diff --git a/src/pymor/algorithms/gram_schmidt.py b/src/pymor/algorithms/gram_schmidt.py
index 2e65b2e15..114a5e06b 100644
--- a/src/pymor/algorithms/gram_schmidt.py
+++ b/src/pymor/algorithms/gram_schmidt.py
@@ -10,10 +10,10 @@ from pymor.core.logger import getLogger
 
 
 @defaults('atol', 'rtol', 'reiterate', 'reiteration_threshold', 'check', 'check_tol')
-def gram_schmidt(A, product=None, atol=1e-13, rtol=1e-13, offset=0,
+def gram_schmidt(A, product=None, return_R=False, atol=1e-13, rtol=1e-13, offset=0,
                  reiterate=True, reiteration_threshold=1e-1, check=True, check_tol=1e-3,
                  copy=True):
-    """Orthonormalize a |VectorArray| using the stabilized Gram-Schmidt algorithm.
+    """Orthonormalize a |VectorArray| using the modified Gram-Schmidt algorithm.
 
     Parameters
     ----------
@@ -22,6 +22,8 @@ def gram_schmidt(A, product=None, atol=1e-13, rtol=1e-13, offset=0,
     product
         The inner product |Operator| w.r.t. which to orthonormalize.
         If `None`, the Euclidean product is used.
+    return_R
+        If `True`, the R matrix from QR decomposition is returned.
     atol
         Vectors of norm smaller than `atol` are removed from the array.
     rtol
@@ -45,7 +47,10 @@ def gram_schmidt(A, product=None, atol=1e-13, rtol=1e-13, offset=0,
 
     Returns
     -------
-    The orthonormalized |VectorArray|.
+    Q
+        The orthonormalized |VectorArray|.
+    R
+        The upper-triangular/trapezoidal matrix (if `compute_R` is `True`).
     """
 
     logger = getLogger('pymor.algorithms.gram_schmidt.gram_schmidt')
@@ -54,7 +59,8 @@ def gram_schmidt(A, product=None, atol=1e-13, rtol=1e-13, offset=0,
         A = A.copy()
 
     # main loop
-    remove = []
+    R = np.eye(len(A))
+    remove = []  # indices of to be removed vectors
     for i in range(offset, len(A)):
         # first calculate norm
         initial_norm = A[i].norm(product)[0]
@@ -65,41 +71,41 @@ def gram_schmidt(A, product=None, atol=1e-13, rtol=1e-13, offset=0,
             continue
 
         if i == 0:
-            A[0].scal(1/initial_norm)
-
+            A[0].scal(1 / initial_norm)
+            R[i, i] = initial_norm
         else:
-            first_iteration = True
             norm = initial_norm
             # If reiterate is True, reiterate as long as the norm of the vector changes
-            # strongly during orthonormalization (due to Andreas Buhr).
-            while first_iteration or reiterate and norm/old_norm < reiteration_threshold:
-
-                if first_iteration:
-                    first_iteration = False
-                else:
-                    logger.info(f'Orthonormalizing vector {i} again')
-
+            # strongly during orthogonalization (due to Andreas Buhr).
+            while True:
                 # orthogonalize to all vectors left
                 for j in range(i):
                     if j in remove:
                         continue
-                    p = A[i].pairwise_inner(A[j], product)[0]
+                    p = A[j].pairwise_inner(A[i], product)[0]
                     A[i].axpy(-p, A[j])
+                    R[j, i] += p
 
                 # calculate new norm
                 old_norm, norm = norm, A[i].norm(product)[0]
 
-                # remove vector if it got too small:
-                if norm / initial_norm < rtol:
-                    logger.info(f"Removing linear dependent vector {i}")
+                # remove vector if it got too small
+                if norm < rtol * initial_norm:
+                    logger.info(f"Removing linearly dependent vector {i}")
                     remove.append(i)
                     break
 
-            if norm > 0:
-                A[i].scal(1 / norm)
+                # check if reorthogonalization should be done
+                if reiterate and norm < reiteration_threshold * old_norm:
+                    logger.info(f"Orthonormalizing vector {i} again")
+                else:
+                    A[i].scal(1 / norm)
+                    R[i, i] = norm
+                    break
 
     if remove:
         del A[remove]
+        R = np.delete(R, remove, axis=0)
 
     if check:
         error_matrix = A[offset:len(A)].inner(A, product)
@@ -107,12 +113,16 @@ def gram_schmidt(A, product=None, atol=1e-13, rtol=1e-13, offset=0,
         if error_matrix.size > 0:
             err = np.max(np.abs(error_matrix))
             if err >= check_tol:
-                raise AccuracyError(f'result not orthogonal (max err={err})')
+                raise AccuracyError(f"result not orthogonal (max err={err})")
 
-    return A
+    if return_R:
+        return A, R
+    else:
+        return A
 
 
-def gram_schmidt_biorth(V, W, product=None, reiterate=True, reiteration_threshold=1e-1, check=True, check_tol=1e-3,
+def gram_schmidt_biorth(V, W, product=None,
+                        reiterate=True, reiteration_threshold=1e-1, check=True, check_tol=1e-3,
                         copy=True):
     """Biorthonormalize a pair of |VectorArrays| using the biorthonormal Gram-Schmidt process.
 
@@ -161,16 +171,10 @@ def gram_schmidt_biorth(V, W, product=None, reiterate=True, reiteration_threshol
         if i == 0:
             V[0].scal(1 / initial_norm)
         else:
-            first_iteration = True
             norm = initial_norm
             # If reiterate is True, reiterate as long as the norm of the vector changes
             # strongly during projection.
-            while first_iteration or reiterate and norm / old_norm < reiteration_threshold:
-                if first_iteration:
-                    first_iteration = False
-                else:
-                    logger.info(f'Projecting vector V[{i}] again')
-
+            while True:
                 for j in range(i):
                     # project by (I - V[j] * W[j]^T * E)
                     p = W[j].pairwise_inner(V[i], product)[0]
@@ -179,8 +183,12 @@ def gram_schmidt_biorth(V, W, product=None, reiterate=True, reiteration_threshol
                 # calculate new norm
                 old_norm, norm = norm, V[i].norm(product)[0]
 
-            if norm > 0:
-                V[i].scal(1 / norm)
+                # check if reorthogonalization should be done
+                if reiterate and norm < reiteration_threshold * old_norm:
+                    logger.info(f"Projecting vector V[{i}] again")
+                else:
+                    V[i].scal(1 / norm)
+                    break
 
         # calculate norm of W[i]
         initial_norm = W[i].norm(product)[0]
@@ -189,16 +197,10 @@ def gram_schmidt_biorth(V, W, product=None, reiterate=True, reiteration_threshol
         if i == 0:
             W[0].scal(1 / initial_norm)
         else:
-            first_iteration = True
             norm = initial_norm
             # If reiterate is True, reiterate as long as the norm of the vector changes
             # strongly during projection.
-            while first_iteration or reiterate and norm / old_norm < reiteration_threshold:
-                if first_iteration:
-                    first_iteration = False
-                else:
-                    logger.info(f'Projecting vector W[{i}] again')
-
+            while True:
                 for j in range(i):
                     # project by (I - W[j] * V[j]^T * E)
                     p = V[j].pairwise_inner(W[i], product)[0]
@@ -207,8 +209,12 @@ def gram_schmidt_biorth(V, W, product=None, reiterate=True, reiteration_threshol
                 # calculate new norm
                 old_norm, norm = norm, W[i].norm(product)[0]
 
-            if norm > 0:
-                W[i].scal(1 / norm)
+                # check if reorthogonalization should be done
+                if reiterate and norm < reiteration_threshold * old_norm:
+                    logger.info(f"Projecting vector W[{i}] again")
+                else:
+                    W[i].scal(1 / norm)
+                    break
 
         # rescale V[i]
         p = W[i].pairwise_inner(V[i], product)[0]
@@ -220,6 +226,6 @@ def gram_schmidt_biorth(V, W, product=None, reiterate=True, reiteration_threshol
         if error_matrix.size > 0:
             err = np.max(np.abs(error_matrix))
             if err >= check_tol:
-                raise AccuracyError(f'Result not biorthogonal (max err={err})')
+                raise AccuracyError(f"result not biorthogonal (max err={err})")
 
     return V, W

diff --git a/src/pymortests/algorithms/gram_schmidt.py b/src/pymortests/algorithms/gram_schmidt.py
index 82debb6aa..285081e30 100644
--- a/src/pymortests/algorithms/gram_schmidt.py
+++ b/src/pymortests/algorithms/gram_schmidt.py
@@ -24,6 +24,22 @@ def test_gram_schmidt(vector_array):
     assert np.all(almost_equal(onb, U))
 
 
+def test_gram_schmidt_with_R(vector_array):
+    U = vector_array
+
+    V = U.copy()
+    onb, R = gram_schmidt(U, return_R=True, copy=True)
+    assert np.all(almost_equal(U, V))
+    assert np.allclose(onb.dot(onb), np.eye(len(onb)))
+    assert np.all(almost_equal(U, onb.lincomb(U.dot(onb)), rtol=1e-13))
+    assert np.all(almost_equal(V, onb.lincomb(R.T)))
+
+    onb2, R2 = gram_schmidt(U, return_R=True, copy=False)
+    assert np.all(almost_equal(onb, onb2))
+    assert np.all(R == R2)
+    assert np.all(almost_equal(onb, U))
+
+
 def test_gram_schmidt_with_product(operator_with_arrays_and_products):
     _, _, U, _, p, _ = operator_with_arrays_and_products
 
@@ -38,6 +54,22 @@ def test_gram_schmidt_with_product(operator_with_arrays_and_products):
     assert np.all(almost_equal(onb, U))
 
 
+def test_gram_schmidt_with_product_and_R(operator_with_arrays_and_products):
+    _, _, U, _, p, _ = operator_with_arrays_and_products
+
+    V = U.copy()
+    onb, R = gram_schmidt(U, product=p, return_R=True, copy=True)
+    assert np.all(almost_equal(U, V))
+    assert np.allclose(p.apply2(onb, onb), np.eye(len(onb)))
+    assert np.all(almost_equal(U, onb.lincomb(p.apply2(U, onb)), rtol=1e-13))
+    assert np.all(almost_equal(U, onb.lincomb(R.T)))
+
+    onb2, R2 = gram_schmidt(U, product=p, return_R=True, copy=False)
+    assert np.all(almost_equal(onb, onb2))
+    assert np.all(R == R2)
+    assert np.all(almost_equal(onb, U))
+
+
 def test_gram_schmidt_biorth(vector_array):
     U = vector_array
     if U.dim < 2:

gram_schmidt doesn't return the R matrix
Currently, `gram_schmidt` only returns the Q matrix. The R matrix is used in the [shift selection for the LRADI method](https://github.com/pymor/pymor/blob/1aeccb428700fa140cb6faa7490bdaf87a9754a3/src/pymor/algorithms/lyapunov.py#L285-L287) and in the [rational Krylov subspace method for Lyapunov equations](http://www.dm.unibo.it/~simoncin/rksm.m) (related to #387).

It might also be useful for computing an SVD of a vectorarray (Q * svd(R)).
src/pymortests/algorithms/gram_schmidt.py::test_gram_schmidt_with_R[0-<lambda>0] src/pymortests/algorithms/gram_schmidt.py::test_gram_schmidt_with_R[0-<lambda>1] src/pymortests/algorithms/gram_schmidt.py::test_gram_schmidt_with_R[0-<lambda>2] src/pymortests/algorithms/gram_schmidt.py::test_gram_schmidt_with_R[0-<lambda>3] src/pymortests/algorithms/gram_schmidt.py::test_gram_schmidt_with_R[0-<lambda>4] src/pymortests/algorithms/gram_schmidt.py::test_gram_schmidt_with_R[0-<lambda>5] src/pymortests/algorithms/gram_schmidt.py::test_gram_schmidt_with_R[0-<lambda>6] src/pymortests/algorithms/gram_schmidt.py::test_gram_schmidt_with_R[0-<lambda>7] src/pymortests/algorithms/gram_schmidt.py::test_gram_schmidt_with_R[0-<lambda>8] src/pymortests/algorithms/gram_schmidt.py::test_gram_schmidt_with_R[0-<lambda>9] src/pymortests/algorithms/gram_schmidt.py::test_gram_schmidt_with_R[0-<lambda>10] src/pymortests/algorithms/gram_schmidt.py::test_gram_schmidt_with_R[0-<lambda>11] src/pymortests/algorithms/gram_schmidt.py::test_gram_schmidt_with_R[0-<lambda>12] src/pymortests/algorithms/gram_schmidt.py::test_gram_schmidt_with_R[0-<lambda>13] src/pymortests/algorithms/gram_schmidt.py::test_gram_schmidt_with_R[0-<lambda>14] src/pymortests/algorithms/gram_schmidt.py::test_gram_schmidt_with_R[0-<lambda>15] src/pymortests/algorithms/gram_schmidt.py::test_gram_schmidt_with_R[0-<lambda>16] src/pymortests/algorithms/gram_schmidt.py::test_gram_schmidt_with_R[1-<lambda>0] src/pymortests/algorithms/gram_schmidt.py::test_gram_schmidt_with_R[1-<lambda>1] src/pymortests/algorithms/gram_schmidt.py::test_gram_schmidt_with_R[1-<lambda>2] src/pymortests/algorithms/gram_schmidt.py::test_gram_schmidt_with_R[1-<lambda>3] src/pymortests/algorithms/gram_schmidt.py::test_gram_schmidt_with_R[1-<lambda>4] src/pymortests/algorithms/gram_schmidt.py::test_gram_schmidt_with_R[1-<lambda>5] src/pymortests/algorithms/gram_schmidt.py::test_gram_schmidt_with_R[1-<lambda>6] 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