diff --git a/array_api_compat/common/_linalg.py b/array_api_compat/common/_linalg.py
index 97c584be..ce5b55d1 100644
--- a/array_api_compat/common/_linalg.py
+++ b/array_api_compat/common/_linalg.py
@@ -11,7 +11,7 @@
 else:
     from numpy.core.numeric import normalize_axis_tuple
 
-from ._aliases import matmul, matrix_transpose, tensordot, vecdot
+from ._aliases import matmul, matrix_transpose, tensordot, vecdot, isdtype
 from .._internal import get_xp
 
 # These are in the main NumPy namespace but not in numpy.linalg
@@ -59,7 +59,10 @@ def svd(x: ndarray, /, xp, *, full_matrices: bool = True, **kwargs) -> SVDResult
 def cholesky(x: ndarray, /, xp, *, upper: bool = False, **kwargs) -> ndarray:
     L = xp.linalg.cholesky(x, **kwargs)
     if upper:
-        return get_xp(xp)(matrix_transpose)(L)
+        U = get_xp(xp)(matrix_transpose)(L)
+        if get_xp(xp)(isdtype)(U.dtype, 'complex floating'):
+            U = xp.conj(U)
+        return U
     return L
 
 # The rtol keyword argument of matrix_rank() and pinv() is new from NumPy.