diff --git a/src/pyrecest/numerics.py b/src/pyrecest/numerics.py
index 2b28e179d..1bd455372 100644
--- a/src/pyrecest/numerics.py
+++ b/src/pyrecest/numerics.py
@@ -256,9 +256,14 @@ def nearest_symmetric_psd(matrix, *, min_eigenvalue: float = 0.0):
     _raise_if_not_square_matrix(arr)
     _raise_if_nonfinite_matrix(arr, "matrix")
     sym = _stable_symmetric_average(arr)
-    eigvals, eigvecs = np.linalg.eigh(sym)
-    clipped = np.maximum(eigvals, min_eigenvalue)
-    repaired = (eigvecs * clipped) @ eigvecs.T
+    if sym.size == 0:
+        return _from_numpy_array(sym)
+
+    scale = max(1.0, float(np.max(np.abs(sym))), min_eigenvalue)
+    scaled_sym = sym / scale
+    eigvals, eigvecs = np.linalg.eigh(scaled_sym)
+    clipped = np.maximum(eigvals, min_eigenvalue / scale)
+    repaired = ((eigvecs * clipped) @ eigvecs.T) * scale
     return _from_numpy_array(_stable_symmetric_average(repaired))
 
 
diff --git a/tests/test_numerics_extreme_psd_projection.py b/tests/test_numerics_extreme_psd_projection.py
new file mode 100644
index 000000000..96317d1ad
--- /dev/null
+++ b/tests/test_numerics_extreme_psd_projection.py
@@ -0,0 +1,19 @@
+import numpy as np
+
+from pyrecest.numerics import nearest_symmetric_psd
+
+
+def test_nearest_symmetric_psd_scales_dense_extreme_covariance_before_eigh():
+    maximum = np.finfo(float).max
+    matrix = np.full((2, 2), maximum)
+
+    with np.errstate(over="raise", invalid="raise"):
+        repaired = np.asarray(nearest_symmetric_psd(matrix))
+
+    assert np.all(np.isfinite(repaired))
+    np.testing.assert_allclose(
+        repaired,
+        matrix,
+        rtol=8.0 * np.finfo(float).eps,
+        atol=0.0,
+    )