diff --git a/cola/linalg/diag_trace.py b/cola/linalg/diag_trace.py
index 195cac48..08296bba 100644
--- a/cola/linalg/diag_trace.py
+++ b/cola/linalg/diag_trace.py
@@ -3,7 +3,7 @@
 from cola.utils import export, dispatch
 from cola.ops import LinearOperator, I_like, Diagonal, Identity
 from cola.ops import BlockDiag, ScalarMul, Sum, Dense, Array
-from cola.ops import Kronecker, KronSum
+from cola.ops import Kronecker, KronSum, Product
 from cola.algorithms import exact_diag, approx_diag
 
 
@@ -141,3 +141,8 @@ def trace(A: LinearOperator, **kwargs):
 @dispatch
 def trace(A: Kronecker, **kwargs):
     return product([trace(M, **kwargs) for M in A.Ms])
+
+
+@dispatch(cond=lambda A, **kwargs: A.Ms[0].shape[0] > A.Ms[0].shape[1])
+def trace(A: Product[LinearOperator, LinearOperator]):
+    return trace(Product(*reversed(A.Ms)))
diff --git a/cola/linalg/inv.py b/cola/linalg/inv.py
index 4a72365d..6e4243d0 100644
--- a/cola/linalg/inv.py
+++ b/cola/linalg/inv.py
@@ -2,8 +2,8 @@
 from plum import dispatch
 from cola.ops import LinearOperator
 from cola.ops import Diagonal, Permutation
-from cola.ops import Identity
-from cola.ops import ScalarMul
+from cola.ops import Identity, Dense
+from cola.ops import ScalarMul, Sum
 from cola.ops import BlockDiag, Triangular
 from cola.ops import Kronecker, Product
 from cola.algorithms.cg import cg
@@ -168,3 +168,16 @@ def inv(A: Diagonal, **kwargs):
 @dispatch
 def inv(A: Triangular, **kwargs):
     return TriangularInv(A)
+
+
+@dispatch
+def inv(A: Sum[Product[Dense,Dense], Diagonal], **kwargs):
+    U, V = A.Ms[0].Ms
+    D_inv = inv(A.Ms[1])
+    I = Identity(shape=(V.shape[0], U.shape[1]), dtype=V.dtype) 
+    return D_inv - D_inv @ U @ inv(V @ D_inv @ U + I) @ V @ D_inv
+
+
+@dispatch
+def inv(A: Sum[Diagonal, Product[Dense,Dense]], **kwargs):
+    return inv(Product(*A.Ms[1].Ms) + A.Ms[0])
\ No newline at end of file
diff --git a/tests/linalg/operator_market.py b/tests/linalg/operator_market.py
index 534270f4..d779186a 100644
--- a/tests/linalg/operator_market.py
+++ b/tests/linalg/operator_market.py
@@ -2,7 +2,7 @@
 from cola.ops import LinearOperator, Tridiagonal, Diagonal, Identity
 from cola.ops import KronSum, Product
 from cola.ops import Triangular, Kronecker, Permutation
-from cola.ops import Dense, BlockDiag, Jacobian, Hessian
+from cola.ops import Dense, BlockDiag, Jacobian, Hessian, Sparse
 from cola.annotations import SelfAdjoint
 from cola.annotations import PSD
 from cola.utils_test import get_xnp
@@ -30,6 +30,8 @@
     'square_product',
     # 'square_sparse',
     'square_tridiagonal',
+    'diagonal_plus_uv',
+    'uv_plus_diagonal',
 }
 
 
@@ -73,6 +75,7 @@ def get_test_operator(backend: str, precision: str, op_name: str,
                     op = BlockDiag(M1, M2, multiplicities=[2, 3])
                 case 'prod':
                     op = M1 @ M1.T
+    
         case ('psd', 'kron'):
             M1 = Dense(xnp.array([[6., 2], [2, 4]], dtype=dtype, device=device))
             M2 = Dense(xnp.array([[7, 6], [6, 8]], dtype=dtype, device=device))
@@ -134,6 +137,18 @@ def get_test_operator(backend: str, precision: str, op_name: str,
             shape = (3, 3)
             sparse = Sparse(data, indices, indptr, shape)
 
+        case ('diagonal', 'plus_uv'):
+            D = Diagonal(xnp.array([.1, .5, .22, 8.], dtype=dtype, device=device))
+            U = Dense(xnp.array([[6., 2], [2, 4], [1.2, 2], [5, 10]], dtype=dtype, device=device))
+            V = Dense(xnp.array([[7, 6, 6, 8.7], [3., .2, 13, 4]], dtype=dtype, device=device))
+            op = D + U @ V
+
+        case ('uv', 'plus_diagonal'):
+            D = Diagonal(xnp.array([.1, .5, .22, 8.], dtype=dtype, device=device))
+            U = Dense(xnp.array([[6., 2], [2, 4], [1.2, 2], [5, 10]], dtype=dtype, device=device))
+            V = Dense(xnp.array([[7, 6, 6, 8.7], [3., .2, 13, 4]], dtype=dtype, device=device))
+            op = U @ V + D
+
     # Check to sure that we hit a case statement
     if op is None:
         raise ValueError(op_name)
diff --git a/tests/linalg/test_diagonal.py b/tests/linalg/test_diagonal.py
index e2511f55..4b4d8f83 100644
--- a/tests/linalg/test_diagonal.py
+++ b/tests/linalg/test_diagonal.py
@@ -63,4 +63,17 @@ def test_diagonal_diag(backend):
             d1 = diag(M, u)
             d2 = xnp.diag(M.to_dense(), u)
             assert d1.shape ==  d2.shape
-            assert relative_error(d1, d2) < 1e-5
\ No newline at end of file
+            assert relative_error(d1, d2) < 1e-5
+
+@parametrize(['torch', 'jax'], ['exact', 'approx']).excluding[:,'approx']
+def test_cyclic_trace(backend, method):
+    xnp = get_xnp(backend)
+    dtype = xnp.float32
+    array_fat = xnp.fixed_normal_samples((100, 200), dtype=dtype, device=None)
+    U = Dense(array_fat)
+    array_tall = xnp.fixed_normal_samples((200, 100), dtype=dtype, device=None)
+    V = Dense(array_tall)
+    A = U @ V
+    d1 = trace(A, method=method, tol=2e-2)
+    d2 = xnp.diag(array_fat @ array_tall).sum()
+    assert relative_error(d1, d2) < (1e-1 if method == 'approx' else 1e-5)
\ No newline at end of file