docs: move notes below the fold and highlight RFC 2119 keywords

src/array_api_stubs/_draft/linear_algebra_functions.py

@@ -8,47 +8,57 @@ def matmul(x1: array, x2: array, /) -> array:
     """
     Computes the matrix product.
 
-    .. note::
-       The ``matmul`` function must implement the same semantics as the built-in ``@`` operator (see `PEP 465 <https://www.python.org/dev/peps/pep-0465>`_).
-
     Parameters
     ----------
     x1: array
-        first input array. Should have a numeric data type. Must have at least one dimension. If ``x1`` is one-dimensional having shape ``(M,)`` and ``x2`` has more than one dimension, ``x1`` must be promoted to a two-dimensional array by prepending ``1`` to its dimensions (i.e., must have shape ``(1, M)``). After matrix multiplication, the prepended dimensions in the returned array must be removed. If ``x1`` has more than one dimension (including after vector-to-matrix promotion), ``shape(x1)[:-2]`` must be compatible with ``shape(x2)[:-2]`` (after vector-to-matrix promotion) (see :ref:`broadcasting`). If ``x1`` has shape ``(..., M, K)``, the innermost two dimensions form matrices on which to perform matrix multiplication.
-    x2: array
-        second input array. Should have a numeric data type. Must have at least one dimension. If ``x2`` is one-dimensional having shape ``(N,)`` and ``x1`` has more than one dimension, ``x2`` must be promoted to a two-dimensional array by appending ``1`` to its dimensions (i.e., must have shape ``(N, 1)``). After matrix multiplication, the appended dimensions in the returned array must be removed. If ``x2`` has more than one dimension (including after vector-to-matrix promotion), ``shape(x2)[:-2]`` must be compatible with ``shape(x1)[:-2]`` (after vector-to-matrix promotion) (see :ref:`broadcasting`). If ``x2`` has shape ``(..., K, N)``, the innermost two dimensions form matrices on which to perform matrix multiplication.
+        first input array. **Should** have a numeric data type. **Must** have at least one dimension.
+
+        -   If ``x1`` is a one-dimensional array having shape ``(M,)`` and ``x2`` has more than one dimension, ``x1`` **must** be promoted to a two-dimensional array by prepending ``1`` to its dimensions (i.e., **must** have shape ``(1, M)``). After matrix multiplication, the prepended dimensions in the returned array **must** be removed.
+        -   If ``x1`` has more than one dimension (including after vector-to-matrix promotion), ``shape(x1)[:-2]`` **must** be compatible with ``shape(x2)[:-2]`` (after vector-to-matrix promotion) (see :ref:`broadcasting`).
+        -   If ``x1`` has shape ``(..., M, K)``, the innermost two dimensions form matrices on which to perform matrix multiplication.
 
+    x2: array
+        second input array. **Should** have a numeric data type. **Must** have at least one dimension.
 
-    .. note::
-       If either ``x1`` or ``x2`` has a complex floating-point data type, neither argument must be complex-conjugated or transposed. If conjugation and/or transposition is desired, these operations should be explicitly performed prior to computing the matrix product.
+        -   If ``x2`` is one-dimensional array having shape ``(N,)`` and ``x1`` has more than one dimension, ``x2`` **must** be promoted to a two-dimensional array by appending ``1`` to its dimensions (i.e., **must** have shape ``(N, 1)``). After matrix multiplication, the appended dimensions in the returned array **must** be removed.
+        -   If ``x2`` has more than one dimension (including after vector-to-matrix promotion), ``shape(x2)[:-2]`` **must** be compatible with ``shape(x1)[:-2]`` (after vector-to-matrix promotion) (see :ref:`broadcasting`).
+        -   If ``x2`` has shape ``(..., K, N)``, the innermost two dimensions form matrices on which to perform matrix multiplication.
 
     Returns
     -------
     out: array
-        -   if both ``x1`` and ``x2`` are one-dimensional arrays having shape ``(N,)``, a zero-dimensional array containing the inner product as its only element.
-        -   if ``x1`` is a two-dimensional array having shape ``(M, K)`` and ``x2`` is a two-dimensional array having shape ``(K, N)``, a two-dimensional array containing the `conventional matrix product <https://en.wikipedia.org/wiki/Matrix_multiplication>`_ and having shape ``(M, N)``.
-        -   if ``x1`` is a one-dimensional array having shape ``(K,)`` and ``x2`` is an array having shape ``(..., K, N)``, an array having shape ``(..., N)`` (i.e., prepended dimensions during vector-to-matrix promotion must be removed) and containing the `conventional matrix product <https://en.wikipedia.org/wiki/Matrix_multiplication>`_.
-        -   if ``x1`` is an array having shape ``(..., M, K)`` and ``x2`` is a one-dimensional array having shape ``(K,)``, an array having shape ``(..., M)`` (i.e., appended dimensions during vector-to-matrix promotion must be removed) and containing the `conventional matrix product <https://en.wikipedia.org/wiki/Matrix_multiplication>`_.
-        -   if ``x1`` is a two-dimensional array having shape ``(M, K)`` and ``x2`` is an array having shape ``(..., K, N)``, an array having shape ``(..., M, N)`` and containing the `conventional matrix product <https://en.wikipedia.org/wiki/Matrix_multiplication>`_ for each stacked matrix.
-        -   if ``x1`` is an array having shape ``(..., M, K)`` and ``x2`` is a two-dimensional array having shape ``(K, N)``, an array having shape ``(..., M, N)`` and containing the `conventional matrix product <https://en.wikipedia.org/wiki/Matrix_multiplication>`_ for each stacked matrix.
-        -   if either ``x1`` or ``x2`` has more than two dimensions, an array having a shape determined by :ref:`broadcasting` ``shape(x1)[:-2]`` against ``shape(x2)[:-2]`` and containing the `conventional matrix product <https://en.wikipedia.org/wiki/Matrix_multiplication>`_ for each stacked matrix.
+        output array.
+
+        -   If both ``x1`` and ``x2`` are one-dimensional arrays having shape ``(N,)``, the returned array **must** be a zero-dimensional array and **must** contain the inner product as its only element.
+        -   If ``x1`` is a two-dimensional array having shape ``(M, K)`` and ``x2`` is a two-dimensional array having shape ``(K, N)``, the returned array **must** be a two-dimensional array and **must** contain the `conventional matrix product <https://en.wikipedia.org/wiki/Matrix_multiplication>`_ and having shape ``(M, N)``.
+        -   If ``x1`` is a one-dimensional array having shape ``(K,)`` and ``x2`` is an array having shape ``(..., K, N)``, the returned array **must** be an array having shape ``(..., N)`` (i.e., prepended dimensions during vector-to-matrix promotion **must** be removed) and **must** contain the `conventional matrix product <https://en.wikipedia.org/wiki/Matrix_multiplication>`_.
+        -   If ``x1`` is an array having shape ``(..., M, K)`` and ``x2`` is a one-dimensional array having shape ``(K,)``, the returned array **must** be an array having shape ``(..., M)`` (i.e., appended dimensions during vector-to-matrix promotion **must** be removed) and **must** contain the `conventional matrix product <https://en.wikipedia.org/wiki/Matrix_multiplication>`_.
+        -   If ``x1`` is a two-dimensional array having shape ``(M, K)`` and ``x2`` is an array having shape ``(..., K, N)``, the returned array **must** be an array having shape ``(..., M, N)`` and **must** contain the `conventional matrix product <https://en.wikipedia.org/wiki/Matrix_multiplication>`_ for each stacked matrix.
+        -   If ``x1`` is an array having shape ``(..., M, K)`` and ``x2`` is a two-dimensional array having shape ``(K, N)``, the returned array **must** be an array having shape ``(..., M, N)`` and **must** contain the `conventional matrix product <https://en.wikipedia.org/wiki/Matrix_multiplication>`_ for each stacked matrix.
+        -   If either ``x1`` or ``x2`` has more than two dimensions, the returned array **must** be an array having a shape determined by :ref:`broadcasting` ``shape(x1)[:-2]`` against ``shape(x2)[:-2]`` and **must** contain the `conventional matrix product <https://en.wikipedia.org/wiki/Matrix_multiplication>`_ for each stacked matrix.
+
+        The returned array **must** have a data type determined by :ref:`type-promotion`.
 
-        The returned array must have a data type determined by :ref:`type-promotion`.
+    Raises
+    ------
+    Exception
+        an exception **should** be raised in the following circumstances:
+
+        -   if either ``x1`` or ``x2`` is a zero-dimensional array.
+        -   if ``x1`` is a one-dimensional array having shape ``(K,)``, ``x2`` is a one-dimensional array having shape ``(L,)``, and ``K != L``.
+        -   if ``x1`` is a one-dimensional array having shape ``(K,)``, ``x2`` is an array having shape ``(..., L, N)``, and ``K != L``.
+        -   if ``x1`` is an array having shape ``(..., M, K)``, ``x2`` is a one-dimensional array having shape ``(L,)``, and ``K != L``.
+        -   if ``x1`` is an array having shape ``(..., M, K)``, ``x2`` is an array having shape ``(..., L, N)``, and ``K != L``.
 
     Notes
     -----
 
-    .. versionchanged:: 2022.12
-       Added complex data type support.
+    -   The ``matmul`` function **must** implement the same semantics as the built-in ``@`` operator (see `PEP 465 <https://www.python.org/dev/peps/pep-0465>`_).
 
-    **Raises**
-
-    -   if either ``x1`` or ``x2`` is a zero-dimensional array.
-    -   if ``x1`` is a one-dimensional array having shape ``(K,)``, ``x2`` is a one-dimensional array having shape ``(L,)``, and ``K != L``.
-    -   if ``x1`` is a one-dimensional array having shape ``(K,)``, ``x2`` is an array having shape ``(..., L, N)``, and ``K != L``.
-    -   if ``x1`` is an array having shape ``(..., M, K)``, ``x2`` is a one-dimensional array having shape ``(L,)``, and ``K != L``.
-    -   if ``x1`` is an array having shape ``(..., M, K)``, ``x2`` is an array having shape ``(..., L, N)``, and ``K != L``.
+    -   If either ``x1`` or ``x2`` has a complex floating-point data type, the function **must not** complex-conjugate or tranpose either argument. If conjugation and/or transposition is desired, a user can explicitly perform these operations prior to computing the matrix product.
 
+    .. versionchanged:: 2022.12
+       Added complex data type support.
     """
 
 
@@ -64,7 +74,7 @@ def matrix_transpose(x: array, /) -> array:
     Returns
     -------
     out: array
-        an array containing the transpose for each matrix and having shape ``(..., N, M)``. The returned array must have the same data type as ``x``.
+        an array containing the transpose for each matrix. The returned array **must** have shape ``(..., N, M)``. The returned array **must** have the same data type as ``x``.
     """
 
 
@@ -78,42 +88,37 @@ def tensordot(
     """
     Returns a tensor contraction of ``x1`` and ``x2`` over specific axes.
 
-    .. note::
-       The ``tensordot`` function corresponds to the generalized matrix product.
-
     Parameters
     ----------
     x1: array
-        first input array. Should have a numeric data type.
+        first input array. **Should** have a numeric data type.
     x2: array
-        second input array. Should have a numeric data type. Corresponding contracted axes of ``x1`` and ``x2`` must be equal.
-
-        .. note::
-           Contracted axes (dimensions) must not be broadcasted.
+        second input array. **Should** have a numeric data type. Corresponding contracted axes of ``x1`` and ``x2`` **must** be equal.
 
     axes: Union[int, Tuple[Sequence[int], Sequence[int]]]
-        number of axes (dimensions) to contract or explicit sequences of axis (dimension) indices for ``x1`` and ``x2``, respectively.
+        number of axes to contract or explicit sequences of axis indices for ``x1`` and ``x2``, respectively.
 
-        If ``axes`` is an ``int`` equal to ``N``, then contraction must be performed over the last ``N`` axes of ``x1`` and the first ``N`` axes of ``x2`` in order. The size of each corresponding axis (dimension) must match. Must be nonnegative.
+        If ``axes`` is an ``int`` equal to ``N``, then contraction **must** be performed over the last ``N`` axes of ``x1`` and the first ``N`` axes of ``x2`` in order. The size of each corresponding axis **must** match. An integer ``axes`` value **must** be nonnegative.
 
-        -   If ``N`` equals ``0``, the result is the tensor (outer) product.
-        -   If ``N`` equals ``1``, the result is the tensor dot product.
-        -   If ``N`` equals ``2``, the result is the tensor double contraction (default).
+        -   If ``N`` equals ``0``, the result **must** be the tensor (outer) product.
+        -   If ``N`` equals ``1``, the result **must** be the tensor dot product.
+        -   If ``N`` equals ``2``, the result **must** be the tensor double contraction (default).
 
-        If ``axes`` is a tuple of two sequences ``(x1_axes, x2_axes)``, the first sequence must apply to ``x1`` and the second sequence to ``x2``. Both sequences must have the same length. Each axis (dimension) ``x1_axes[i]`` for ``x1`` must have the same size as the respective axis (dimension) ``x2_axes[i]`` for ``x2``. Each index referred to in a sequence must be unique. If ``x1`` has rank (i.e, number of dimensions) ``N``, a valid ``x1`` axis must reside on the half-open interval ``[-N, N)``. If ``x2`` has rank ``M``, a valid ``x2`` axis must reside on the half-open interval ``[-M, M)``.
+        If ``axes`` is a tuple of two sequences ``(x1_axes, x2_axes)``, the first sequence **must** apply to ``x1`` and the second sequence **must** apply to ``x2``. Both sequences **must** have the same length. Each axis ``x1_axes[i]`` for ``x1`` **must** have the same size as the respective axis ``x2_axes[i]`` for ``x2``. Each index referred to in a sequence **must** be unique. A valid axis **must** be an integer on the interval ``[-S, S)``, where ``S`` is the number of axes in respective array. Hence, if ``x1`` has ``N`` axes, a valid ``x1`` axes **must** be an integer on the interval ``[-N, N)``. If ``x2`` has ``M`` axes, a valid ``x2`` axes **must** be an integer on the interval ``[-M, M)``. If an axis is specified as a negative integer, the function **must** determine the axis along which to perform the operation by counting backward from the last axis (where ``-1`` refers to the last axis). If provided an invalid axis, the function **must** raise an exception.
 
 
-    .. note::
-       If either ``x1`` or ``x2`` has a complex floating-point data type, neither argument must be complex-conjugated or transposed. If conjugation and/or transposition is desired, these operations should be explicitly performed prior to computing the generalized matrix product.
-
     Returns
     -------
     out: array
-        an array containing the tensor contraction whose shape consists of the non-contracted axes (dimensions) of the first array ``x1``, followed by the non-contracted axes (dimensions) of the second array ``x2``. The returned array must have a data type determined by :ref:`type-promotion`.
+        an array containing the tensor contraction. The returned array **must** have a shape which consists of the non-contracted axes of the first array ``x1``, followed by the non-contracted axes of the second array ``x2``. The returned array **must** have a data type determined by :ref:`type-promotion`.
 
     Notes
     -----
 
+    -   The ``tensordot`` function corresponds to the generalized matrix product.
+    -   Contracted axes **must** not be broadcasted.
+    -   If either ``x1`` or ``x2`` has a complex floating-point data type, the function **must not** complex-conjugate or transpose either argument. If conjugation and/or transposition is desired, a user can explicitly perform these operations prior to computing the generalized matrix product.
+
     .. versionchanged:: 2022.12
        Added complex data type support.
 
@@ -131,32 +136,33 @@ def vecdot(x1: array, x2: array, /, *, axis: int = -1) -> array:
     .. math::
        \mathbf{a} \cdot \mathbf{b} = \sum_{i=0}^{n-1} \overline{a_i}b_i
 
-    over the dimension specified by ``axis`` and where :math:`n` is the dimension size and :math:`\overline{a_i}` denotes the complex conjugate if :math:`a_i` is complex and the identity if :math:`a_i` is real-valued.
+    over the axis specified by ``axis`` and where :math:`n` is the axis size and :math:`\overline{a_i}` denotes the complex conjugate if :math:`a_i` is complex and the identity if :math:`a_i` is real-valued.
 
     Parameters
     ----------
     x1: array
-        first input array. Should have a floating-point data type.
+        first input array. **Should** have a floating-point data type.
     x2: array
-        second input array. Must be compatible with ``x1`` for all non-contracted axes (see :ref:`broadcasting`). The size of the axis over which to compute the dot product must be the same size as the respective axis in ``x1``. Should have a floating-point data type.
-
-        .. note::
-           The contracted axis (dimension) must not be broadcasted.
-
+        second input array. **Must** be compatible with ``x1`` for all non-contracted axes (see :ref:`broadcasting`). The size of the axis over which to compute the dot product **must** be the same size as the respective axis in ``x1``. **Should** have a floating-point data type.
     axis: int
-        the axis (dimension) of ``x1`` and ``x2`` containing the vectors for which to compute the dot product. Should be an integer on the interval ``[-N, -1]``, where ``N`` is ``min(x1.ndim, x2.ndim)``. The function must determine the axis along which to compute the dot product by counting backward from the last dimension (where ``-1`` refers to the last dimension). By default, the function must compute the dot product over the last axis. Default: ``-1``.
+        axis of ``x1`` and ``x2`` containing the vectors for which to compute the dot product. **Should** be an integer on the interval ``[-N, -1]``, where ``N`` is ``min(x1.ndim, x2.ndim)``. The function **must** determine the axis along which to perform the operation by counting backward from the last axis (where ``-1`` refers to the last axis). By default, the function **must** compute the dot product over the last axis. Default: ``-1``.
 
     Returns
     -------
     out: array
-        if ``x1`` and ``x2`` are both one-dimensional arrays, a zero-dimensional containing the dot product; otherwise, a non-zero-dimensional array containing the dot products and having rank ``N-1``, where ``N`` is the rank (number of dimensions) of the shape determined according to :ref:`broadcasting` along the non-contracted axes. The returned array must have a data type determined by :ref:`type-promotion`.
+        if ``x1`` and ``x2`` are both one-dimensional arrays, a zero-dimensional containing the dot product; otherwise, a non-zero-dimensional array containing the dot products and having ``N-1`` axes, where ``N`` is number of axes in the shape determined according to :ref:`broadcasting` along the non-contracted axes. The returned array **must** have a data type determined by :ref:`type-promotion`.
+
+    Raises
+    ------
+    Exception
+        an exception **should** be raised in the following circumstances:
+
+        -   if the size of the axis over which to compute the dot product is not the same (before broadcasting) for both ``x1`` and ``x2``.
 
     Notes
     -----
 
-    **Raises**
-
-    -   if the size of the axis over which to compute the dot product is not the same (before broadcasting) for both ``x1`` and ``x2``.
+    -   The contracted axis **must** not be broadcasted.
 
     .. versionchanged:: 2022.12
        Added complex data type support.