diff --git a/lineax/_operator.py b/lineax/_operator.py
index ad1e4f5..d8ab745 100644
--- a/lineax/_operator.py
+++ b/lineax/_operator.py
@@ -733,7 +733,14 @@ def mv(self, vector):
             return jtu.tree_unflatten(treedef, shaped)
 
     def as_matrix(self):
-        return jnp.eye(self.out_size(), self.in_size())
+        leaves = jtu.tree_leaves(self.in_structure())
+        with jax.numpy_dtype_promotion("standard"):
+            dtype = (
+                default_floating_dtype()
+                if len(leaves) == 0
+                else jnp.result_type(*leaves)
+            )
+        return jnp.eye(self.out_size(), self.in_size(), dtype=dtype)
 
     def transpose(self):
         return IdentityLinearOperator(self.out_structure(), self.in_structure())
diff --git a/tests/helpers.py b/tests/helpers.py
index 32f81d0..ddbdaef 100644
--- a/tests/helpers.py
+++ b/tests/helpers.py
@@ -198,6 +198,12 @@ def make_diagonal_operator(getkey, matrix, tags):
     return lx.DiagonalLinearOperator(diag)
 
 
+@_operators_append
+def make_identity_operator(getkey, matrix, tags):
+    in_struct = jax.ShapeDtypeStruct((matrix.shape[-1],), matrix.dtype)
+    return lx.IdentityLinearOperator(input_structure=in_struct)
+
+
 @_operators_append
 def make_tridiagonal_operator(getkey, matrix, tags):
     diag1 = jnp.diag(matrix)
diff --git a/tests/test_operator.py b/tests/test_operator.py
index e0e1e7a..06e4ddf 100644
--- a/tests/test_operator.py
+++ b/tests/test_operator.py
@@ -23,22 +23,37 @@
 
 from .helpers import (
     make_diagonal_operator,
+    make_identity_operator,
     make_operators,
     make_tridiagonal_operator,
     tree_allclose,
 )
 
 
-def test_ops(getkey):
-    matrix1 = lx.MatrixLinearOperator(jr.normal(getkey(), (3, 3)))
-    matrix2 = lx.MatrixLinearOperator(jr.normal(getkey(), (3, 3)))
-    scalar = jr.normal(getkey(), ())
+@pytest.mark.parametrize("make_operator", make_operators)
+@pytest.mark.parametrize("dtype", (jnp.float64, jnp.complex128))
+def test_ops(make_operator, getkey, dtype):
+    if (
+        make_operator is make_diagonal_operator
+        or make_operator is make_identity_operator
+    ):
+        matrix = jnp.eye(3, dtype=dtype)
+        tags = lx.diagonal_tag
+    elif make_operator is make_tridiagonal_operator:
+        matrix = jnp.eye(3, dtype=dtype)
+        tags = lx.tridiagonal_tag
+    else:
+        matrix = jr.normal(getkey(), (3, 3), dtype=dtype)
+        tags = ()
+    matrix1 = make_operator(getkey, matrix, tags)
+    matrix2 = lx.MatrixLinearOperator(jr.normal(getkey(), (3, 3), dtype=dtype))
+    scalar = jr.normal(getkey(), (), dtype=dtype)
     add = matrix1 + matrix2
     composed = matrix1 @ matrix2
     mul = matrix1 * scalar
     rmul = cast(lx.AbstractLinearOperator, scalar * matrix1)
     div = matrix1 / scalar
-    vec = jr.normal(getkey(), (3,))
+    vec = jr.normal(getkey(), (3,), dtype=dtype)
 
     assert tree_allclose(matrix1.mv(vec) + matrix2.mv(vec), add.mv(vec))
     assert tree_allclose(matrix1.mv(matrix2.mv(vec)), composed.mv(vec))
@@ -66,7 +81,10 @@ def test_ops(getkey):
 
 @pytest.mark.parametrize("make_operator", make_operators)
 def test_structures_vector(make_operator, getkey):
-    if make_operator is make_diagonal_operator:
+    if (
+        make_operator is make_diagonal_operator
+        or make_operator is make_identity_operator
+    ):
         matrix = jnp.eye(4)
         tags = lx.diagonal_tag
         in_size = out_size = 4
@@ -96,6 +114,12 @@ def _setup(getkey, matrix, tag: Union[object, frozenset[object]] = frozenset()):
             lx.symmetric_tag,
         ):
             continue
+        if make_operator is make_identity_operator and tag not in (
+            lx.tridiagonal_tag,
+            lx.diagonal_tag,
+            lx.symmetric_tag,
+        ):
+            continue
         operator = make_operator(getkey, matrix, tag)
         yield operator
 
@@ -327,7 +351,7 @@ def test_identity_with_different_structures():
 
     assert op1.T == op2
     # assert op2.transpose((True, False)) == op3
-    assert jnp.array_equal(op1.as_matrix(), jnp.eye(5, 7))
+    assert jnp.array_equal(op1.as_matrix(), jnp.eye(5, 7, dtype=jnp.float32))
     assert op1.in_size() == 7
     assert op1.out_size() == 5
     vec1 = (
@@ -356,7 +380,7 @@ def test_identity_with_different_structures_complex():
 
     assert op1.T == op2
     # assert op2.transpose((True, False)) == op3
-    assert jnp.array_equal(op1.as_matrix(), jnp.eye(5, 7))
+    assert jnp.array_equal(op1.as_matrix(), jnp.eye(5, 7, dtype=jnp.complex128))
     assert op1.in_size() == 7
     assert op1.out_size() == 5
     vec1 = (