diff --git a/python/tests/test_relatedness_vector.py b/python/tests/test_relatedness_vector.py
index f765c75c9f..6488fa7b24 100644
--- a/python/tests/test_relatedness_vector.py
+++ b/python/tests/test_relatedness_vector.py
@@ -460,7 +460,7 @@ def check_relatedness_vector(
     return R
 
 
-class TestExamples:
+class TestRelatednessVector:
 
     def test_bad_weights(self):
         n = 5
@@ -737,3 +737,147 @@ def test_disconnected_non_sample_topology(self, centre):
             ts2, internal_checks=True, centre=centre, do_nodes=False
         )
         np.testing.assert_array_almost_equal(D1, D2)
+
+
+def pca(ts, windows, centre):
+    drop_dimension = windows is None
+    if drop_dimension:
+        windows = [0, ts.sequence_length]
+    Sigma = relatedness_matrix(ts=ts, windows=windows, centre=centre)
+    U, S, _ = np.linalg.svd(Sigma, hermitian=True)
+    if drop_dimension:
+        U = U[0]
+        S = S[0]
+    return U, S
+
+
+def allclose_up_to_sign(x, y, **kwargs):
+    # check if two vectors are the same up to sign
+    x_const = np.isclose(np.std(x), 0)
+    y_const = np.isclose(np.std(y), 0)
+    if x_const or y_const:
+        if np.allclose(x, 0):
+            r = 1.0
+        else:
+            r = np.mean(x / y)
+    else:
+        r = np.sign(np.corrcoef(x, y)[0, 1])
+    return np.allclose(x, r * y, **kwargs)
+
+
+def assert_pcs_equal(U, D, U_full, D_full, rtol=1e-05, atol=1e-08):
+    # check that the PCs in U, D occur in U_full, D_full
+    # accounting for sign and ordering
+    assert len(D) <= len(D_full)
+    assert U.shape[0] == U_full.shape[0]
+    assert U.shape[1] == len(D)
+    for k in range(len(D)):
+        u = U[:, k]
+        d = D[k]
+        (ii,) = np.where(np.isclose(D_full, d, rtol=rtol, atol=atol))
+        assert len(ii) > 0, f"{k}th singular value {d} not found in {D_full}."
+        found_it = False
+        for i in ii:
+            if allclose_up_to_sign(u, U_full[:, i], rtol=rtol, atol=atol):
+                found_it = True
+                break
+        assert found_it, f"{k}th singular vector {u} not found in {U_full}."
+
+
+class TestPCA:
+
+    def verify_pca(self, ts, num_windows, n_components, centre):
+        if num_windows == 0:
+            windows = None
+        elif num_windows % 2 == 0:
+            windows = np.linspace(
+                0.2 * ts.sequence_length, 0.8 * ts.sequence_length, num_windows + 1
+            )
+        else:
+            windows = np.linspace(0, ts.sequence_length, num_windows + 1)
+        ts_U, ts_D = ts.pca(
+            windows=windows, n_components=n_components, centre=centre, random_seed=123
+        )
+        num_rows = ts.num_samples
+        if windows is None:
+            assert ts_U.shape == (num_rows, n_components)
+            assert ts_D.shape == (n_components,)
+        else:
+            assert ts_U.shape == (num_windows, num_rows, n_components)
+            assert ts_D.shape == (num_windows, n_components)
+        U, D = pca(ts=ts, windows=windows, centre=centre)
+        if windows is None:
+            np.testing.assert_allclose(ts_D, D[:n_components], atol=1e-8)
+            assert_pcs_equal(ts_U, ts_D, U, D)
+        else:
+            for w in range(num_windows):
+                np.testing.assert_allclose(ts_D[w], D[w, :n_components], atol=1e-8)
+                assert_pcs_equal(ts_U[w], ts_D[w], U[w], D[w])
+
+    def test_bad_windows(self):
+        ts = msprime.sim_ancestry(
+            3,
+            ploidy=2,
+            sequence_length=10,
+            random_seed=123,
+        )
+        for bad_w in ([], [1]):
+            with pytest.raises(ValueError, match="Number of windows"):
+                ts.pca(n_components=2, windows=bad_w)
+        for bad_w in ([1, 0], [-3, 10]):
+            with pytest.raises(tskit.LibraryError, match="TSK_ERR_BAD_WINDOWS"):
+                ts.pca(n_components=2, windows=bad_w)
+
+    def test_bad_num_components(self):
+        ts = msprime.sim_ancestry(
+            3,
+            ploidy=2,
+            sequence_length=10,
+            random_seed=123,
+        )
+        with pytest.raises(ValueError, match="Number of components"):
+            ts.pca(n_components=ts.num_samples + 1)
+        with pytest.raises(ValueError, match="Number of components"):
+            ts.pca(n_components=4, samples=[0, 1, 2])
+        with pytest.raises(ValueError, match="Number of components"):
+            ts.pca(n_components=4, individuals=[0, 1])
+
+    def test_indivs_and_samples(self):
+        ts = msprime.sim_ancestry(
+            3,
+            ploidy=2,
+            sequence_length=10,
+            random_seed=123,
+        )
+        with pytest.raises(ValueError, match="Samples and individuals"):
+            ts.pca(n_components=2, samples=[0, 1, 2, 3], individuals=[0, 1, 2])
+
+    def test_modes(self):
+        ts = msprime.sim_ancestry(
+            3,
+            ploidy=2,
+            sequence_length=10,
+            random_seed=123,
+        )
+        for bad_mode in ("site", "node"):
+            with pytest.raises(
+                tskit.LibraryError, match="TSK_ERR_UNSUPPORTED_STAT_MODE"
+            ):
+                ts.pca(n_components=2, mode=bad_mode)
+
+    @pytest.mark.parametrize("n", [2, 3, 5, 15])
+    @pytest.mark.parametrize("centre", (True, False))
+    @pytest.mark.parametrize("num_windows", (0, 1, 2, 3))
+    @pytest.mark.parametrize("n_components", (1, 3))
+    def test_simple_sims(self, n, centre, num_windows, n_components):
+        ploidy = 1
+        nc = min(n_components, n * ploidy)
+        ts = msprime.sim_ancestry(
+            n,
+            ploidy=ploidy,
+            population_size=20,
+            sequence_length=100,
+            recombination_rate=0.01,
+            random_seed=12345,
+        )
+        self.verify_pca(ts, num_windows=num_windows, n_components=nc, centre=centre)
diff --git a/python/tskit/trees.py b/python/tskit/trees.py
index 7d3bbca7b9..6064e7556f 100644
--- a/python/tskit/trees.py
+++ b/python/tskit/trees.py
@@ -8592,6 +8592,216 @@ def genetic_relatedness_vector(
         )
         return out
 
+    def pca(
+        self,
+        num_components: int,
+        windows: list = None,
+        samples: np.ndarray = None,
+        individuals: np.ndarray = None,
+        mode: str = "branch",
+        centre: bool = True,
+        iterated_power: int = 5,
+        num_oversamples: int = 10,
+        random_seed: int = None,
+        range_sketch: np.ndarray = None,
+    ) -> (np.ndarray, np.ndarray, np.ndarray):
+        """
+        Run randomized singular value decomposition (rSVD) to obtain principal
+        components.
+        API partially adopted from `scikit-learn`'s
+        `sklearn.decomposition.PCA.html`
+
+        By default, performs PCA for the samples, so output has one coordinate
+        for each sample), but alternatively either a list of sample IDs or a
+        list of individual IDs can be provided (but not both).
+
+        TODO: say exactly what is returned (and relationship to
+        :meth:`genetic_relatedness <.TreeSequence.genetic_relatedness>`).
+
+        TODO: say what algorithms are used.
+
+        :param int num_components: Number of principal components.
+        :param list windows: An increasing list of breakpoints between the windows
+            to compute the statistic in.
+        :param np.ndarray samples: Samples to perform PCA with.
+        :param np.ndarray individuals: Individuals to perform PCA with. Cannot specify
+            both `samples` and `individuals`.
+        :param str mode: A string giving the "type" of relatedness to be computed
+            (defaults to "branch"; see
+            :meth:`genetic_relatedness_vector
+            <.TreeSequence.genetic_relatedness_vector>`)
+        :param bool centre: Centre the genetic relatedness matrix.
+        :param int iterated_power: Number of power iteration of range finder.
+        :param int num_oversamples: Number of additional test vectors.
+        :param int random_seed: The random seed. If this is None, a random seed will
+            be automatically generated. Valid random seeds must be between 1 and
+            :math:`2^32 − 1`.
+        :param np.ndarray range_sketch: Sketch matrix for each window. Default is None.
+        :return: A tuple (U, D, Q) of ndarrays, with the principal component loadings in U
+            and the principal values in D. Q is the range sketch array for each window.
+        """
+
+        if samples is None and individuals is None:
+            samples = self.samples()
+
+        if samples is not None and individuals is not None:
+            raise ValueError("Samples and individuals cannot be used at the same time")
+        elif samples is not None:
+            output_type = "node"
+            dim = len(samples)
+        else:
+            assert individuals is not None
+            output_type = "individual"
+            dim = len(individuals)
+        
+        if range_sketch is not None:
+            if windows is not None:
+                assert range_sketch.shape[0] == len(windows) - 1
+            elif windows is None:
+                range_sketch = np.expand_dims(range_sketch, 0)
+
+        if num_components > dim:
+            raise ValueError(
+                "Number of components must be less than or equal to "
+                "the number of samples (or individuals, if specified)."
+            )
+
+        random_state = np.random.default_rng(random_seed)
+
+        def _rand_pow_range_finder(
+            operator,
+            operator_dim: int,
+            rank: int,
+            depth: int,
+            num_vectors: int,
+            rng: np.random.Generator,
+            range_sketch: np.ndarray = None,
+        ) -> np.ndarray:
+            """
+            Algorithm 9 in https://arxiv.org/pdf/2002.01387
+            """
+            assert num_vectors >= rank > 0, "num_vectors should be larger than rank"
+            if range_sketch is None:
+                test_vectors = rng.normal(size=(operator_dim, num_vectors))
+                Q = test_vectors
+            else:
+                Q = range_sketch
+            for _ in range(depth):
+                Q = np.linalg.qr(Q).Q
+                Q = operator(Q)
+            Q = np.linalg.qr(Q).Q
+            return Q[:, :rank]
+
+        def _rand_svd(
+            operator,
+            operator_dim: int,
+            rank: int,
+            depth: int,
+            num_vectors: int,
+            rng: np.random.Generator,
+            range_sketch: np.ndarray = None,
+        ) -> (np.ndarray, np.ndarray, np.ndarray, float):
+            """
+            Algorithm 8 in https://arxiv.org/pdf/2002.01387
+            """
+            assert num_vectors >= rank > 0
+            Q = _rand_pow_range_finder(
+                operator, operator_dim, num_vectors, depth, num_vectors, rng, range_sketch
+            )
+            C = operator(Q).T
+            U_hat, D, V = np.linalg.svd(C, full_matrices=False)
+            U = Q @ U_hat
+
+            error_factor = np.power(
+                    1 + 4 * np.sqrt(2 * operator_dim / (rank - 1)), 
+                    1 / (2 * depth + 1))
+            error_bound = D[-1] * (2 + error_factor)
+            return U[:, :rank], D[:rank], V[:rank], Q, error_bound
+
+        def _genetic_relatedness_vector_individual(
+            arr: np.ndarray,
+            centre: bool = True,
+            windows=None,
+        ) -> np.ndarray:
+            ij = np.vstack(
+                [
+                    [n, k]
+                    for k, i in enumerate(individuals)
+                    for n in self.individual(i).nodes
+                ]
+            )
+            samples, sample_individuals = (
+                ij[:, 0],
+                ij[:, 1],
+            )  # sample node index, individual of those nodes
+            x = (
+                arr - arr.mean(axis=0) if centre else arr
+            )  # centering within index in rows
+            x = self.genetic_relatedness_vector(
+                W=x[sample_individuals],
+                windows=windows,
+                mode=mode,
+                centre=False,
+                nodes=samples,
+            )[0]
+
+            def bincount_fn(w):
+                return np.bincount(sample_individuals, w)
+
+            x = np.apply_along_axis(bincount_fn, axis=0, arr=x)
+            x = x - x.mean(axis=0) if centre else x  # centering within index in cols
+
+            return x
+
+        def _genetic_relatedness_vector_node(
+            arr: np.ndarray,
+            centre: bool = True,
+            windows=None,
+        ) -> np.ndarray:
+            x = arr - arr.mean(axis=0) if centre else arr
+            x = self.genetic_relatedness_vector(
+                W=x, windows=windows, mode=mode, centre=False, nodes=samples
+            )[0]
+            x = x - x.mean(axis=0) if centre else x
+
+            return x
+
+        drop_windows = windows is None
+        windows = self.parse_windows(windows)
+        num_windows = len(windows) - 1
+        if num_windows < 1:
+            raise ValueError("Number of windows must be at least 1.")
+
+        U = np.empty((num_windows, dim, num_components))
+        D = np.empty((num_windows, num_components))
+        Q = np.empty((num_windows, dim, num_components + num_oversamples))
+        E = np.empty(num_windows)
+        for i in range(num_windows):
+            this_window = windows[i : i + 2]
+            _f = (
+                _genetic_relatedness_vector_node
+                if output_type == "node"
+                else _genetic_relatedness_vector_individual
+            )
+
+            def _G(x):
+                return _f(x, centre=centre, windows=this_window)  # NOQA: B023
+
+            U[i], D[i], _, Q[i], E[i] = _rand_svd(
+                operator=_G,
+                operator_dim=dim,
+                rank=num_components,
+                depth=iterated_power,
+                num_vectors=num_components + num_oversamples,
+                rng=random_state,
+                range_sketch=None if range_sketch is None else range_sketch[i],
+            )
+
+        if drop_windows:
+            U, D, Q = U[0], D[0], Q[0]
+
+        return U, D, Q, E
+
     def trait_covariance(self, W, windows=None, mode="site", span_normalise=True):
         """
         Computes the mean squared covariances between each of the columns of ``W``