sagemathgh-37606: Border matrix
    
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Plain TeX users may remember `\bordermatrix`. Here we implement this as
options of the `Matrix` class's `str` method.
```
            sage: M = matrix([[1,2,3], [4,5,6], [7,8,9]])
            sage: M.subdivide(None, 2)
            sage: print(M.str(unicode=True,
            ....:             top_border=['ab', 'cde', 'f'],
            ....:             bottom_border=['*', '', ''],
            ....:             left_border=[1, 10, 100],
            ....:             right_border=['', ' <', '']))
                        ab cde   f
                     1⎛  1   2│  3⎞
                    10⎜  4   5│  6⎟ <
                   100⎝  7   8│  9⎠
                         *
```

Follow-up PR: As the guiding application for this feature, we equip
finite-dimensional modules with basis with methods `_repr_matrix`,
`_ascii_art_matrix`, `_unicode_art_matrix` that can be used as in this
example:
```
                sage: M = matrix(ZZ, [[1, 0, 0], [0, 1, 0]],
                ....:            column_keys=['a', 'b', 'c'],
                ....:            row_keys=['v', 'w']); M
                Generic morphism:
                From: Free module generated by {'a', 'b', 'c'} over
Integer Ring
                To:   Free module generated by {'v', 'w'} over Integer
Ring
                sage: M._unicode_art_ = M._unicode_art_matrix
                sage: unicode_art(M)                            #
indirect doctest
                    a b c
                  v⎛1 0 0⎞
                  w⎝0 1 0⎠
```

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- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
- [ ] I have linked a relevant issue or discussion.
- [ ] I have created tests covering the changes.
- [ ] I have updated the documentation accordingly.

### ⌛ Dependencies

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URL: sagemath#37606
Reported by: Matthias Köppe
Reviewer(s): David Coudert, Matthias Köppe

Author: Release Manager

src/sage/matrix/matrix0.pyx

@@ -1796,7 +1796,9 @@ cdef class Matrix(sage.structure.element.Matrix):
         return self.str()
 
     def str(self, rep_mapping=None, zero=None, plus_one=None, minus_one=None,
-            *, unicode=False, shape=None, character_art=False):
+            *, unicode=False, shape=None, character_art=False,
+            left_border=None, right_border=None,
+            top_border=None, bottom_border=None):
         r"""
         Return a nice string representation of the matrix.
 
@@ -1846,6 +1848,16 @@ cdef class Matrix(sage.structure.element.Matrix):
           :class:`~sage.typeset.unicode_art.UnicodeArt` which support line
           breaking of wide matrices that exceed the window width
 
+        - ``left_border``, ``right_border`` -- sequence (default: ``None``);
+          if not ``None``, call :func:`str` on the elements and use the
+          results as labels for the rows of the matrix. The labels appear
+          outside of the parentheses.
+
+        - ``top_border``, ``bottom_border`` -- sequence (default: ``None``);
+          if not ``None``, call :func:`str` on the elements and use the
+          results as labels for the columns of the matrix. The labels appear
+          outside of the parentheses.
+
         EXAMPLES::
 
             sage: R = PolynomialRing(QQ,6,'z')
@@ -1921,6 +1933,21 @@ cdef class Matrix(sage.structure.element.Matrix):
             [ 0.333333333333333   66.6666666666667]
             [ -3.00000000000000 1.00000000000000e6]
 
+        Matrices with borders::
+
+            sage: M = matrix([[1,2,3], [4,5,6], [7,8,9]])
+            sage: M.subdivide(None, 2)
+            sage: print(M.str(unicode=True,
+            ....:             top_border=['ab', 'cde', 'f'],
+            ....:             bottom_border=['*', '', ''],
+            ....:             left_border=[1, 10, 100],
+            ....:             right_border=['', ' <', '']))
+                 ab cde   f
+              1⎛  1   2│  3⎞
+             10⎜  4   5│  6⎟ <
+            100⎝  7   8│  9⎠
+                  *
+
         TESTS:
 
         Prior to :issue:`11544` this could take a full minute to run (2011). ::
@@ -1958,6 +1985,32 @@ cdef class Matrix(sage.structure.element.Matrix):
             sage: matrix(R, [[2/3 - 10^6 * x^3 + 3 * y + O(x, y)^4]])
             [2/3 + 3*y - 1000000*x^3 + O(x, y)^4]
             sage: matrix.options._reset()
+
+        Edge cases of matrices with borders::
+
+            sage: print(matrix(ZZ, 0, 0).str(
+            ....:     top_border=[], bottom_border=[], left_border=[], right_border=[]))
+            []
+            sage: print(matrix(ZZ, 0, 4).str(
+            ....:     unicode=True,
+            ....:     top_border='abcd', bottom_border=range(4)))
+            ()
+            sage: print(matrix(ZZ, 1, 4).str(
+            ....:     unicode=True,
+            ....:     top_border='abcd', bottom_border=range(4)))
+             a b c d
+            (0 0 0 0)
+             0 1 2 3
+            sage: print(matrix(ZZ, 2, 4).str(
+            ....:     unicode=True,
+            ....:     top_border='abcd', bottom_border=range(4), left_border='uv'))
+              a b c d
+            u⎛0 0 0 0⎞
+            v⎝0 0 0 0⎠
+              0 1 2 3
+            sage: print(matrix(ZZ, 2, 0).str(
+            ....:     top_border='', left_border='uv', right_border=['*', '']))
+              []
         """
         cdef Py_ssize_t nr, nc, r, c
         nr = self._nrows
@@ -2063,6 +2116,12 @@ cdef class Matrix(sage.structure.element.Matrix):
 
         # compute column widths
         S = []
+        if top_border is not None:
+            for x in top_border:
+                S.append(str(x))
+            top_count = 1
+        else:
+            top_count = 0
         for x in entries:
             # Override the usual representations with those specified
             if callable(rep_mapping):
@@ -2075,39 +2134,83 @@ cdef class Matrix(sage.structure.element.Matrix):
             else:
                 rep = repr(x)
             S.append(rep)
+        if bottom_border is not None:
+            for x in bottom_border:
+                S.append(str(x))
+            bottom_count = 1
+        else:
+            bottom_count = 0
 
         width = max(map(len, S))
+        left = []
         rows = []
+        right = []
 
         hline = cl.join(hl * ((width + 1)*(b - a) - 1)
-                       for a,b in zip([0] + col_divs, col_divs + [nc]))
+                        for a,b in zip([0] + col_divs, col_divs + [nc]))
 
         # compute rows
-        for r from 0 <= r < nr:
-            rows += [hline] * row_divs.count(r)
+        for r in range(-top_count, nr + bottom_count):
+            if 0 <= r < nr:
+                n = row_divs.count(r)
+                if n:
+                    left.extend([""] * n)
+                    rows.extend([hline] * n)
+                    right.extend([""] * n)
+            if left_border is not None and 0 <= r < nr:
+                left.append(str(left_border[r]))
+            else:
+                left.append("")
             s = ""
             for c from 0 <= c < nc:
                 if col_div_counts[c]:
-                    sep = vl * col_div_counts[c]
+                    if 0 <= r < nr:
+                        sep = vl * col_div_counts[c]
+                    else:
+                        sep = " " * col_div_counts[c]
                 elif c == 0:
                     sep = ""
                 else:
                     sep = " "
-                entry = S[r * nc + c]
+                entry = S[(r + top_count) * nc + c]
                 entry = " " * (width - len(entry)) + entry
                 s = s + sep + entry
-            s = s + vl * col_div_counts[nc]
+            else:
+                if 0 <= r < nr:
+                    s = s + vl * col_div_counts[nc]
+                else:
+                    s = s + " " * col_div_counts[nc]
             rows.append(s)
-        rows += [hline] * row_divs.count(nr)
-
-        last_row = len(rows) - 1
-        if last_row == 0:
-            rows[0] = slb + rows[0] + srb
+            if right_border is not None and 0 <= r < nr:
+                right.append(str(right_border[r]))
+            else:
+                right.append("")
+        else:
+            if nr == nr + bottom_count:
+                n = row_divs.count(nr)
+                if n:
+                    left.extend([""] * n)
+                    rows.extend([hline] * n)
+                    right.extend([""] * n)
+
+        # left and right brackets
+        for i in range(top_count):
+            rows[i] = " "*len(slb) + rows[i] + " "*len(srb)
+        if len(rows) == top_count + 1 + bottom_count:
+            rows[top_count] = slb + rows[top_count] + srb
         else:
-            rows[0] = tlb + rows[0] + trb
-            for r from 1 <= r < last_row:
-                rows[r] = mlb + rows[r] + mrb
-            rows[last_row] = blb + rows[last_row] + brb
+            rows[top_count] = tlb + rows[top_count] + trb
+            for i in range(top_count + 1, len(rows) - bottom_count - 1):
+                rows[i] = mlb + rows[i] + mrb
+            rows[-1 - bottom_count] = blb + rows[-1 - bottom_count] + brb
+        for i in range(bottom_count):
+            rows[-1 - i] = " "*len(slb) + rows[-1 - i] + " "*len(srb)
+
+        # left and right border
+        left_width = max(len(s) for s in left)
+        right_width = max(len(s) for s in right)
+        for i in range(len(rows)):
+            rows[i] = left[i].rjust(left_width) + rows[i] + right[i].rjust(right_width)
 
         if character_art:
             breakpoints = []

src/sage/matrix/matrix_mod2_dense.pyx

@@ -346,7 +346,9 @@ cdef class Matrix_mod2_dense(matrix_dense.Matrix_dense):   # dense or sparse
 
 
     def str(self, rep_mapping=None, zero=None, plus_one=None, minus_one=None,
-            *, unicode=False, shape=None, character_art=False):
+            *, unicode=False, shape=None, character_art=False,
+            left_border=None, right_border=None,
+            top_border=None, bottom_border=None):
         r"""
         Return a nice string representation of the matrix.
 
@@ -384,6 +386,16 @@ cdef class Matrix_mod2_dense(matrix_dense.Matrix_dense):   # dense or sparse
           :class:`~sage.typeset.unicode_art.UnicodeArt` which support line
           breaking of wide matrices that exceed the window width
 
+        - ``left_border``, ``right_border`` -- sequence (default: ``None``);
+          if not ``None``, call :func:`str` on the elements and use the
+          results as labels for the rows of the matrix. The labels appear
+          outside of the parentheses.
+
+        - ``top_border``, ``bottom_border`` -- sequence (default: ``None``);
+          if not ``None``, call :func:`str` on the elements and use the
+          results as labels for the columns of the matrix. The labels appear
+          outside of the parentheses.
+
         EXAMPLES::
 
             sage: B = matrix(GF(2), 3, 3, [0, 1, 0, 0, 1, 1, 0, 0, 0])
@@ -416,13 +428,19 @@ cdef class Matrix_mod2_dense(matrix_dense.Matrix_dense):   # dense or sparse
         # Set the mapping based on keyword arguments
         # We ignore minus_one (it's only there for compatibility with Matrix)
         if (rep_mapping is not None or zero is not None or plus_one is not None
-                or unicode or shape is not None or character_art):
+                or unicode or shape is not None or character_art
+                or left_border is not None or right_border is not None
+                or top_border is not None or bottom_border is not None):
             # Shunt mappings off to the generic code since they might not be
             # single characters
             return matrix_dense.Matrix_dense.str(self, rep_mapping=rep_mapping,
                                                  zero=zero, plus_one=plus_one,
                                                  unicode=unicode, shape=shape,
-                                                 character_art=character_art)
+                                                 character_art=character_art,
+                                                 left_border=left_border,
+                                                 right_border=right_border,
+                                                 top_border=top_border,
+                                                 bottom_border=bottom_border)
 
         if self._nrows == 0 or self._ncols == 0:
             return "[]"