Add complex number support to linalg.cholesky (#443)

* Add equations

* Fix syntax

* Update description and equations

* Comment out MathJax config

* Remove config due to failing build

This config does not seem to be needed. SymPy does not include it
in its docs. Seems fine for us to omit, as well, given that it
causes builds to fail.

* Use raw string

* Remove backlash

Author: kgryte

spec/API_specification/array_api/linalg.py

@@ -2,26 +2,41 @@
 from .constants import inf
 
 def cholesky(x: array, /, *, upper: bool = False) -> array:
-    """
-    Returns the lower (upper) Cholesky decomposition x = LLᵀ (x = UᵀU) of a symmetric positive-definite matrix (or a stack of matrices) ``x``, where ``L`` is a lower-triangular matrix or a stack of matrices (``U`` is an upper-triangular matrix or a stack of matrices).
+    r"""
+    Returns the lower (upper) Cholesky decomposition of a complex Hermitian or real symmetric positive-definite matrix ``x``.
 
-    ..
-      NOTE: once complex numbers are supported, each square matrix must be Hermitian.
+    If ``x`` is real-valued, let :math:`\mathbb{K}` be the set of real numbers $\mathbb{R}$, and, if ``x`` is complex-valued, let $\mathbb{K}$ be the set of complex numbers $\mathbb{C}$.
+
+    The lower **Cholesky decomposition** of a complex Hermitian or real symmetric positive-definite matrix :math:`x \in\ \mathbb{K}^{n \times n}` is defined as
+
+    .. math::
+       x = LL^{H} \qquad \text{L $\in\ \mathbb{K}^{n \times n}$}
+
+    where :math:`L` is a lower triangular matrix and :math:`L^{H}` is the conjugate transpose when :math:`L` is complex-valued and the transpose when :math:`L` is real-valued.
+
+    The upper Cholesky decomposition is defined similarly
+
+    .. math::
+       x = UU^{H} \qquad \text{U $\in\ \mathbb{K}^{n \times n}$}
+
+    where :math:`U` is an upper triangular matrix.
+
+    When ``x`` is a stack of matrices, the function must compute the Cholesky decomposition for each matrix in the stack.
 
     .. note::
-       Whether an array library explicitly checks whether an input array is a symmetric positive-definite matrix (or a stack of matrices) is implementation-defined.
+       Whether an array library explicitly checks whether an input array is Hermitian or a symmetric positive-definite matrix (or a stack of matrices) is implementation-defined.
 
     Parameters
     ----------
     x: array
-        input array having shape ``(..., M, M)`` and whose innermost two dimensions form square symmetric positive-definite matrices. Should have a real-valued floating-point data type.
+        input array having shape ``(..., M, M)`` and whose innermost two dimensions form square complex Hermitian or real symmetric positive-definite matrices. Should have a floating-point data type.
     upper: bool
-        If ``True``, the result must be the upper-triangular Cholesky factor ``U``. If ``False``, the result must be the lower-triangular Cholesky factor ``L``. Default: ``False``.
+        If ``True``, the result must be the upper-triangular Cholesky factor :math:`U`. If ``False``, the result must be the lower-triangular Cholesky factor :math:`L`. Default: ``False``.
 
     Returns
     -------
     out: array
-        an array containing the Cholesky factors for each square matrix. If ``upper`` is ``False``, the returned array must contain lower-triangular matrices; otherwise, the returned array must contain upper-triangular matrices. The returned array must have a real-valued floating-point data type determined by :ref:`type-promotion` and must have the same shape as ``x``.
+        an array containing the Cholesky factors for each square matrix. If ``upper`` is ``False``, the returned array must contain lower-triangular matrices; otherwise, the returned array must contain upper-triangular matrices. The returned array must have a floating-point data type determined by :ref:`type-promotion` and must have the same shape as ``x``.
     """
 
 def cross(x1: array, x2: array, /, *, axis: int = -1) -> array:

spec/conf.py

@@ -50,14 +50,7 @@
 napoleon_custom_sections = [('Returns', 'params_style')]
 default_role = 'code'
 
-# Mathjax configuration
-mathjax_config = {
-    'tex2jax': {
-        'inlineMath': [ ["\\(","\\)"] ],
-        'displayMath': [["\\[","\\]"] ],
-    },
-}
-
+# Mathjax configuration:
 mathjax3_config = {
   "tex": {
     "inlineMath": [['\\(', '\\)']],