code: def _householder_vector(x):
      if not x.cols == 1:
          raise ValueError("Input must be a column matrix")
      v = x.copy()
      v_plus = x.copy()
      v_minus = x.copy()
      q = x[0, 0] / abs(x[0, 0])
      norm_x = x.norm()
      v_plus[0, 0] = x[0, 0] + q * norm_x
      v_minus[0, 0] = x[0, 0] - q * norm_x
      if x[1:, 0].norm() == 0:
          bet = 0
          v[0, 0] = 1
      else:
          if v_plus.norm() <= v_minus.norm():
              v = v_plus
          else:
              v = v_minus
          v = v / v[0]
          bet = 2 / (v.norm() ** 2)
      return v, bet
  
imports: from types import FunctionType
  from collections import Counter
  from mpmath import mp, workprec
  from mpmath.libmp.libmpf import prec_to_dps
  from sympy.core.sorting import default_sort_key
  from sympy.core.evalf import DEFAULT_MAXPREC, PrecisionExhausted
  from sympy.core.logic import fuzzy_and, fuzzy_or
  from sympy.core.numbers import Float
  from sympy.core.sympify import _sympify
  from sympy.functions.elementary.miscellaneous import sqrt
  from sympy.polys import roots, CRootOf, ZZ, QQ, EX
  from sympy.polys.matrices import DomainMatrix
  from sympy.polys.matrices.eigen import dom_eigenvects, dom_eigenvects_to_sympy
  from sympy.polys.polytools import gcd
  from .exceptions import MatrixError, NonSquareMatrixError
  from .determinant import _find_reasonable_pivot
  from .utilities import _iszero, _simplify
label_desc: Use image node_id 13 for calling a global function with example usage: _householder_vector(x) and returns: v,  bet