Step by Step solution about How to resolve the algorithm Matrix-exponentiation operator step by step in the Go programming language

The provided Go code defines a matrix type and its operations, including multiplication and exponentiation. Here's a detailed explanation:

1. Type Definitions:

   • vector: A type alias for a slice of float64 values, representing a vector.
   • matrix: A type alias for a slice of vector values, representing a matrix.

2. Matrix Multiplication (Method mul):

   • The mul method multiplies two matrices m1 and m2 and returns the result as a new matrix.
   • It verifies that the number of columns in m1 matches the number of rows in m2 (a multiplication requirement).
   • The method iterates through the elements of the result matrix, calculating each element as the dot product of the corresponding row in m1 and column in m2.

3. Identity Matrix Function (identityMatrix):

   • This function creates and returns an identity matrix of size n.
   • An identity matrix is a square matrix with 1s on the diagonal and 0s everywhere else.

4. Matrix Exponentiation (Method pow):

   • The pow method raises a matrix m to the power n and returns the result.
   • It first checks if m is a square matrix (i.e., the number of rows equals the number of columns).
   • If n is less than 0, it panics since negative exponents are not supported.
   • If n is 0, it returns an identity matrix of the same size as m.
   • If n is 1, it returns m itself.
   • For other n values, it employs the square-and-multiply algorithm to efficiently compute the result.
   • The algorithm uses the binary representation of n to determine the sequence of matrix multiplications required to calculate m^n.

5. Main Function:

   • In the main function:
     • A matrix m is defined.
     • A loop from 0 to 10 is used to compute and print the powers of m from m^0 to m^10.

This code serves as a useful tool for anyone working with matrices and linear algebra operations in Go. It allows for convenient matrix multiplication and exponentiation, common tasks in various scientific and engineering applications.

package main

import "fmt"

type vector = []float64
type matrix []vector

func (m1 matrix) mul(m2 matrix) matrix {
    rows1, cols1 := len(m1), len(m1[0])
    rows2, cols2 := len(m2), len(m2[0])
    if cols1 != rows2 {
        panic("Matrices cannot be multiplied.")
    }
    result := make(matrix, rows1)
    for i := 0; i < rows1; i++ {
        result[i] = make(vector, cols2)
        for j := 0; j < cols2; j++ {
            for k := 0; k < rows2; k++ {
                result[i][j] += m1[i][k] * m2[k][j]
            }
        }
    }
    return result
}

func identityMatrix(n int) matrix {
    if n < 1 {
        panic("Size of identity matrix can't be less than 1")
    }
    ident := make(matrix, n)
    for i := 0; i < n; i++ {
        ident[i] = make(vector, n)
        ident[i][i] = 1
    }
    return ident
}

func (m matrix) pow(n int) matrix {
    le := len(m)
    if le != len(m[0]) {
        panic("Not a square matrix")
    }
    switch {
    case n < 0:
        panic("Negative exponents not supported")
    case n == 0:
        return identityMatrix(le)
    case n == 1:
        return m
    }
    pow := identityMatrix(le)
    base := m
    e := n
    for e > 0 {
        if (e & 1) == 1 {
            pow = pow.mul(base)
        }
        e >>= 1
        base = base.mul(base)
    }
    return pow
}

func main() {
    m := matrix{{3, 2}, {2, 1}}
    for i := 0; i <= 10; i++ {
        fmt.Println("** Power of", i, "**")
        fmt.Println(m.pow(i))
        fmt.Println()
    }
}