diff --git a/array_api_tests/test_linalg.py b/array_api_tests/test_linalg.py
index 40fe035d..c997948b 100644
--- a/array_api_tests/test_linalg.py
+++ b/array_api_tests/test_linalg.py
@@ -45,6 +45,7 @@
 from . import _array_module as xp
 from ._array_module import linalg
 
+
 def assert_equal(x, y, msg_extra=None):
     extra = '' if not msg_extra else f' ({msg_extra})'
     if x.dtype in dh.all_float_dtypes:
@@ -60,6 +61,7 @@ def assert_equal(x, y, msg_extra=None):
     else:
         assert_exactly_equal(x, y, msg_extra=msg_extra)
 
+
 def _test_stacks(f, *args, res=None, dims=2, true_val=None,
                  matrix_axes=(-2, -1),
                  res_axes=None,
@@ -106,6 +108,7 @@ def _test_stacks(f, *args, res=None, dims=2, true_val=None,
         if true_val:
             assert_equal(decomp_res_stack, true_val(*x_stacks, **kw), msg_extra)
 
+
 def _test_namedtuple(res, fields, func_name):
     """
     Test that res is a namedtuple with the correct fields.
@@ -121,6 +124,7 @@ def _test_namedtuple(res, fields, func_name):
         assert hasattr(res, field), f"{func_name}() result namedtuple doesn't have the '{field}' field"
         assert res[i] is getattr(res, field), f"{func_name}() result namedtuple '{field}' field is not in position {i}"
 
+
 @pytest.mark.unvectorized
 @pytest.mark.xp_extension('linalg')
 @given(
@@ -901,6 +905,15 @@ def true_trace(x_stack, offset=0):
 
     _test_stacks(linalg.trace, x, **kw, res=res, dims=0, true_val=true_trace)
 
+
+def _conj(x):
+    # XXX: replace with xp.dtype when all array libraries implement it 
+    if x.dtype in (xp.complex64, xp.complex128):
+        return xp.conj(x)
+    else:
+        return x
+
+
 def _test_vecdot(namespace, x1, x2, data):
     vecdot = namespace.vecdot
     broadcasted_shape = sh.broadcast_shapes(x1.shape, x2.shape)
@@ -925,11 +938,8 @@ def _test_vecdot(namespace, x1, x2, data):
     ph.assert_result_shape("vecdot", in_shapes=[x1.shape, x2.shape],
                            out_shape=res.shape, expected=expected_shape)
 
-    if x1.dtype in dh.int_dtypes:
-        def true_val(x, y, axis=-1):
-            return xp.sum(xp.multiply(x, y), dtype=res.dtype)
-    else:
-        true_val = None
+    def true_val(x, y, axis=-1):
+        return xp.sum(xp.multiply(_conj(x), y), dtype=res.dtype)
 
     _test_stacks(vecdot, x1, x2, res=res, dims=0,
                  matrix_axes=(axis,), true_val=true_val)
@@ -944,6 +954,7 @@ def true_val(x, y, axis=-1):
 def test_linalg_vecdot(x1, x2, data):
     _test_vecdot(linalg, x1, x2, data)
 
+
 @pytest.mark.unvectorized
 @given(
     *two_mutual_arrays(dh.numeric_dtypes, mutually_broadcastable_shapes(2, min_dims=1)),
@@ -952,10 +963,12 @@ def test_linalg_vecdot(x1, x2, data):
 def test_vecdot(x1, x2, data):
     _test_vecdot(_array_module, x1, x2, data)
 
+
 # Insanely large orders might not work. There isn't a limit specified in the
 # spec, so we just limit to reasonable values here.
 max_ord = 100
 
+
 @pytest.mark.unvectorized
 @pytest.mark.xp_extension('linalg')
 @given(