scikit-learn
diff --git a/‎doc/whats_new/v0.23.rst
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diff --git a/‎sklearn/decomposition/_kernel_pca.py
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diff --git a/‎sklearn/decomposition/tests/test_kernel_pca.py
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@@ -142,6 +142,10 @@ Changelog
   :func:`decomposition.non_negative_factorization` now preserves float32 dtype.
   :pr:`16280` by :user:`Jeremie du Boisberranger <jeremiedbb>`.
 
+- |Fix| :class:`decomposition.KernelPCA` method ``inverse_transform`` now
+  applies the correct inverse transform to the transformed data. :pr:`16655`
+  by :user:`Lewis Ball <lrjball>`.
+
 :mod:`sklearn.ensemble`
 .......................
 
 
@@ -358,5 +358,6 @@ def inverse_transform(self, X):
                                  "the inverse transform is not available.")
 
         K = self._get_kernel(X, self.X_transformed_fit_)
-
+        n_samples = self.X_transformed_fit_.shape[0]
+        K.flat[::n_samples + 1] += self.alpha
         return np.dot(K, self.dual_coef_)
@@ -7,6 +7,7 @@
 
 from sklearn.decomposition import PCA, KernelPCA
 from sklearn.datasets import make_circles
+from sklearn.datasets import make_blobs
 from sklearn.linear_model import Perceptron
 from sklearn.pipeline import Pipeline
 from sklearn.model_selection import GridSearchCV
@@ -282,3 +283,15 @@ def test_kernel_conditioning():
     # check that the small non-zero eigenvalue was correctly set to zero
     assert kpca.lambdas_.min() == 0
     assert np.all(kpca.lambdas_ == _check_psd_eigenvalues(kpca.lambdas_))
+
+
+@pytest.mark.parametrize("kernel",
+                         ["linear", "poly", "rbf", "sigmoid", "cosine"])
+def test_kernel_pca_inverse_transform(kernel):
+    X, *_ = make_blobs(n_samples=100, n_features=4, centers=[[1, 1, 1, 1]],
+                       random_state=0)
+
+    kp = KernelPCA(n_components=2, kernel=kernel, fit_inverse_transform=True)
+    X_trans = kp.fit_transform(X)
+    X_inv = kp.inverse_transform(X_trans)
+    assert_allclose(X, X_inv)