#### Matrix Exponentiation Recursive

P
```/*
Source:
https://en.wikipedia.org/wiki/Exponentiation_by_squaring

Complexity:
O(d^3 log n)
where: d is the dimension of the square matrix
n is the power the matrix is raised to
*/

const Identity = (n) => {
// Input: n: int
// Output: res: Identity matrix of size n x n
// Complexity: O(n^2)
const res = []
for (let i = 0; i < n; i++) {
res[i] = []
for (let j = 0; j < n; j++) {
res[i][j] = i === j ? 1 : 0
}
}
return res
}

const MatMult = (matrixA, matrixB) => {
// Input: matrixA: 2D Array of Numbers of size n x n
//        matrixB: 2D Array of Numbers of size n x n
// Output: matrixA x matrixB: 2D Array of Numbers of size n x n
// Complexity: O(n^3)
const n = matrixA.length
const matrixC = []
for (let i = 0; i < n; i++) {
matrixC[i] = []
for (let j = 0; j < n; j++) {
matrixC[i][j] = 0
}
}
for (let i = 0; i < n; i++) {
for (let j = 0; j < n; j++) {
for (let k = 0; k < n; k++) {
matrixC[i][j] += matrixA[i][k] * matrixB[k][j]
}
}
}
return matrixC
}

export const MatrixExponentiationRecursive = (mat, m) => {
// Input: mat: 2D Array of Numbers of size n x n
// Output: mat^n: 2D Array of Numbers of size n x n
// Complexity: O(n^3 log m)
if (m === 0) {
// return identity matrix of size n x n
return Identity(mat.length)
} else if (m % 2 === 1) {
// tmp = mat ^ m-1
const tmp = MatrixExponentiationRecursive(mat, m - 1)
/// return tmp * mat = (mat ^ m-1) * mat = mat ^ m
return MatMult(tmp, mat)
} else {
// tmp = mat ^ m/2
const tmp = MatrixExponentiationRecursive(mat, m >> 1)
// return tmp * tmp = (mat ^ m/2) ^ 2 = mat ^ m
return MatMult(tmp, tmp)
}
}

// const mat = [[1, 0, 2], [2, 1, 0], [0, 2, 1]]

// // mat ^ 0 = [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ]
// MatrixExponentiationRecursive(mat, 0)

// // mat ^ 1 = [ [ 1, 0, 2 ], [ 2, 1, 0 ], [ 0, 2, 1 ] ]
// MatrixExponentiationRecursive(mat, 1)

// // mat ^ 2 = [ [ 1, 4, 4 ], [ 4, 1, 4 ], [ 4, 4, 1 ] ]
// MatrixExponentiationRecursive(mat, 2)

// // mat ^ 5 = [ [ 1, 4, 4 ], [ 4, 1, 4 ], [ 4, 4, 1 ] ]
// MatrixExponentiationRecursive(mat, 5)
```