How to resolve the algorithm Matrix-exponentiation operator step by step in the Mathematica/Wolfram Language programming language

Published on 22 June 2024 08:30 PM

How to resolve the algorithm Matrix-exponentiation operator step by step in the Mathematica/Wolfram Language programming language

Table of Contents

Problem Statement

Most programming languages have a built-in implementation of exponentiation for integers and reals only.

Demonstrate how to implement matrix exponentiation as an operator.

Let's start with the solution:

Step by Step solution about How to resolve the algorithm Matrix-exponentiation operator step by step in the Mathematica/Wolfram Language programming language

Matrix Power Function

The code snippet demonstrates the use of the MatrixPower function in Wolfram programming language, which calculates the power of a matrix. Here's a detailed explanation:

Input:

a = {{3, 2}, {4, 1}};

This line creates a matrix a with the following values:

| 3 2 |
| 4 1 |

Matrix Power Calculations:

MatrixPower[a, 0]

This line calculates the matrix a raised to the power of 0. In matrix theory, any matrix raised to the power of 0 is the identity matrix, which is a square matrix with 1s on the diagonal and 0s everywhere else. So, the result is:

| 1 0 |
| 0 1 |
MatrixPower[a, 1]

This line calculates the matrix a raised to the power of 1. This is equivalent to the original matrix a:

| 3 2 |
| 4 1 |
MatrixPower[a, -1]

This line calculates the matrix a raised to the power of -1. This is the inverse of the matrix a, which means that it "undoes" the effects of matrix a. The inverse is:

| -0.5 1.0 |
|  2.0 -0.5 |
MatrixPower[a, 4]

This line calculates the matrix a raised to the power of 4. This is calculated by multiplying the matrix a with itself 4 times:

| 76 40 |
| 96 51 |
MatrixPower[a, 1/2]

This line calculates the matrix a raised to the power of 1/2. This is equivalent to finding the square root of the matrix a. This can be approximated using numerical methods:

| 1.68604 -0.369494 |
| 2.32712  0.435685 | (approximate values)
MatrixPower[a, Pi]

This line calculates the matrix a raised to the power of Pi, where Pi is the mathematical constant approximately equal to 3.14159. This is an example of raising a matrix to a non-integer power. The result is a complex matrix:

| -1.28302 -2.63051 I |
|  2.14656 +2.12936 I | (approximate values)

Matrix Power with Simplification:

MatrixPower[{{i, j}, {k, l}}, m] // Simplify

This line calculates the matrix power of a symbolic matrix with variables i, j, k, and l raised to the power of m. The // Simplify operator is used to simplify the result. For example:

MatrixPower[{{i, j}, {k, l}}, 2] // Simplify

will output:

{{i^2 + j^2, i j + j l}, {k i + l i, k j + l^2}}

Source code in the wolfram programming language

a = {{3, 2}, {4, 1}};
MatrixPower[a, 0]
MatrixPower[a, 1]
MatrixPower[a, -1]
MatrixPower[a, 4]
MatrixPower[a, 1/2]
MatrixPower[a, Pi]


MatrixPower[{{i, j}, {k, l}}, m] // Simplify


  

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