diff --git a/curvlinops/inverse.py b/curvlinops/inverse.py
index 8b49b24..7a2bb8b 100644
--- a/curvlinops/inverse.py
+++ b/curvlinops/inverse.py
@@ -1,7 +1,7 @@
 """Implements linear operator inverses."""
 
 from math import sqrt
-from typing import Any, Callable, Dict, List, Optional, Tuple, Union
+from typing import Any, Callable, Dict, List, Optional, Tuple, TypeVar, Union
 from warnings import warn
 
 from einops import einsum, rearrange
@@ -10,9 +10,11 @@
 from torch import Tensor, cat, cholesky_inverse, eye, float64, outer
 from torch.linalg import cholesky, eigh
 
-from curvlinops.kfac import KFACLinearOperator
+from curvlinops.kfac import KFACLinearOperator, ParameterMatrixType
 
-KFAC_INV_TYPE = Union[Optional[Tensor], Tuple[Optional[Tensor], Optional[Tensor]]]
+KFACInvType = TypeVar(
+    "KFACInvType", Optional[Tensor], Tuple[Optional[Tensor], Optional[Tensor]]
+)
 
 
 class _InverseLinearOperator(LinearOperator):
@@ -360,8 +362,8 @@ def __init__(
         self._use_exact_damping = use_exact_damping
         self._cache = cache
         self._retry_double_precision = retry_double_precision
-        self._inverse_input_covariances: Dict[str, KFAC_INV_TYPE] = {}
-        self._inverse_gradient_covariances: Dict[str, KFAC_INV_TYPE] = {}
+        self._inverse_input_covariances: Dict[str, KFACInvType] = {}
+        self._inverse_gradient_covariances: Dict[str, KFACInvType] = {}
 
     def _compute_damping(
         self, aaT: Optional[Tensor], ggT: Optional[Tensor]
@@ -407,7 +409,7 @@ def _damped_cholesky(self, M: Tensor, damping: float) -> Tensor:
 
     def _compute_inverse_factors(
         self, aaT: Optional[Tensor], ggT: Optional[Tensor]
-    ) -> Tuple[KFAC_INV_TYPE, KFAC_INV_TYPE]:
+    ) -> Tuple[KFACInvType, KFACInvType]:
         """Compute the inverses of the Kronecker factors for a given layer.
 
         Args:
@@ -478,7 +480,7 @@ def _compute_inverse_factors(
 
     def _compute_or_get_cached_inverse(
         self, name: str
-    ) -> Tuple[KFAC_INV_TYPE, KFAC_INV_TYPE]:
+    ) -> Tuple[KFACInvType, KFACInvType]:
         """Invert the Kronecker factors of the KFACLinearOperator or retrieve them.
 
         Args:
@@ -505,7 +507,7 @@ def _compute_or_get_cached_inverse(
         return aaT_inv, ggT_inv
 
     def _left_and_right_multiply(
-        self, M_joint: Tensor, aaT_inv: KFAC_INV_TYPE, ggT_inv: KFAC_INV_TYPE
+        self, M_joint: Tensor, aaT_inv: KFACInvType, ggT_inv: KFACInvType
     ) -> Tensor:
         """Left and right multiply matrix with inverse Kronecker factors.
 
@@ -541,8 +543,8 @@ def _separate_left_and_right_multiply(
         self,
         M_torch: Tensor,
         param_pos: Dict[str, int],
-        aaT_inv: KFAC_INV_TYPE,
-        ggT_inv: KFAC_INV_TYPE,
+        aaT_inv: KFACInvType,
+        ggT_inv: KFACInvType,
     ) -> Tensor:
         """Multiply matrix with inverse Kronecker factors for separated weight and bias.
 
@@ -598,9 +600,7 @@ def _separate_left_and_right_multiply(
 
         return M_torch
 
-    def torch_matmat(
-        self, M_torch: Union[Tensor, List[Tensor]]
-    ) -> Union[Tensor, List[Tensor]]:
+    def torch_matmat(self, M_torch: ParameterMatrixType) -> ParameterMatrixType:
         """Apply the inverse of KFAC to a matrix (multiple vectors) in PyTorch.
 
         This allows for matrix-matrix products with the inverse KFAC approximation in
@@ -650,9 +650,7 @@ def torch_matmat(
 
         return M_torch
 
-    def torch_matvec(
-        self, v_torch: Union[Tensor, List[Tensor]]
-    ) -> Union[Tensor, List[Tensor]]:
+    def torch_matvec(self, v_torch: ParameterMatrixType) -> ParameterMatrixType:
         """Apply the inverse of KFAC to a vector in PyTorch.
 
         This allows for matrix-vector products with the inverse KFAC approximation in
diff --git a/curvlinops/kfac.py b/curvlinops/kfac.py
index 19a79a6..052a64c 100644
--- a/curvlinops/kfac.py
+++ b/curvlinops/kfac.py
@@ -22,7 +22,7 @@
 from enum import Enum
 from functools import partial
 from math import sqrt
-from typing import Any, Callable, Dict, Iterable, List, Optional, Tuple, Union
+from typing import Any, Callable, Dict, Iterable, List, Optional, Tuple, TypeVar, Union
 
 from einops import einsum, rearrange, reduce
 from numpy import ndarray
@@ -46,6 +46,11 @@
     loss_hessian_matrix_sqrt,
 )
 
+# Type for a matrix/vector that can be represented as a list of tensors with the same
+# shape as the parameters, or a single matrix/vector of shape `[D, D]`/`[D]` where `D`
+# is the number of parameters.
+ParameterMatrixType = TypeVar("ParameterMatrixType", Tensor, List[Tensor])
+
 
 class FisherType(str, Enum):
     """Enum for the Fisher type."""
@@ -342,7 +347,7 @@ def _torch_preprocess(self, M: Tensor) -> List[Tensor]:
         return [res.T.reshape(shape) for res, shape in zip(result, shapes)]
 
     def _check_input_type_and_preprocess(
-        self, M_torch: Union[Tensor, List[Tensor]]
+        self, M_torch: ParameterMatrixType
     ) -> Tuple[bool, List[Tensor]]:
         """Check input type and maybe preprocess to list format.
 
@@ -392,9 +397,7 @@ def _check_input_type_and_preprocess(
             M_torch = self._torch_preprocess(M_torch)
         return return_tensor, M_torch
 
-    def torch_matmat(
-        self, M_torch: Union[Tensor, List[Tensor]]
-    ) -> Union[Tensor, List[Tensor]]:
+    def torch_matmat(self, M_torch: ParameterMatrixType) -> ParameterMatrixType:
         """Apply KFAC to a matrix (multiple vectors) in PyTorch.
 
         This allows for matrix-matrix products with the KFAC approximation in PyTorch
@@ -459,9 +462,7 @@ def torch_matmat(
 
         return M_torch
 
-    def torch_matvec(
-        self, v_torch: Union[Tensor, List[Tensor]]
-    ) -> Union[Tensor, List[Tensor]]:
+    def torch_matvec(self, v_torch: ParameterMatrixType) -> ParameterMatrixType:
         """Apply KFAC to a vector in PyTorch.
 
         This allows for matrix-vector products with the KFAC approximation in PyTorch