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