diff --git a/include/ceres/internal/port.h b/include/ceres/internal/port.h
index f4dcaee..8660a66 100644
--- a/include/ceres/internal/port.h
+++ b/include/ceres/internal/port.h
@@ -51,9 +51,13 @@
 
 // We allocate some Eigen objects on the stack and other places they
 // might not be aligned to 16-byte boundaries.  If we have C++11, we
-// can specify their alignment anyway, and thus can safely enable
-// vectorization on those matrices; in C++99, we are out of luck.  Figure out
-// what case we're in and write macros that do the right thing.
+// can specify their alignment (which is desirable, as it means we can safely
+// enable vectorisation on matrices).  However, the standard gives wide
+// lattitude as to what alignments are legal.  It must be the case that
+// alignments up to alignof(std::max_align_t) are valid, but this might be < 16
+// on some platforms, in which case even if using C++11, on these platforms
+// we should not attempt to align to 16-byte boundaries.  If using < C++11,
+// we cannot specify the alignment.
 #ifdef CERES_USE_CXX11
 namespace port_constants {
 static constexpr size_t kMaxAlignBytes =
diff --git a/include/ceres/jet.h b/include/ceres/jet.h
index be0ccb0..e6adaa0 100644
--- a/include/ceres/jet.h
+++ b/include/ceres/jet.h
@@ -229,11 +229,13 @@
 
   // The infinitesimal part.
 
-  // We allocate Jets on the stack and other places they
-  // might not be aligned to 16-byte boundaries.  If we have C++11, we
-  // can specify their alignment anyway, and thus can safely enable
-  // vectorization on those matrices; in C++99, we are out of luck.  Figure out
-  // what case we're in and do the right thing.
+  // We allocate Jets on the stack and other places they might not be aligned
+  // to 16-byte boundaries, which would prevent the safe use of vectorisation.
+  // If we have C++11, we can specify the alignment.  However, the standard
+  // gives wide lattitude as to what alignments are valid, and it might be that
+  // the maximum supported alignment is < 16, in which case even with C++11, we
+  // cannot specify 16-byte alignment.  If using < C++11, we cannot specify
+  // alignment.
 #ifndef CERES_USE_CXX11
   // fall back to safe version:
   Eigen::Matrix<T, N, 1, Eigen::DontAlign> v;
@@ -242,7 +244,9 @@
       16 <= ::ceres::port_constants::kMaxAlignBytes;
   static constexpr int kAlignHint = kShouldAlignMatrix ?
       Eigen::AutoAlign : Eigen::DontAlign;
-  static constexpr size_t kAlignment = kShouldAlignMatrix ? 16 : 1;
+  // alignas(0) should always be ignored, in which case this definition of
+  // v should be equivalent to the non-C++11 case.
+  static constexpr size_t kAlignment = kShouldAlignMatrix ? 16 : 0;
   alignas(kAlignment) Eigen::Matrix<T, N, 1, kAlignHint> v;
 #endif
 };