def minor_at_element(matrix, row, column):
# Handle input errors
square_matrix(matrix)
whole_number(row, 'second')
whole_number(column, 'third')
# Create intermediary dictionary and list to return
storage = {}
result = []
# Iterate over outer lists of input
for m in range(len(matrix)):
if m != row:
storage[m] = []
# Iterate over inner lists of input
for n in range(len(matrix[0])):
if n != column:
storage[m].append(matrix[m][n])
# Iterate over keys in dictionary
for key in storage:
result.append(storage[key])
# Return result
return result
def linear_determinant(matrix, result=0):
"""
Calculate the determinant of a matrix
Parameters
----------
matrix : list of lists of int or float
List of lists of numbers representing a matrix
Raises
------
TypeError
First argument must be a 2-dimensional list
TypeError
Elements nested within first argument must be integers or floats
ValueError
First argument must contain the same amount of lists as the amount of elements contained within its first list
Returns
-------
determinant : float
Determinant of a matrix
Warning
-------
Function has factorial time complexity; not recommended for matrices larger than 7-by-7
See Also
--------
:func:`~regressions.matrices.minors.matrix_of_minors`, :func:`~regressions.matrices.inverse.inverse_matrix`
Notes
-----
- Original matrix: :math:`\\mathbf{A} = \\begin{bmatrix} a_{1,1} & a_{1,2} & \\cdots & a_{1,j} & a_{1,n} \\\\ a_{2,1} & a_{2,2} & \\cdots & a_{2,j} & a_{2,n} \\\\ a_{i,1} & a_{i,2} & \\cdots & a_{i,j} & a_{i,n} \\\\ \\cdots & \\cdots & \\cdots & \\cdots & \\cdots \\\\ a_{m,1} & a_{m,2} & \\cdots & a_{m,j} & a_{m,n} \\end{bmatrix}`
- Determinant of matrix (if :math:`\\mathbf{A}` contains an odd number of columns): :math:`|\\mathbf{A}| = a_{1,1}\\cdot{|\\mathbf{A}_{1,1}|} - a_{1,2}\\cdot{|\\mathbf{A}_{1,2}|} + \\cdots - a_{1,j}\\cdot{|\\mathbf{A}_{1,j}|} + a_{1,n}\\cdot{|\\mathbf{A}_{1,n}|}`
- Determinant of matrix (if :math:`\\mathbf{A}` contains an even number of columns): :math:`|\\mathbf{A}| = a_{1,1}\\cdot{|\\mathbf{A}_{1,1}|} - a_{1,2}\\cdot{|\\mathbf{A}_{1,2}|} + \\cdots + a_{1,j}\\cdot{|\\mathbf{A}_{1,j}|} - a_{1,n}\\cdot{|\\mathbf{A}_{1,n}|}`
- |determinant|
Examples
--------
Import `linear_determinant` function from `regressions` library
>>> from regressions.matrices.determinant import linear_determinant
Calculate the determinant of [[1, 2], [3, 4]]
>>> determinant_2x2 = linear_determinant([[1, 2], [3, 4]])
>>> print(determinant_2x2)
-2.0
Calculate the determinant of [[2, 3, 5], [7, 11, 13], [17, 19, 23]]
>>> determinant_3x3 = linear_determinant([[2, 3, 5], [7, 11, 13], [17, 19, 23]])
>>> print(determinant_3x3)
-78.0
"""
# Handle input errors
square_matrix(matrix)
whole_number(result, 'second')
# Handle base case
if len(matrix) == 1:
result += float(matrix[0][0])
return result
# Handle recursive case
else:
# Create intermediary variables
alternating = []
minors = []
leads = matrix[0]
# Iterate over first inner list of input
for i in range(len(leads)):
minors.append(minor_at_element(matrix, 0, i))
if i % 2 == 0:
alternating.append(leads[i])
else:
alternating.append(-1 * leads[i])
# Iterate over alternating
for j in range(len(alternating)):
result += alternating[j] * linear_determinant(minors[j])
# Convert result to float
floated_result = float(result)
return floated_result
def test_whole_array_raises(self):
with self.assertRaises(Exception) as context:
whole_number(choices, 'second')
self.assertEqual(type(context.exception), ValueError)
self.assertEqual(str(context.exception),
'Second argument must be a whole number')
def test_whole_negative_raises(self):
with self.assertRaises(Exception) as context:
whole_number(good_integer, 'second')
self.assertEqual(type(context.exception), ValueError)
self.assertEqual(str(context.exception),
'Second argument must be a whole number')
def test_whole_natural(self):
whole_natural = whole_number(good_positive, 'third')
self.assertEqual(whole_natural, 'Third argument is a whole number')
def test_whole_zero(self):
whole_zero = whole_number(good_whole, 'second')
self.assertEqual(whole_zero, 'Second argument is a whole number')