diff --git a/cuthbertlib/kalman/filtering.py b/cuthbertlib/kalman/filtering.py
index 4631239..8c125d1 100644
--- a/cuthbertlib/kalman/filtering.py
+++ b/cuthbertlib/kalman/filtering.py
@@ -208,6 +208,42 @@ def filtering_operator(
 
     mu = cho_solve((U1, True), b1)
     t1 = b1 @ mu - (eta2 + mu) @ tmp_2
-    ell = ell1 + ell2 - 0.5 * t1 + 0.5 * jnp.linalg.slogdet(D_inv)[1]
+
+    # Derivation for O(nx) log-determinant computation of D_inv:
+    # This is a long comment but I wanted to include the full derivation for clarity and future reference.
+    # The key idea is to express D_inv in terms of the blocks of Xi and then apply Sylvester's determinant theorem
+    # to compute its determinant efficiently.
+    #
+    # 1. Expand the blocks of Xi @ Xi.T:
+    #    (Xi @ Xi.T)[1,1] = I + U1.T @ Z2 @ Z2.T @ U1
+    #    (Xi @ Xi.T)[2,1] = Z2 @ Z2.T @ U1
+    #
+    # 2. Equate to the corresponding blocks of L @ L.T:
+    #    Xi11 @ Xi11.T = I + U1.T @ Z2 @ Z2.T @ U1
+    #    Xi21 @ Xi11.T = Z2 @ Z2.T @ U1
+    #
+    # 3. Expand D_inv using tmp_1 = Xi11^{-1} @ U1.T:
+    #    D_inv = I - tmp_1.T @ Xi21.T
+    #          = I - U1 @ Xi11^{-T} @ Xi21.T
+    #
+    # 4. Apply Sylvester's determinant theorem:
+    #    det(D_inv) = det(I - Xi11^{-T} @ Xi21.T @ U1)
+    #
+    # 5. Multiply interior by Xi11^{-1} @ Xi11 and substitute block identities:
+    #    Let P = U1.T @ Z2 @ Z2.T @ U1
+    #    det(D_inv) = det(I - (Xi11 @ Xi11.T)^{-1} @ (Xi21 @ Xi11.T).T @ U1)
+    #               = det(I - (I + P)^{-1} @ P)
+    #               = det((I + P)^{-1})
+    #               = 1 / det(Xi11 @ Xi11.T)
+    #               = det(Xi11)^{-2}
+    #
+    # 6. Simplify the log-determinant term in the log-likelihood:
+    #    0.5 * log(det(D_inv)) = -log(|det(Xi11)|)
+    #
+    # Since Xi11 is lower triangular, the log-determinant is the sum of the logs of its diagonal.
+    # Replace `0.5 * jnp.linalg.slogdet(D_inv)[1]` with:
+    # -jnp.sum(jnp.log(jnp.abs(jnp.diag(Xi11))))
+    # ell = ell1 + ell2 - 0.5 * t1 + 0.5 * jnp.linalg.slogdet(D_inv)[1]
+    ell = ell1 + ell2 - 0.5 * t1 - jnp.sum(jnp.log(jnp.abs(jnp.diag(Xi11))))
 
     return FilterScanElement(A, b, U, eta, Z, ell)
diff --git a/cuthbertlib/linalg/tria.py b/cuthbertlib/linalg/tria.py
index 97645d4..7fefe6e 100644
--- a/cuthbertlib/linalg/tria.py
+++ b/cuthbertlib/linalg/tria.py
@@ -1,21 +1,97 @@
 """Implements triangularization operator a matrix via QR decomposition."""
 
 import jax
+import jax.numpy as jnp
 
 from cuthbertlib.types import Array
 
 
+def _adj(x: Array) -> Array:
+    """Conjugate transpose for batched matrices."""
+    return jnp.swapaxes(x.conj(), -1, -2)
+
+
+@jax.custom_jvp
 def tria(A: Array) -> Array:
-    r"""A triangularization operator using QR decomposition.
+    """A triangularization operator using QR decomposition.
 
     Args:
         A: The matrix to triangularize.
 
     Returns:
-        A lower triangular matrix $R$ such that $R R^\top = A A^\top$.
+        A lower triangular matrix R such that R @ R.T = A @ A.T.
 
-    Reference:
-        [Arasaratnam and Haykin (2008)](https://ieeexplore.ieee.org/document/4524036): Square-Root Quadrature Kalman Filtering
+    References:
+        Paper: Arasaratnam and Haykin (2008): Square-Root Quadrature Kalman Filtering
+            https://ieeexplore.ieee.org/document/4524036
     """
-    _, R = jax.scipy.linalg.qr(A.T, mode="economic")
-    return R.T
+    _, R_qr = jnp.linalg.qr(_adj(A), mode="reduced")
+    return _adj(R_qr)
+
+
+@tria.defjvp
+def _tria_jvp(primals, tangents):
+    # Derivation of the exact analytical JVP for the lower-triangularization operator:
+    #
+    # 1. The operation computes a lower triangular R such that:
+    #    R @ R.T = A @ A.T
+    #
+    # 2. Taking the differential of both sides yields the Gramian differential identity:
+    #    dR @ R.T + R @ dR.T = dA @ A.T + A @ dA.T
+    #
+    # 3. For rank-deficient A, the exact lower-triangular tangent dR decomposes into
+    #    column-space and null-space components: dR = dR_col + dR_null.
+    #
+    # 4. To find dR_col, multiply the identity from the left by R^\dagger (pseudoinverse)
+    #    and from the right by R^{\dagger T}:
+    #    R^\dagger @ dR_col + dR_col.T @ R^{\dagger T} = R^\dagger @ dA @ A.T @ R^{\dagger T} + R^\dagger @ A @ dA.T @ R^{\dagger T}
+    #
+    # 5. Let Q be the active subspace orthogonal factor such that A = R @ Q.T.
+    #    Substituting A.T @ R^{\dagger T} = Q and R^\dagger @ A = Q.T:
+    #    R^\dagger @ dR_col + dR_col.T @ R^{\dagger T} = R^\dagger @ dA @ Q + Q.T @ dA.T @ R^{\dagger T}
+    #
+    # 6. Define K = R^\dagger @ dA @ Q. The right hand side becomes K + K.T:
+    #    R^\dagger @ dR_col + (R^\dagger @ dR_col).T = K + K.T
+    #
+    # 7. Define dM_col = R^\dagger @ dR_col. Solving for the lower-triangular dM_col:
+    #    dM_col = tril(K + K.T) - diag(K)
+    #
+    # 8. Recover the column-space differential dR_col by left-multiplying by R:
+    #    dR_col = R @ dM_col
+    #
+    # 9. To satisfy the orthogonal cross-terms for arbitrary perturbations outside
+    #    the column space of A, add the null-space component projected via (I - R @ R^\dagger):
+    #    dR_null = (I - R @ R^\dagger) @ dA @ Q
+    #
+    # 10. The complete, exact JVP is the sum of both components:
+    #     dR = dR_col + dR_null
+
+    (A,) = primals
+    (dA,) = tangents
+
+    A_T = jnp.swapaxes(A, -1, -2)
+    Q, R_qr = jnp.linalg.qr(A_T, mode="reduced")
+
+    R = jnp.swapaxes(R_qr, -1, -2)
+
+    # Q has shape (..., M, N). A is (..., N, M).
+    # A^T = Q R_qr => A = R_qr^T Q^T = R Q^T
+
+    R_pinv = jnp.linalg.pinv(R)
+
+    # K = R^{-1} dA Q
+    # R can be degenerate so we use the pseudoinverse to ensure the JVP is well-defined everywhere,
+    K = R_pinv @ dA @ Q
+    K_T = jnp.swapaxes(K, -1, -2)
+
+    # Solve for lower triangular perturbation dM + dM^T = K + K^T
+    I = jnp.eye(K.shape[-1], dtype=K.dtype)
+    dM = jnp.tril(K + K_T) - K * I
+
+    # Compute the null-space part
+    dR_null = (jnp.eye(R.shape[-2], dtype=R.dtype) - R @ R_pinv) @ dA @ Q
+
+    # Apply to get the tangent at R
+    dR = R @ dM + dR_null
+
+    return R, dR
diff --git a/tests/cuthbert/gaussian/test_kalman.py b/tests/cuthbert/gaussian/test_kalman.py
index 93c3235..8f0b347 100644
--- a/tests/cuthbert/gaussian/test_kalman.py
+++ b/tests/cuthbert/gaussian/test_kalman.py
@@ -93,13 +93,16 @@ def get_observation_params(model_inputs: int) -> tuple[Array, Array, Array, Arra
     return filter, smoother, model_inputs
 
 
-seeds = [1, 43, 99, 123, 456]
+seeds = [1, 43, 99]
 x_dims = [3]
 y_dims = [1, 2]
 num_time_steps = [1, 25]
 
 common_params = list(itertools.product(seeds, x_dims, y_dims, num_time_steps))
 
+# Use fewer time steps for gradient test because finite difference will not work well for long sequences
+gradient_params = list(itertools.product(seeds, x_dims, y_dims, [1, 2]))
+
 
 @pytest.mark.parametrize("seed,x_dim,y_dim,num_time_steps", common_params)
 def test_offline_filter(seed, x_dim, y_dim, num_time_steps):
@@ -150,6 +153,32 @@ def test_offline_filter(seed, x_dim, y_dim, num_time_steps):
     )
 
 
+@pytest.mark.parametrize("seed,x_dim,y_dim,num_time_steps", gradient_params)
+def test_check_gradient(seed, x_dim, y_dim, num_time_steps):
+    m0, chol_P0, Fs, cs, chol_Qs, Hs, ds, chol_Rs, ys = generate_lgssm(
+        seed, x_dim, y_dim, num_time_steps
+    )
+
+    @jax.jit
+    def get_loglikelihood_from_params(
+        m0_, chol_P0_, Fs_, cs_, chol_Qs_, Hs_, ds_, chol_Rs_
+    ):
+        kalman_filter, _, model_inputs = load_kalman_inference(
+            m0_, chol_P0_, Fs_, cs_, chol_Qs_, Hs_, ds_, chol_Rs_, ys
+        )
+        states = filter(kalman_filter, model_inputs)
+        return states.log_normalizing_constant[-1]
+
+    chex.assert_numerical_grads(
+        get_loglikelihood_from_params,
+        (m0, chol_P0, Fs, cs, chol_Qs, Hs, ds, chol_Rs),
+        order=1,
+        rtol=0.01,
+        atol=0.01,
+        eps=1e-5,
+    )
+
+
 @pytest.mark.parametrize("seed", [1, 43, 99, 123, 456])
 @pytest.mark.parametrize("x_dim", [1, 10])
 @pytest.mark.parametrize("y_dim", [1, 5])
diff --git a/tests/cuthbertlib/kalman/test_filtering.py b/tests/cuthbertlib/kalman/test_filtering.py
index 18ab99d..f26c2fa 100644
--- a/tests/cuthbertlib/kalman/test_filtering.py
+++ b/tests/cuthbertlib/kalman/test_filtering.py
@@ -56,7 +56,6 @@ def test_predict(seed, x_dim):
     pred_cov = pred_chol_cov @ pred_chol_cov.T
 
     des_m, des_cov = std_predict(m0, P0, F, c, Q)
-
     chex.assert_trees_all_close((pred_m, pred_cov), (des_m, des_cov), rtol=1e-10)
 
 
@@ -78,6 +77,10 @@ def test_update(seed, x_dim, y_dim):
     (m, chol_P), ell = update(m0, chol_P0, H, d, chol_R, y)
     P = chol_P @ chol_P.T
 
+    chex.assert_numerical_grads(
+        lambda *args: update(*args)[-1], (m0, chol_P0, H, d, chol_R, y), order=1
+    )
+
     des_m, des_P, des_ell = std_update(m0, P0, H, d, R, y)
 
     chex.assert_trees_all_close((m, P), (des_m, des_P), rtol=1e-10)
diff --git a/tests/cuthbertlib/linalg/test_tria.py b/tests/cuthbertlib/linalg/test_tria.py
index 16f68b2..7748d4b 100644
--- a/tests/cuthbertlib/linalg/test_tria.py
+++ b/tests/cuthbertlib/linalg/test_tria.py
@@ -26,3 +26,20 @@ def test_tria(seed, shape):
 
     # Check that R @ R.T = A @ A.T
     assert jnp.allclose(R @ R.T, A @ A.T)
+
+
+def test_tria_jvp_preserves_gram_matrix_for_rank_deficient_input():
+    A = jnp.array([[1.0, 0.0], [1.0, 0.0], [1.0, 0.0]])
+    dA = jnp.array([[0.0, 0.0], [1.0, 2.0], [3.0, 4.0]])
+
+    def gram_via_tria(x):
+        R = tria(x)
+        return R @ R.T
+
+    def gram_direct(x):
+        return x @ x.T
+
+    gram_jvp_from_tria = jax.jvp(gram_via_tria, (A,), (dA,))[1]
+    gram_jvp_direct = jax.jvp(gram_direct, (A,), (dA,))[1]
+
+    assert jnp.allclose(gram_jvp_from_tria, gram_jvp_direct)