Table of Contents (click to navigate)

• Matrix Functions in the MATLAB Ecosystem
• An Introduction to Matrix Functions
• Introduction to Matrix Arithmetic
• The Identity Matrix and the Inverse Matrix
• Matrix Applications Conclusions
  • Determination of an Inverse Matrix in the MATLAB R2018b Environment
  • Example
• Conclusions

Inverse Matrix Function Basics: Brief Tutorial

What is MATLAB R2018b?

MATLAB is a software simulation tool used internationally by millions of scientists and engineers for their research and development and operational needs. If you have the desire to analyze and design scientific and engineering systems, MATLAB is the ideal tool to facilitate the complex mathematical transformations that will be critical in your research and development exercises, prior to actual laboratory and pilot project execution.

With powerful calculation algorithms, and a user-friendly Graphical User Interface, visualizing of project to facilitate project insights from the comfort of your desktop or laptop ensures the appropriate environment for your detailed project data analysis. With years of experience under the team’s belt, you can guarantee that your projects will not only be works that can be validated, but those that can facilitate easy transfer to the rest of your project team via MATLAB’s ability to be integrated with other systems. A visual of the Graphical User Interface is shown below:

To the right of the screen is the MATLAB command window, where the relevant instruction code is entered into the system, and the left is the final display once an action has been executed by the system. O

What’s in this paper?

1. Introduction to Inverse Matrix Functions
2. Inverse Matrix Function Basics: Brief Tutorial

Matrix Functions in the MATLAB Ecosystem

MATLAB is the most popular matrix- based computation packages in the scientific and engineering community. With its transferrable language, it is the choice of most educational and industrial facilities as a baseline for project research and development. With its massive popularity, knowing the language is an investment that is well worth it. As it has gained in popularity MATLAB R2018b has developed into not only a mathematical package, but an impressive simulation suite as well. Today, we’ll look at some of the basics of the MATLAB R2018b Matrix Functions, including how to calculate a simple Inverse Matrix Function.

An Introduction to Matrix Functions

Matrix functions are utilized in the lives of scientists and engineers to facilitate a multitude of computational purposes. With applications in several branches of mathematics and science, the matrix function is an aid to convert the physical environment into a mathematical model, and thus facilitate a digital transformation of that model, allow the analyst to determine the outcome, and then identify the real-world implications of that change.

Matrix mathematics is utilized in various computations in computer graphics, and influences image perspectives on screens and facilitates the appropriate adjustments from the user perspective. In fields such as optics, matrices were utilized to facilitate the prediction of reflection and refraction patterns.

For those who are electronically inclined, there are detailed computations that are involved in the determination of system electrical circuits, even down to the size of a microchip. With the aid of matrix technology, the appropriate system properties can be determined by engineers before the systems are built. With the appropriate modeling tools, it is easy to visualize the electrical system and also to determine the system robustness before implementation in the field.

In the business world, quants depend on statistics and probabilities to determine the odds of certain occurrences, and their financial implications. In healthcare, using personal data certain insurance premium calculations can be determined. In the financial sector, financial models and investment portfolios, as well as investment transactions can be determined by computation. All of these calculations are based on probability vector calculations, which require the use of matrix computations in order to facilitate their appropriate, predictable outcomes.

At its core, one can see that Matrix Mathematics touches our digital lives in a myriad of ways. They provide means via which linear algebra computations can be significantly simplified.

Introduction to Matrix Arithmetic

Mathematically, a matrix is a means via which a numerical data set can be organized and represented by an ordered row and column of variables. [2]

Organized as a “2×3” matrix, the observer can see that the matrix has two rows and three columns. MATLAB utilizes this baseline for some of the detailed calculations that it computes. The types of matrix inputs that can generated in the MATLAB environment are shown below:

This list is by no means exhaustive, and via the MATLAB support system, the system will indicate to you how best to represent the specific data that you’ll need to compute your data in a representative way.

The general types of matrices you’ll encounter in your career include:

1. The square matrix which contains equal numbers of columns and rows

The Identity Matrix and the Inverse Matrix

Critical to this tutorial is the introduction of the identity matrix and the inverse matrix. The identity matrix is an nXn square matrix that has ones on its diagonal. The identity matrix is highlighted below:

Using this matrix as a multiplier, the effect is similar to multiplying its complementary matrix by one.

In the case of the inverse matrix if there are two matrices that multiply each other to give the matrix above, they are inverses of each other. Numerically this is depicted by two matrices A and B, such that AB=I_n.  B is identified as the inverse matrix of A and denoted as A^-1.

Matrix Applications Conclusions

The preceding paragraphs introduced you to the functions that matrix mathematics has played in our daily lives.  Various bodies of science are integrated to facilitate the advent of digital modeling in order to lower the cost of experimentation, and also increase the variability of experiments that can be attempted by scientists, engineers and mathematicians.

By being able to model the physical environment around them, and expressing data as matrices, the transformations needed to simulate an actual environment change can easily be computed. Today’s tutorial introduces you to some of these transformations.

Determination of an Inverse Matrix in the MATLAB R2018b Environment

The MATLAB Graphical User Interface is designed for easy computation of mathematical representations of the physical world. The general GUI is shown below:

MATLAB is often attained via one of two main ways: Via Organizational license or Individual License. If you’re affiliated with an organization such as a University or Industrial company, the IT packages will facilitate multiple licenses that are available to organizational users. With the single access identification system, via the appropriate portals, users can find the link to access MATLAB R2018b. For individual users, there is the option to purchase the system via their website: https://www.mathworks.com/

Once the access has been granted, the preceding image is what the user will see as they navigate the Graphical User Interface (GUI). Designed to be intuitive, the GUI elements include:

1. An interactive ribbon at the top of the page containing various system functions
2. The command window to the right of the screen. This environment is where the system code will be entered into the system. Much like the command prompt in any operating system, this aspect of MATLAB is where the critical system code will be entered, in order to facilitate the background program algorithms to compute the desired tasks,
3. The display windows to the left of the screen. As a graphical system, if the user so desires it will be possible for the system to display a visual of the desired mathematical function to be displayed. For functions such as the calculation of population densities and other 3-D functions, the power of the MATLAB environment is easily felt in the outcome of the computation.

Knowing these baselines, let us identify how to manipulate the MATLAB environment to facilitate the computation of an inverse function. From the tutorial above, it was identified that if two matrices multiply each other to equal the identify matrix, they are inverses of each other. The following is the relationship:

AA⁻¹ = A⁻¹A = 1

Within the MATLAB Command Center, the user can attempt to compute this function. The user must be aware that not all matrices, due to the nature of the desired computations, will have an inverse. MATLAB has a detailed coding system, and the desired function for the calculation of an inverse function is denoted by the inv function. The following example is an ideal way to show the computational power of MATLAB at work:

Example

Create a script file and type the following code −

a = [ 1 2 3; 2 3 4; 1 2 5]

inv(a)

When you run the file, it displays the following result –

a =

   1     2     3

   2     3     4

   1     2     5

ans =

   -3.5000    2.0000    0.5000

   3.0000   -1.0000    -1.0000

   -0.5000      0       0.5000

Conclusions

From the computations and detailed theory and background above, it can easily be identified that MATLAB R2018b is a powerful simulation tool. The system does require specialized knowledge in order to properly code and navigate the system, and has an initially steep learning curve. Fortunately, there are plenty of support tools and tutorials such as ours that are available online, that will facilitate a smooth transition to the world of MATLAB. As gamification is on the rise, users will enjoy the challenge that the system will provide in order to help them achieve in a fun and interactive manner the means via which their systems can be modeled before actual computation.

Matrix computations are ubiquitous, and they can only help us to advance science and technology. Use the tutorial above to help with your scientific and engineering needs.

Image Source: https://www.tutorialspoint.com/matlab/matlab_matrix_inverse.htm

Detailed Matrix Uses: https://www.decodedscience.org/practical-uses-matrix-mathematics/40494