Added matrix_exponentiation in algorithms/matrix

Author: c-utkarsh

README.md

@@ -209,6 +209,7 @@ If you want to uninstall algorithms, it is as simple as:
     - [gcd/lcm](algorithms/maths/gcd.py)
     - [generate_strobogrammtic](algorithms/maths/generate_strobogrammtic.py)
     - [is_strobogrammatic](algorithms/maths/is_strobogrammatic.py)
+    - [matrix_exponentiation](algorithms/matrix/matrix_exponentiation.py)
     - [modular_exponential](algorithms/maths/modular_exponential.py)
     - [next_bigger](algorithms/maths/next_bigger.py)
     - [next_perfect_square](algorithms/maths/next_perfect_square.py)

algorithms/matrix/__init__.py

@@ -1 +1 @@
-
+from .matrix_exponentiation import *

algorithms/matrix/matrix_exponentiation.py

@@ -0,0 +1,60 @@
+def multiply(matA: list, matB: list) -> list:
+    """
+    Multiplies two square matrices matA and matB od size n x n
+    Time Complexity: O(n^3)
+    """
+    n = len(matA)
+    matC = [[0 for i in range(n)] for j in range(n)]
+
+    for i in range(n):
+        for j in range(n):
+            for k in range(n):
+                matC[i][j] += matA[i][k] * matB[k][j]
+
+    return matC
+
+def identity(n: int) -> list:
+    """
+    Returns the Identity matrix of size n x n
+    Time Complecity: O(n^2)
+    """
+    I = [[0 for i in range(n)] for j in range(n)]
+    
+    for i in range(n):
+        I[i][i] = 1
+    
+    return I
+
+def matrix_exponentiation(mat: list, n: int) -> list:
+    """
+    Calculates mat^n by repeated squaring
+    Time Complexity: O(d^3 log(n))
+                     d: dimesion of the square matrix mat
+                     n: power the matrix is raised to
+    """
+    if n == 0:
+        return identity(len(mat))
+    elif n % 2 == 1:
+        return multiply(matrix_exponentiation(mat, n - 1), mat)
+    else:
+        tmp = matrix_exponentiation(mat, n // 2)
+        return multiply(tmp, tmp)
+
+if __name__ == "__main__":
+    mat = [[1, 0, 2], [2, 1, 0], [0, 2, 1]]
+    
+    res0 = matrix_exponentiation(mat, 0)
+    assert res0 == [[1, 0, 0], [0, 1, 0], [0, 0, 1]]
+    print(f"{mat}^0 = {res0}")
+
+    res1 = matrix_exponentiation(mat, 1)
+    assert res1 == [[1, 0, 2], [2, 1, 0], [0, 2, 1]]
+    print(f"{mat}^1 = {res1}")
+
+    res2 = matrix_exponentiation(mat, 2)
+    assert res2 == [[1, 4, 4], [4, 1, 4], [4, 4, 1]]
+    print(f"{mat}^2 = {res2}")
+    
+    res5 = matrix_exponentiation(mat, 5)
+    assert res5 == [[81, 72, 90], [90, 81, 72], [72, 90, 81]]
+    print(f"{mat}^5 = {res5}")