diff --git a/mlx/backend/cpu/eigh.cpp b/mlx/backend/cpu/eigh.cpp
index d457c1fd99..88bcc87e2e 100644
--- a/mlx/backend/cpu/eigh.cpp
+++ b/mlx/backend/cpu/eigh.cpp
@@ -155,10 +155,12 @@ void eigh_impl(
   auto& encoder = cpu::get_command_encoder(stream);
   encoder.set_output_array(vectors);
   encoder.set_output_array(values);
+  // LAPACK reads the row-major input as its (conjugate) transpose, so the
+  // requested triangle is the opposite one in LAPACK's view.
   encoder.dispatch([vec_ptr,
                     eig_ptr,
                     jobz,
-                    uplo = uplo[0],
+                    uplo = uplo[0] == 'L' ? 'U' : 'L',
                     N = vectors.shape(-1),
                     size = vectors.size()]() mutable {
     // Work query
@@ -202,6 +204,15 @@ void Eigh::eval_cpu(
       a.flags().row_contiguous ? CopyType::Vector : CopyType::General,
       stream());
 
+  // Nothing to decompose for n = 0; LAPACK rejects lda = 0.
+  if (a.shape(-1) == 0) {
+    if (!compute_eigenvectors_) {
+      auto& encoder = cpu::get_command_encoder(stream());
+      encoder.add_temporary(vectors);
+    }
+    return;
+  }
+
   if (compute_eigenvectors_) {
     // Set the strides and flags so the eigenvectors
     // are in the columns of the output
diff --git a/mlx/backend/cpu/svd.cpp b/mlx/backend/cpu/svd.cpp
index ca01a0a65a..808ff92787 100644
--- a/mlx/backend/cpu/svd.cpp
+++ b/mlx/backend/cpu/svd.cpp
@@ -206,6 +206,14 @@ void svd_impl(
 
   using R = typename SVDWork<T>::R;
 
+  // Nothing to decompose for empty matrices; LAPACK rejects lda = 0.
+  if (M == 0 || N == 0) {
+    for (auto& o : outputs) {
+      o.set_data(allocator::malloc(o.nbytes()));
+    }
+    return;
+  }
+
   size_t num_matrices = a.size() / (M * N);
 
   // lapack clobbers the input, so we have to make a copy.
diff --git a/python/tests/test_linalg.py b/python/tests/test_linalg.py
index afdf75d7c8..fc66dd2b14 100644
--- a/python/tests/test_linalg.py
+++ b/python/tests/test_linalg.py
@@ -188,6 +188,13 @@ def test_svd_decomposition(self):
                 )
             )
 
+        # Zero-size inputs
+        for shape in [(0, 4, 4), (3, 0, 0)]:
+            U, S, Vt = mx.linalg.svd(mx.zeros(shape), stream=mx.cpu)
+            mx.eval(U, S, Vt)
+            self.assertEqual(S.shape, shape[:-1])
+            self.assertEqual(U.shape, shape)
+
         # Test float64 - use CPU stream since float64 is not supported on GPU
         with mx.stream(mx.cpu):
             A_f64 = mx.array(
@@ -493,6 +500,21 @@ def check_eigs_and_vecs(A_np, kwargs={}):
         A_np = A_np + A_np.T.conj()
         check_eigs_and_vecs(A_np)
 
+        # UPLO picks the triangle like numpy; only observable when the two
+        # triangles disagree
+        A_np = np.array([[1.0, 999.0], [2.0, 3.0]], dtype=np.float32)
+        for uplo in ("L", "U"):
+            w = mx.linalg.eigvalsh(mx.array(A_np), UPLO=uplo, stream=mx.cpu)
+            w_np = np.linalg.eigvalsh(A_np, UPLO=uplo)
+            self.assertTrue(np.allclose(w, w_np, atol=1e-5))
+
+        # Zero-size inputs
+        for shape in [(0, 4, 4), (3, 0, 0)]:
+            w, v = mx.linalg.eigh(mx.zeros(shape), stream=mx.cpu)
+            mx.eval(w, v)
+            self.assertEqual(w.shape, shape[:-1])
+            self.assertEqual(v.shape, shape)
+
         # Test error cases
         with self.assertRaises(ValueError):
             mx.linalg.eigh(mx.array([1.0, 2.0]))  # 1D array