diff --git a/src/traceax/_trace.py b/src/traceax/_trace.py
index 8dd2f45..2b73b2d 100644
--- a/src/traceax/_trace.py
+++ b/src/traceax/_trace.py
@@ -400,22 +400,87 @@ def _remove_undefined_primal(x):
         return
 
 
-def _build_diagonal(ct_result: float, op: lx.AbstractLinearOperator) -> lx.AbstractLinearOperator:
+# def _build_diagonal(ct_result: float, op: lx.AbstractLinearOperator) -> lx.AbstractLinearOperator:
+#     operator_struct = jtu.tree_map(_remove_undefined_primal, op, is_leaf=_is_undefined)
+#     if isinstance(op, lx.MatrixLinearOperator):
+#         in_size = eqx.filter_eval_shape(lambda o: o.in_size(), operator_struct)
+#         diag = ct_result * jnp.ones(in_size)
+#         return lx.MatrixLinearOperator(jnp.diag(diag), tags=operator_struct.tags)
+#     elif isinstance(op, lx.DiagonalLinearOperator):
+#         in_size = eqx.filter_eval_shape(lambda o: o.in_size(), operator_struct)
+#         diag = ct_result * jnp.ones(in_size)
+#         return lx.DiagonalLinearOperator(diag)
+#     elif isinstance(op, lx.MulLinearOperator):
+#         inner_op = _build_diagonal(ct_result, op.operator)
+#         scalar = op.scalar
+#         return scalar * inner_op  # type: ignore
+#     else:
+#         raise ValueError("Unsupported type!")
+
+
+# replaces _build_diagonal
+@ft.singledispatch
+def _make_identity(op: lx.AbstractLinearOperator, ct_result: float) -> lx.AbstractLinearOperator:
+    raise ValueError("Unsupported type!")
+
+
+@_make_identity.register
+def _(op: lx.MatrixLinearOperator, ct_result: float) -> lx.AbstractLinearOperator:
     operator_struct = jtu.tree_map(_remove_undefined_primal, op, is_leaf=_is_undefined)
-    if isinstance(op, lx.MatrixLinearOperator):
-        in_size = eqx.filter_eval_shape(lambda o: o.in_size(), operator_struct)
-        diag = ct_result * jnp.ones(in_size)
-        return lx.MatrixLinearOperator(jnp.diag(diag), tags=operator_struct.tags)
-    elif isinstance(op, lx.DiagonalLinearOperator):
-        in_size = eqx.filter_eval_shape(lambda o: o.in_size(), operator_struct)
-        diag = ct_result * jnp.ones(in_size)
-        return lx.DiagonalLinearOperator(diag)
-    elif isinstance(op, lx.MulLinearOperator):
-        inner_op = _build_diagonal(ct_result, op.operator)
-        scalar = op.scalar
-        return scalar * inner_op  # type: ignore
-    else:
-        raise ValueError("Unsupported type!")
+    in_size = eqx.filter_eval_shape(lambda o: o.in_size(), operator_struct)
+    diag = ct_result * jnp.ones(in_size)
+    return lx.MatrixLinearOperator(jnp.diag(diag), tags=operator_struct.tags)
+
+
+@_make_identity.register
+def _(op: lx.DiagonalLinearOperator, ct_result: float) -> lx.AbstractLinearOperator:
+    operator_struct = jtu.tree_map(_remove_undefined_primal, op, is_leaf=_is_undefined)
+    in_size = eqx.filter_eval_shape(lambda o: o.in_size(), operator_struct)
+    diag = ct_result * jnp.ones(in_size)
+    return lx.DiagonalLinearOperator(diag)
+
+
+@_make_identity.register
+def _(op: lx.MulLinearOperator, ct_result: float) -> lx.AbstractLinearOperator:
+    inner_op = _make_identity(op.operator, ct_result)
+    scalar = op.scalar
+    return lx.MulLinearOperator(inner_op, scalar)
+
+
+@_make_identity.register
+def _(op: lx.TridiagonalLinearOperator, ct_result: float) -> lx.AbstractLinearOperator:
+    operator_struct = jtu.tree_map(_remove_undefined_primal, op, is_leaf=_is_undefined)
+    in_size = eqx.filter_eval_shape(lambda o: o.in_size(), operator_struct)
+    diag = ct_result * jnp.ones(in_size)
+    off_diag = jnp.zeros(in_size - 1)
+    return lx.TridiagonalLinearOperator(diag, off_diag, off_diag)
+
+
+@_make_identity.register
+def _(op: lx.AddLinearOperator, ct_result: float) -> lx.AbstractLinearOperator:
+    inner_op1 = _make_identity(op.operator1, ct_result)
+    inner_op2 = _make_identity(op.operator2, ct_result)
+    return lx.AddLinearOperator(inner_op1, inner_op2)
+
+
+@_make_identity.register
+def _(op: lx.NegLinearOperator, ct_result: float) -> lx.AbstractLinearOperator:
+    inner_op = _make_identity(op.operator, ct_result)
+    return lx.NegLinearOperator(inner_op)
+
+
+@_make_identity.register
+def _(op: lx.DivLinearOperator, ct_result: float) -> lx.AbstractLinearOperator:
+    inner_op = _make_identity(op.operator, ct_result)
+    scalar = op.scalar
+    return lx.DivLinearOperator(inner_op, scalar)
+
+
+@_make_identity.register
+def _(op: lx.ComposedLinearOperator, ct_result: float) -> lx.AbstractLinearOperator:
+    inner_op1 = _make_identity(op.operator1, ct_result)
+    inner_op2 = _make_identity(op.operator2, ct_result)
+    return lx.ComposedLinearOperator(inner_op1, inner_op2)
 
 
 @eqxi.filter_primitive_transpose(materialise_zeros=True)  # pyright: ignore
@@ -431,7 +496,8 @@ def _estimate_trace_transpose(inputs, cts_out):
 
     # the internals of the operator are UndefinedPrimal leaves so
     # we need to rely on abstract values to pull structure info
-    op_t = _build_diagonal(cts_result, operator)
+    # op_t = _build_diagonal(cts_result, operator)
+    op_t = _make_identity(operator, cts_result)
 
     key_none = jtu.tree_map(lambda _: None, key)
     # state_none = jtu.tree_map(lambda _: None, state)