diff --git a/matfree/backend/typing.py b/matfree/backend/typing.py
index c57b00b..48256c4 100644
--- a/matfree/backend/typing.py
+++ b/matfree/backend/typing.py
@@ -2,7 +2,15 @@
 
 # fmt: off
 from collections.abc import Callable  # noqa: F401
-from typing import Any, Generic, Iterable, Sequence, Tuple, TypeVar  # noqa: F401, UP035
+from typing import (  # noqa: F401, UP035
+    Any,
+    Generic,
+    Iterable,
+    Sequence,
+    Tuple,
+    TypeVar,
+    Union,
+)
 
 from jax import Array  # noqa: F401
 
diff --git a/matfree/decomp.py b/matfree/decomp.py
index 8e2613c..f7147f3 100644
--- a/matfree/decomp.py
+++ b/matfree/decomp.py
@@ -9,12 +9,20 @@
 """
 
 from matfree.backend import containers, control_flow, func, linalg, np, tree_util
-from matfree.backend.typing import Array, Callable
+from matfree.backend.typing import Array, Callable, Union
 
 
 class _DecompResult(containers.NamedTuple):
-    Q_tall: Array | tuple[Array, ...]
-    J_small: Array | tuple[Array, ...]
+    # If an algorithm returns a single Q, place it here.
+    # If it returns multiple Qs, stack them
+    # into a tuple and place them here.
+    Q_tall: Union[Array, tuple[Array, ...]]
+
+    # If an algorithm returns a materialized matrix,
+    # place it here. If it returns a sparse representation
+    # (e.g. two vectors representing diagonals), place it here
+    J_small: Union[Array, tuple[Array, ...]]
+
     residual: Array
     init_length_inv: Array