diff --git a/tessellate_ipu/linalg/__init__.py b/tessellate_ipu/linalg/__init__.py
index 659a04f..cd0b099 100644
--- a/tessellate_ipu/linalg/__init__.py
+++ b/tessellate_ipu/linalg/__init__.py
@@ -4,5 +4,5 @@
 from .tile_linalg_hessenberg import ipu_hessenberg
 from .tile_linalg_jacobi import ipu_eigh
 from .tile_linalg_qr import ipu_qr
-from .tile_linalg_tridiagonal_eigh import ipu_eigh_hess, ipu_tridiagonal_eigh
+from .tile_linalg_tridiagonal_eigh import ipu_hess_eigh, ipu_tridiagonal_eigh
 from .tile_linalg_tridiagonal_solver import ipu_tridiag_solve
diff --git a/tessellate_ipu/linalg/tile_linalg_tridiagonal_eigenvalue.py b/tessellate_ipu/linalg/tile_linalg_tridiagonal_eigenvalue.py
index 8f2a249..ea83bcb 100755
--- a/tessellate_ipu/linalg/tile_linalg_tridiagonal_eigenvalue.py
+++ b/tessellate_ipu/linalg/tile_linalg_tridiagonal_eigenvalue.py
@@ -9,6 +9,16 @@
 
 jax.config.FLAGS.jax_platform_name = "cpu"
 
+vertex_filename = osp.join(osp.dirname(__file__), "../core", "vertex", "tile_tridiagonal_eigh.cpp")
+grad = create_ipu_tile_primitive(
+    "Sturm",
+    "Sturm",
+    inputs=["alpha", "beta_sq", "pivmin", "alpha0_pertubation", "x", "id", "out_shape", "lower", "mid", "upper"],
+    outputs={"lower_out": 7, "mid_out": 8, "upper_out": 9},
+    gp_filename=vertex_filename,
+    perf_estimate=100,
+)
+
 
 def ipu_tridiagonal_eigenvalue(d, e, *, select="a", select_range=None, tol=None):
     alpha, beta = jnp.asarray(d), jnp.asarray(e)
@@ -64,16 +74,6 @@ def ipu_tridiagonal_eigenvalue(d, e, *, select="a", select_range=None, tol=None)
     pivmin = jnp.broadcast_to(pivmin, target_shape)
     alpha0_perturbation = jnp.broadcast_to(alpha0_perturbation, target_shape)
 
-    vertex_filename = osp.join(osp.dirname(__file__), "../core", "vertex", "tile_tridiagonal_eigh.cpp")
-    grad = create_ipu_tile_primitive(
-        "Sturm",
-        "Sturm",
-        inputs=["alpha", "beta_sq", "pivmin", "alpha0_pertubation", "x", "id", "out_shape", "lower", "mid", "upper"],
-        outputs={"lower_out": 7, "mid_out": 8, "upper_out": 9},
-        gp_filename=vertex_filename,
-        perf_estimate=100,
-    )
-
     x = mid
     n = x.shape[0]
     tiles = tuple(range(n))
@@ -106,7 +106,6 @@ def body(j, args):
     import jax
     import scipy
 
-    np.random.seed(42)
     jax.config.FLAGS.jax_platform_name = "cpu"
     np.random.seed(42)
 
diff --git a/tessellate_ipu/linalg/tile_linalg_tridiagonal_eigh.py b/tessellate_ipu/linalg/tile_linalg_tridiagonal_eigh.py
index b3162f8..1db999e 100644
--- a/tessellate_ipu/linalg/tile_linalg_tridiagonal_eigh.py
+++ b/tessellate_ipu/linalg/tile_linalg_tridiagonal_eigh.py
@@ -11,7 +11,7 @@
 jax.config.update("jax_enable_x64", False)
 
 
-def ipu_tridiagonal_eigh(d, e, n_iter=2, seed=42):
+def ipu_tridiagonal_eigh(d, e, num_iters=2, seed=42):
 
     N = d.shape[0]
     eig = ipu_tridiagonal_eigenvalue(d, e)[:N]
@@ -33,19 +33,19 @@ def inverse_iteration(i, x):
         x /= jnp.linalg.norm(x, axis=1)[:, jnp.newaxis]
         return x
 
-    x = jax.lax.fori_loop(0, n_iter, inverse_iteration, x)
+    x = jax.lax.fori_loop(0, num_iters, inverse_iteration, x)
 
     return x, eig
 
 
-def ipu_eigh_hess(M):
+def ipu_hess_eigh(M, num_iters=2):
 
     Q, M_tri_ = ipu_hessenberg(M)
     M_tri = M_tri_.array
     d, e = jnp.diag(M_tri), jnp.diag(M_tri, k=1)
 
-    x, eig = ipu_tridiagonal_eigh(d, e)
-    return x @ Q.array.T, eig
+    x, eig = ipu_tridiagonal_eigh(d, e, num_iters)
+    return eig, x @ Q.array.T
 
 
 if __name__ == "__main__":
@@ -96,7 +96,7 @@ def ipu_eigh_hess(M):
         print("Specify one of the options -r or -f")
         sys.exit(1)
 
-    x, eig = jax.jit(ipu_eigh_hess, backend="ipu")(mat)
+    eig, x = jax.jit(ipu_hess_eigh, backend="ipu")(mat)
 
     x = np.array(x)
 
diff --git a/tests/linalg/test_tile_linalg_eig_hess.py b/tests/linalg/test_tile_linalg_eig_hess.py
new file mode 100644
index 0000000..4daca94
--- /dev/null
+++ b/tests/linalg/test_tile_linalg_eig_hess.py
@@ -0,0 +1,50 @@
+# Copyright (c) 2022 Graphcore Ltd. All rights reserved.
+import unittest
+
+import chex
+import jax
+import numpy as np
+import numpy.testing as npt
+import pytest
+from absl.testing import parameterized
+
+from tessellate_ipu.linalg.tile_linalg_tridiagonal_eigh import ipu_hess_eigh
+from tessellate_ipu.utils import IpuTargetType
+
+# Skipping some tests if no local IPU hardware.
+ipu_hw_available = len(jax.devices("ipu")) > 0 and jax.devices("ipu")[0].target_type == IpuTargetType.IPU
+ipu_num_tiles = jax.devices("ipu")[0].num_tiles
+
+
+@pytest.mark.ipu_hardware
+class IpuTileLinalgHessEigh(chex.TestCase, parameterized.TestCase):
+    def setUp(self):
+        self.device = jax.devices("ipu")[0]
+        self.num_tiles = self.device.num_tiles
+        np.random.seed(42)
+
+    @unittest.skipUnless(ipu_num_tiles >= 16, "Requires IPU with 16 tiles")
+    @parameterized.parameters(
+        {"N": 4},
+        # {"N": 512},
+    )
+    def test__hess_eigh_raw__proper_eigh_result(self, N):
+        x = np.random.randn(N, N).astype(np.float32)
+        x = (x + x.T) / 2.0
+
+        hess_eigh_fn = jax.jit(ipu_hess_eigh, backend="ipu")
+        # Should be enough iterations...
+        eigvalues, VT = hess_eigh_fn(x, num_iters=2)
+        eigvalues = np.asarray(eigvalues).reshape(-1)
+        VT = np.asarray(VT)
+        # Expected eigen values and vectors (from Lapack?)
+        expected_eigvalues, expected_eigvectors = np.linalg.eigh(x)
+
+        # Order raw outputs.
+        indices = np.argsort(eigvalues)
+        eigvalues_sorted = eigvalues[indices]
+        eigvectors_sorted = VT[indices].T
+        npt.assert_array_almost_equal(eigvalues_sorted, expected_eigvalues, decimal=5)
+        npt.assert_array_almost_equal(np.abs(eigvectors_sorted), np.abs(expected_eigvectors), decimal=5)
+
+    # TODO: Performance test