#### Strassen Matrix Multiplication

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```from __future__ import annotations

import math

def default_matrix_multiplication(a: list, b: list) -> list:
"""
Multiplication only for 2x2 matrices
"""
if len(a) != 2 or len(a[0]) != 2 or len(b) != 2 or len(b[0]) != 2:
raise Exception("Matrices are not 2x2")
new_matrix = [
[a[0][0] * b[0][0] + a[0][1] * b[1][0], a[0][0] * b[0][1] + a[0][1] * b[1][1]],
[a[1][0] * b[0][0] + a[1][1] * b[1][0], a[1][0] * b[0][1] + a[1][1] * b[1][1]],
]
return new_matrix

def matrix_addition(matrix_a: list, matrix_b: list):
return [
[matrix_a[row][col] + matrix_b[row][col] for col in range(len(matrix_a[row]))]
for row in range(len(matrix_a))
]

def matrix_subtraction(matrix_a: list, matrix_b: list):
return [
[matrix_a[row][col] - matrix_b[row][col] for col in range(len(matrix_a[row]))]
for row in range(len(matrix_a))
]

def split_matrix(a: list) -> tuple[list, list, list, list]:
"""
Given an even length matrix, returns the top_left, top_right, bot_left, bot_right
quadrant.

>>> split_matrix([[4,3,2,4],[2,3,1,1],[6,5,4,3],[8,4,1,6]])
([[4, 3], [2, 3]], [[2, 4], [1, 1]], [[6, 5], [8, 4]], [[4, 3], [1, 6]])
>>> split_matrix([
...     [4,3,2,4,4,3,2,4],[2,3,1,1,2,3,1,1],[6,5,4,3,6,5,4,3],[8,4,1,6,8,4,1,6],
...     [4,3,2,4,4,3,2,4],[2,3,1,1,2,3,1,1],[6,5,4,3,6,5,4,3],[8,4,1,6,8,4,1,6]
... ])  # doctest: +NORMALIZE_WHITESPACE
([[4, 3, 2, 4], [2, 3, 1, 1], [6, 5, 4, 3], [8, 4, 1, 6]], [[4, 3, 2, 4],
[2, 3, 1, 1], [6, 5, 4, 3], [8, 4, 1, 6]], [[4, 3, 2, 4], [2, 3, 1, 1],
[6, 5, 4, 3], [8, 4, 1, 6]], [[4, 3, 2, 4], [2, 3, 1, 1], [6, 5, 4, 3],
[8, 4, 1, 6]])
"""
if len(a) % 2 != 0 or len(a[0]) % 2 != 0:
raise Exception("Odd matrices are not supported!")

matrix_length = len(a)
mid = matrix_length // 2

top_right = [[a[i][j] for j in range(mid, matrix_length)] for i in range(mid)]
bot_right = [
[a[i][j] for j in range(mid, matrix_length)] for i in range(mid, matrix_length)
]

top_left = [[a[i][j] for j in range(mid)] for i in range(mid)]
bot_left = [[a[i][j] for j in range(mid)] for i in range(mid, matrix_length)]

return top_left, top_right, bot_left, bot_right

def matrix_dimensions(matrix: list) -> tuple[int, int]:
return len(matrix), len(matrix[0])

def print_matrix(matrix: list) -> None:
print("\n".join(str(line) for line in matrix))

def actual_strassen(matrix_a: list, matrix_b: list) -> list:
"""
Recursive function to calculate the product of two matrices, using the Strassen
Algorithm.  It only supports even length matrices.
"""
if matrix_dimensions(matrix_a) == (2, 2):
return default_matrix_multiplication(matrix_a, matrix_b)

a, b, c, d = split_matrix(matrix_a)
e, f, g, h = split_matrix(matrix_b)

t1 = actual_strassen(a, matrix_subtraction(f, h))
t2 = actual_strassen(matrix_addition(a, b), h)
t3 = actual_strassen(matrix_addition(c, d), e)
t4 = actual_strassen(d, matrix_subtraction(g, e))
t5 = actual_strassen(matrix_addition(a, d), matrix_addition(e, h))
t6 = actual_strassen(matrix_subtraction(b, d), matrix_addition(g, h))
t7 = actual_strassen(matrix_subtraction(a, c), matrix_addition(e, f))

top_left = matrix_addition(matrix_subtraction(matrix_addition(t5, t4), t2), t6)
top_right = matrix_addition(t1, t2)
bot_left = matrix_addition(t3, t4)
bot_right = matrix_subtraction(matrix_subtraction(matrix_addition(t1, t5), t3), t7)

# construct the new matrix from our 4 quadrants
new_matrix = []
for i in range(len(top_right)):
new_matrix.append(top_left[i] + top_right[i])
for i in range(len(bot_right)):
new_matrix.append(bot_left[i] + bot_right[i])
return new_matrix

def strassen(matrix1: list, matrix2: list) -> list:
"""
>>> strassen([[2,1,3],[3,4,6],[1,4,2],[7,6,7]], [[4,2,3,4],[2,1,1,1],[8,6,4,2]])
[[34, 23, 19, 15], [68, 46, 37, 28], [28, 18, 15, 12], [96, 62, 55, 48]]
>>> strassen([[3,7,5,6,9],[1,5,3,7,8],[1,4,4,5,7]], [[2,4],[5,2],[1,7],[5,5],[7,8]])
[[139, 163], [121, 134], [100, 121]]
"""
if matrix_dimensions(matrix1)[1] != matrix_dimensions(matrix2)[0]:
msg = (
"Unable to multiply these matrices, please check the dimensions.\n"
f"Matrix A: {matrix1}\n"
f"Matrix B: {matrix2}"
)
raise Exception(msg)
dimension1 = matrix_dimensions(matrix1)
dimension2 = matrix_dimensions(matrix2)

if dimension1[0] == dimension1[1] and dimension2[0] == dimension2[1]:
return [matrix1, matrix2]

maximum = max(*dimension1, *dimension2)
maxim = int(math.pow(2, math.ceil(math.log2(maximum))))
new_matrix1 = matrix1
new_matrix2 = matrix2

# Adding zeros to the matrices so that the arrays dimensions are the same and also
# power of 2
for i in range(maxim):
if i < dimension1[0]:
for _ in range(dimension1[1], maxim):
new_matrix1[i].append(0)
else:
new_matrix1.append([0] * maxim)
if i < dimension2[0]:
for _ in range(dimension2[1], maxim):
new_matrix2[i].append(0)
else:
new_matrix2.append([0] * maxim)

final_matrix = actual_strassen(new_matrix1, new_matrix2)

# Removing the additional zeros
for i in range(maxim):
if i < dimension1[0]:
for _ in range(dimension2[1], maxim):
final_matrix[i].pop()
else:
final_matrix.pop()
return final_matrix

if __name__ == "__main__":
matrix1 = [
[2, 3, 4, 5],
[6, 4, 3, 1],
[2, 3, 6, 7],
[3, 1, 2, 4],
[2, 3, 4, 5],
[6, 4, 3, 1],
[2, 3, 6, 7],
[3, 1, 2, 4],
[2, 3, 4, 5],
[6, 2, 3, 1],
]
matrix2 = [[0, 2, 1, 1], [16, 2, 3, 3], [2, 2, 7, 7], [13, 11, 22, 4]]
print(strassen(matrix1, matrix2))
```
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