diff --git a/CHANGELOG.md b/CHANGELOG.md
index a110b54..1e34e3e 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -5,6 +5,17 @@ All notable changes to **Tensor** are documented in this file.
 The format is based on [Keep a Changelog](https://keepachangelog.com/en/1.1.0/),
 and this project adheres to [Semantic Versioning](https://semver.org/spec/v2.0.0.html).
 
+## [development]
+
+### Fixed
+
+**Linear algebra.**
+
+- `Cholesky`, `Eigen`, `EigenComplex` and the Cholesky rank-one
+  update refuse a matrix or vector holding a NaN or infinity instead
+  of returning an all-NaN factor or spectrum with a nil error, the
+  refusal the sparse solvers already make.
+
 ## [1.0.0] - 2026-09-03
 
 The initial release of Tensor, a scientific computing library in pure
diff --git a/docs/API.md b/docs/API.md
index 2259fb0..40ec571 100644
--- a/docs/API.md
+++ b/docs/API.md
@@ -629,7 +629,7 @@ neighbours are the way in.
 | `Inv(a)` | returns the inverse of a square matrix, through the same LU kernel. |
 | `Det(a)` | returns the determinant of a square real matrix; a singular matrix gives 0, not an error. |
 | `DetComplex(a)` | the same for a square complex matrix, as a `complex128`. |
-| `Cholesky(a)` | returns the lower factor L with a = L·Lᵀ for a symmetric positive definite a; a non-positive pivot is refused. |
+| `Cholesky(a)` | returns the lower factor L with a = L·Lᵀ for a symmetric positive definite a; a non-positive pivot and a non-finite entry are refused. |
 | `CholeskyUpdate(l, x)` | returns the lower factor of A + x·xᵀ from the factor of A, by orthogonal rotations. |
 | `CholeskyDowndate(l, x)` | returns the lower factor of A − x·xᵀ, by hyperbolic rotations; leaving the positive definite cone is an error. |
 | `QR(a)` | returns q (m×m orthogonal) and r (m×n upper triangular) with a = q·r, for m ≥ n. |
@@ -641,8 +641,8 @@ neighbours are the way in.
 
 | Call | What it does |
 |---|---|
-| `Eigen(a)` | returns the eigenvalues of a real symmetric matrix, ascending, and the orthonormal eigenvectors as columns; asymmetry beyond a scale-relative 1e-12 is refused. |
-| `EigenComplex(a)` | the same for a complex Hermitian matrix; values are real, vectors complex, and the same 1e-12 Hermitian tolerance applies. |
+| `Eigen(a)` | returns the eigenvalues of a real symmetric matrix, ascending, and the orthonormal eigenvectors as columns; asymmetry beyond a scale-relative 1e-12 and a non-finite entry are refused. |
+| `EigenComplex(a)` | the same for a complex Hermitian matrix; values are real, vectors complex, the same 1e-12 Hermitian tolerance applies, and a non-finite entry is refused. |
 | `EigenGeneral(a)` | the eigenvalues and eigenvectors of a square matrix of any dtype; real and int inputs promote to `complex128`, values descend by magnitude, and a real matrix may carry conjugate pairs. |
 | `EigenGeneralised(a, b)` | solves A·v = λ·B·v for symmetric a and symmetric positive definite b, through the Cholesky factor of b; values ascend, eigenvectors are B-orthonormal columns. |
 | `SVD(a)` | the thin decomposition a = U·Σ·Vᵀ for m ≥ n: U is m×n with orthonormal columns, Σ a length-n vector descending, Vᵀ n×n orthogonal; a wide matrix is transposed first and the factors are swapped back. |
@@ -898,7 +898,7 @@ The conditions this package reports:
 - a modification that needs fill the factor does not hold:
   `SparseCholesky.Update` and `Downdate` refuse it and leave the
   factor as it was, and `CholeskyUpdate` refuses a rank-1 factor, a
-  mismatched vector or a complex input.
+  mismatched vector, a complex input or a non-finite entry.
 - a complex or wrong-length right-hand side to a sparse solve, and a
   preconditioner built for another dimension.
 - non-finite input to `NewSparseLU`, `NewSparseILU` and the sparse
diff --git a/linalg/cholupdate.go b/linalg/cholupdate.go
index 9808250..dbac60b 100644
--- a/linalg/cholupdate.go
+++ b/linalg/cholupdate.go
@@ -71,14 +71,25 @@ func cholRankOne(l, x *core.Array, name string, sign int) (*core.Array, error) {
 	}
 	// The rotations below assume a lower triangular factor; a nonzero
 	// strict upper triangle would silently corrupt the sweep, so it is
-	// refused up front.
+	// refused up front, and so is a non-finite entry anywhere: the
+	// diagonal test cannot see a NaN, and the hyperbolic square of an
+	// Inf overflows mid-sweep. The sparse twin refuses the same input.
 	for i := range n {
-		for j := i + 1; j < n; j++ {
-			if out.RawFloats()[i*n+j] != 0 {
+		for j := range n {
+			w := out.RawFloats()[i*n+j]
+			if math.IsNaN(w) || math.IsInf(w, 0) {
+				return nil, base.Errf("%s: factor entry [%d,%d] is not finite", name, i, j)
+			}
+			if j > i && w != 0 {
 				return nil, base.Errf("%s: the factor must be lower triangular (nonzero entry at row %d, column %d)", name, i, j)
 			}
 		}
 	}
+	for i := range n {
+		if math.IsNaN(v[i]) || math.IsInf(v[i], 0) {
+			return nil, base.Errf("%s: entry %d is not finite", name, i)
+		}
+	}
 	// The modified matrix reads [L, x]·J·[L, x]ᵀ with J = diag(I, ±1):
 	// one sweep of (hyperbolic for −1, orthogonal for +1) rotations
 	// eliminates the vector column, and the surviving columns are the
diff --git a/linalg/cholupdate_test.go b/linalg/cholupdate_test.go
index 762e6a4..0fd8aa2 100644
--- a/linalg/cholupdate_test.go
+++ b/linalg/cholupdate_test.go
@@ -5,8 +5,10 @@ package linalg
 
 import (
 	"math"
-	"sourcedock.dev/petrbalvin/tensor/internal/core"
+	"strings"
 	"testing"
+
+	"sourcedock.dev/petrbalvin/tensor/internal/core"
 )
 
 // spdSample builds a deterministic symmetric positive definite matrix:
@@ -156,3 +158,25 @@ func TestCholeskyRankOneErrors(t *testing.T) {
 		t.Fatal("expected an error for a complex vector")
 	}
 }
+
+// TestCholeskyRankOneRefusesNonFinite pins the refusal of a poisoned
+// factor or vector: the diagonal test cannot see a NaN, so the sweep
+// would answer an all-NaN factor with a nil error, where the sparse
+// twin refuses the same modification.
+func TestCholeskyRankOneRefusesNonFinite(t *testing.T) {
+	l, err := Cholesky(mustFromFloats(t, []float64{4, 0, 0, 1}, 2, 2))
+	if err != nil {
+		t.Fatalf("Cholesky: %v", err)
+	}
+	badVec := floatsToArray([]float64{math.NaN(), 0.1}, []int{2})
+	if _, err := CholeskyUpdate(l, badVec); err == nil || !strings.Contains(err.Error(), "not finite") {
+		t.Fatalf("CholeskyUpdate(NaN vector): %v", err)
+	}
+	if _, err := CholeskyDowndate(l, badVec); err == nil || !strings.Contains(err.Error(), "not finite") {
+		t.Fatalf("CholeskyDowndate(NaN vector): %v", err)
+	}
+	badFac := mustFromFloats(t, []float64{4, 0, math.Inf(1), 1}, 2, 2)
+	if _, err := CholeskyUpdate(badFac, floatsToArray([]float64{0.1, 0.1}, []int{2})); err == nil || !strings.Contains(err.Error(), "not finite") {
+		t.Fatalf("CholeskyUpdate(Inf factor): %v", err)
+	}
+}
diff --git a/linalg/decomp.go b/linalg/decomp.go
index d84c61c..3fdf99e 100644
--- a/linalg/decomp.go
+++ b/linalg/decomp.go
@@ -48,7 +48,19 @@ func Cholesky(a *core.Array) (*core.Array, error) {
 		return nil, base.Errf("Cholesky: needs a square 2-D matrix, got shape %s", base.ShapeText(a.Shape()))
 	}
 	n := a.Shape()[0]
-	lMat, err := denseCholFactor(denseFloats(a, n, n), n)
+	aMat := denseFloats(a, n, n)
+	// A non-finite entry is neither positive nor definite, but the pivot
+	// test below cannot see it: NaN fails every comparison, so the sweep
+	// would return an all-NaN factor with a nil error. The sparse twin
+	// refuses the same input, and so does this one.
+	for i := range n {
+		for j := range n {
+			if math.IsNaN(aMat[i*n+j]) || math.IsInf(aMat[i*n+j], 0) {
+				return nil, base.Errf("Cholesky: entry [%d,%d] is not finite", i, j)
+			}
+		}
+	}
+	lMat, err := denseCholFactor(aMat, n)
 	if err != nil {
 		return nil, err
 	}
diff --git a/linalg/decomp2.go b/linalg/decomp2.go
index 125d8a2..58999b6 100644
--- a/linalg/decomp2.go
+++ b/linalg/decomp2.go
@@ -706,6 +706,18 @@ func Eigen(a *core.Array) (values, vectors *core.Array, err error) {
 	}
 	n := a.Shape()[0]
 	mat := denseFloats(a, n, n)
+	// A non-finite entry slips through the symmetry test below: the
+	// difference of two NaNs never exceeds the tolerance, so a poisoned
+	// matrix reads as symmetric and the sweep hands back a NaN spectrum
+	// with a nil error. The sparse twin refuses the same input, and so
+	// does this one.
+	for i := range n {
+		for j := range n {
+			if math.IsNaN(mat[i*n+j]) || math.IsInf(mat[i*n+j], 0) {
+				return nil, nil, base.Errf("Eigen: entry [%d,%d] is not finite", i, j)
+			}
+		}
+	}
 	// A matrix outside the safe window would drive the reflector norms
 	// and the sweep's squared accumulation past the representable range;
 	// the tridiagonalisation and the QR sweep run on it scaled into the
diff --git a/linalg/decomp3.go b/linalg/decomp3.go
index 1c8197e..3714860 100644
--- a/linalg/decomp3.go
+++ b/linalg/decomp3.go
@@ -55,6 +55,18 @@ func EigenComplex(a *core.Array) (values, vectors *core.Array, err error) {
 	if !hermitianOK(h, n) {
 		return nil, nil, base.Errf("EigenComplex: matrix is not Hermitian within 1e-12 tolerance")
 	}
+	// A non-finite entry slips through the Hermitian test above: the
+	// difference of a NaN and its conjugate never exceeds the tolerance.
+	// The sparse twin refuses it in its own guard, and so does this one,
+	// before the Jacobi sweep burns its passes on a poisoned matrix.
+	for i := range n {
+		for j := range n {
+			z := h[i*n+j]
+			if math.IsNaN(real(z)) || math.IsNaN(imag(z)) || math.IsInf(real(z), 0) || math.IsInf(imag(z), 0) {
+				return nil, nil, base.Errf("EigenComplex: entry [%d,%d] is not finite", i, j)
+			}
+		}
+	}
 	// The Jacobi sweep's convergence test is a sum of squared
 	// magnitudes: outside the safe window it overflows to +Inf, which
 	// makes every sweep look already converged, or underflows to 0,
diff --git a/linalg/decomp3_test.go b/linalg/decomp3_test.go
index 76b3c08..e9e4ed6 100644
--- a/linalg/decomp3_test.go
+++ b/linalg/decomp3_test.go
@@ -6,9 +6,11 @@ package linalg
 import (
 	"math"
 	"math/cmplx"
+	"strings"
+	"testing"
+
 	"sourcedock.dev/petrbalvin/tensor/internal/base"
 	"sourcedock.dev/petrbalvin/tensor/internal/core"
-	"testing"
 )
 
 func mustComplexes(t *testing.T, vals []complex128, shape ...int) *core.Array {
@@ -291,3 +293,20 @@ func subComplex(a, b []complex128) []complex128 {
 	}
 	return out
 }
+
+// TestEigenComplexRefusesNonFinite pins that a poisoned matrix never
+// reads as Hermitian: the mirror comparison cannot see a NaN
+// difference, so the entry is refused outright, the way the sparse
+// sibling's Hermitian check refuses it.
+func TestEigenComplexRefusesNonFinite(t *testing.T) {
+	cv := []complex128{complex(math.NaN(), 0), 0, 0, 1}
+	a := mustComplexes(t, cv, 2, 2)
+	if _, _, err := EigenComplex(a); err == nil || !strings.Contains(err.Error(), "not finite") {
+		t.Fatalf("EigenComplex(NaN): %v", err)
+	}
+	cv2 := []complex128{complex(0, math.Inf(1)), 0, 0, 1}
+	b := mustComplexes(t, cv2, 2, 2)
+	if _, _, err := EigenComplex(b); err == nil || !strings.Contains(err.Error(), "not finite") {
+		t.Fatalf("EigenComplex(Inf): %v", err)
+	}
+}
diff --git a/linalg/decomp_test.go b/linalg/decomp_test.go
index 0effd7b..3d99af2 100644
--- a/linalg/decomp_test.go
+++ b/linalg/decomp_test.go
@@ -5,8 +5,10 @@ package linalg
 
 import (
 	"math"
-	"sourcedock.dev/petrbalvin/tensor/internal/core"
+	"strings"
 	"testing"
+
+	"sourcedock.dev/petrbalvin/tensor/internal/core"
 )
 
 // TestSVDReconstruction checks that A = U · Σ · Vᵀ reconstructs A
@@ -477,3 +479,18 @@ func TestHouseholderVectorIntoLargeScale(t *testing.T) {
 		t.Fatalf("beta = %v for a zero vector, want 0", empty.beta)
 	}
 }
+
+// TestEigenRefusesNonFinite pins that a poisoned matrix never reads as
+// symmetric: the guard's comparison cannot see a NaN difference, so the
+// entry is refused outright, the way the sparse sibling's symmetry
+// check refuses it.
+func TestEigenRefusesNonFinite(t *testing.T) {
+	nan := mustFromFloats(t, []float64{math.NaN(), 0, 0, 1}, 2, 2)
+	if _, _, err := Eigen(nan); err == nil || !strings.Contains(err.Error(), "not finite") {
+		t.Fatalf("Eigen(NaN): %v", err)
+	}
+	inf := mustFromFloats(t, []float64{math.Inf(1), 0, 0, 1}, 2, 2)
+	if _, _, err := Eigen(inf); err == nil || !strings.Contains(err.Error(), "not finite") {
+		t.Fatalf("Eigen(Inf): %v", err)
+	}
+}
diff --git a/linalg/decompositions_test.go b/linalg/decompositions_test.go
index ff35cab..cd85e80 100644
--- a/linalg/decompositions_test.go
+++ b/linalg/decompositions_test.go
@@ -691,3 +691,18 @@ func TestCholeskyBlockedReconstruction(t *testing.T) {
 		}
 	}
 }
+
+// TestCholeskyRefusesNonFinite pins the refusal of a poisoned matrix:
+// the pivot test cannot see a NaN (it fails every comparison), so
+// without the gate the sweep would answer an all-NaN factor with a nil
+// error, where the sparse sibling refuses the same input.
+func TestCholeskyRefusesNonFinite(t *testing.T) {
+	nan := mustFromFloats(t, []float64{math.NaN(), 0, 0, 1}, 2, 2)
+	if _, err := Cholesky(nan); err == nil || !strings.Contains(err.Error(), "not finite") {
+		t.Fatalf("Cholesky(NaN): %v", err)
+	}
+	inf := mustFromFloats(t, []float64{0, 0, 0, math.Inf(1)}, 2, 2)
+	if _, err := Cholesky(inf); err == nil || !strings.Contains(err.Error(), "not finite") {
+		t.Fatalf("Cholesky(Inf): %v", err)
+	}
+}