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Author: jpa99

Miscellaneous/Matrix_Exponentiation.java

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+import java.util.Arrays;
+
+public class Matrix_Exponentiation {
+	
+	/*Matrix Exponentiation Algorithm
+
+	 Problem (informal): Given a matrix M, find the product: M^n = M x M x ... x M
+	 
+	 Algorithm: Brute force algorithm for repeated multiplication
+	 
+	 Complexity:
+	 	* Time - O(n*i*j*k) where a has dimensions i x j and b has dimensions j x k to compute M^n
+	 	* Space - O(i*j) where M has i rows and j columns
+	 	
+	 Functions Defined:
+	 	* matrix_multiply() - returns matrix product of matrices a and b
+	 	* matrix_exponentation() - returns M^n via repeated multiplication of M for square matrix M
+	 	
+	 */
+	
+	
+	
+	public static void main(String[] args){
+		int[][] a = {{1, 2, 3, 2}, {2, 3, 4, 8}, {4, 5, 3, 6}}; //3 x 4
+		int[][] b = {{6, 1, 4}, {9, 0, 6}, {6, 4, 5}, {3, 1, 1}}; // 4 x 3
+		int[][] m = matrix_multiply(a, b);
+		System.out.println(Arrays.deepToString(m));
+		
+		//testing staticness with multiplication with identity
+		int[][] identity = {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}};
+		int[][] m2 = matrix_multiply(m, identity);
+		System.out.println(Arrays.deepToString(m2));
+		
+		//matrix exponentiation
+		m2 = matrix_exponentiation(m2, 2);
+		
+		System.out.println(Arrays.deepToString(m2));
+	}
+	
+	//assumes that matrix b has j rows and a has j columns
+	//O(ijk) where a has dimensions i x j and b has dimensions j x k
+	public static int[][] matrix_multiply(int[][] a, int[][] b){
+		int[][] m = new int[a.length][b[0].length];
+		for(int i=0;i<a.length;i++){
+			for(int j = 0; j < b[0].length;j++){
+				//m[i][j] is product of ith row of A and jth column of B
+				int sum = 0;
+				for(int k = 0; k < b.length; k++){
+					sum+= a[i][k]*b[k][j];
+				}
+				m[i][j] = sum;
+			}
+		}
+		return m;
+	}
+		
+	//O(n*ijk) where a has dimensions i x j and b has dimensions j x k to compute M^n
+	public static int[][] matrix_exponentiation(int[][] m, int n){
+		while(n-- > 1){
+			m = matrix_multiply(m, m);
+		}
+		return m;
+	}
+}