert.Equal(new FiniteSet(1, 2, 11), MathS.Matrix(new Entity[,] { { 2, 0, 0 }, { 0, 3, 4 }, { 0, 4, 9 } }).Eigenvalues);
+
+        [Fact]
+        public void ARepeatedRootIsOneMember()
+            => Assert.Equal(new FiniteSet(1), MathS.Matrix(new Entity[,] { { 1, 1 }, { 0, 1 } }).Eigenvalues);
+
+        [Fact]
+        public void OfASymbolicMatrixCarryNoPivotCondition()
+        {
+            var eigenvalues = MathS.Matrix(new Entity[,] { { 0, "J" }, { "J", 0 } }).Eigenvalues;
+            Assert.Equal(new FiniteSet("sqrt(J^2)", "-sqrt(J^2)"), eigenvalues);
+            Assert.DoesNotContain("provided", eigenvalues!.Stringize());
+        }
+
+        [Fact]
+        public void OfANonSquareMatrixAreNull()
+            => Assert.Null(MathS.Matrix(new Entity[,] { { 1, 2, 3 }, { 4, 5, 6 } }).Eigenvalues);
+
+        [Theory]
+        [InlineData(1, 2, 3, 4)]
+        [InlineData(2, -1, 5, 7)]
+        [InlineData(0, 1, -1, 0)]
+        public void EachMakesTheShiftedMatrixSingular(int a, int b, int c, int d)
+        {
+            var matrix = MathS.Matrix(new Entity[,] { { a, b }, { c, d } });
+            var eigenvalues = Assert.IsType<FiniteSet>(matrix.Eigenvalues);
+            Assert.NotEmpty(eigenvalues);
+            foreach (var lambda in eigenvalues)
+            {
+                var shifted = MathS.Matrix(new Entity[,] { { a - lambda, b }, { c, d - lambda } });
+                Assert.True(shifted.Determinant!.EvalNumerical().Abs().EDecimal.ToDouble() < 1e-12,
+                    $"det(A - {lambda} I) is {shifted.Determinant.EvalNumerical()}");
+            }
+        }
+    }
+}

Sources/Tests/UnitTests/Common/PublicApi.txt

@@ -1415,6 +1415,7 @@ AngouriMath.Entity+Matrix.CharacteristicPolynomial(AngouriMath.Entity+Variable)
 AngouriMath.Entity+Matrix.Codomain { } : AngouriMath.Core.Domain
 AngouriMath.Entity+Matrix.ColumnCount { } : System.Int32
 AngouriMath.Entity+Matrix.Determinant { } : AngouriMath.Entity
+AngouriMath.Entity+Matrix.Eigenvalues { } : AngouriMath.Entity+Set
 AngouriMath.Entity+Matrix.EqualityContract { } : System.Type
 AngouriMath.Entity+Matrix.Equals(AngouriMath.Entity) : System.Boolean
 AngouriMath.Entity+Matrix.Equals(AngouriMath.Entity+Matrix) : System.Boolean