Matrix Algebra

The application of matrix has been used widely in mathematics and in various other subjects. It assists in solving linear equations and it can be of numerous settings. The matrices are used in several subjects related to science, mathematics, and engineering mathematics. It is used in statistics to elaborate the collection of numeric information. The identifiers of rows and columns are taken away in order to complete the conversion to matrix algebra. The numbers that are present in a matrix are known as the matrix element.

Type of matrices

There are various types of matrices in linear algebra.

Row matrix

Matrices that have only one row are referred to as row matrices. It can have any number of columns.

Column matrix

The matrix that contains only one column is known as the column matrix. Column matrix can consist of any number of rows.

Null matrix

In case, all the elements within a matrix are 0, the matrix is called zero or null matrix. It is also denoted by the number 0.

Square matrix

The matrix that has an equal number of columns and rows is called a square matrix.

Rectangular matrix 

The matrices that do not have an equal number of rows and columns are called rectangular matrices.

Diagonal matrix

A Square matrix where all elements are 0 except the number that is diagonally presented is known as the diagonal matrix.

Scalar matrix

The scalar matrix is a type of diagonal matrix where the value of the elements is equal and the other elements are 0.

Unit matrix

In case, the value of the elements is 1 in a diagonal matrix, it is called unit matrix.

Equal matrix

The matrices which have elements of equal value are referred to as equal matrices.

Triangular matrix

In a square matrix, when the elements that are present above or below the principal diagonal are zero, referred to as the triangular matrix.

Application of matrix algebra

Matrix algebra can be applied in commerce, social science, and various subjects related to science. It is used in many fields such as,

1. Optics
2. Mathematics
3. Computer graphics
4. Finance ices
5. Cryptography
6. Geology
7. Economics
8. Signal processing and wireless communication
9. Chemistry
10.  Animation and robotics

The cartoons, automation, and architecture are usually done by hand drawings. Nowadays, computer graphics are used in this procedure. The linear transformation of objects is readily represented by the square matrices. It is used to describe three-dimensional pictures into two-dimensional planes. In the first step, the digital image is considered as a matrix in graphics. The columns and rows of the pixels are represented as the columns and rows of the matrix. The numerical values of the matrix are described by the color values of the pixels. In graphics of video games, a point can be manipulated by using matrices.

Matrices can be used to express graphs also. The columns and rows of the matrix can represent the node and the value of intersection symbolizes the connection. Various matrix operations such as rotation, scaling, and translation are also used in graphics. The process of encryption is also developed by using matrices. Digital video or audio are analyzed to form a sequence of numbers that represents the several variations that occur for a long period of time. The filtering procedure that is used for audio is based on matrix multiplication. It is also used in creating wireless signals and the process of optimization of these signals.

Matrices are used for extracting, detecting, and processing collected data that is embedded in the signal. Matrix algebra plays an important role in identifying issues and estimating signals. It is also used in optics to assess the amount of reflection and refraction. It is also used in aerospace engineering, quantum mechanics, electrical circuits, and “resistor conversion of electrical energy”.

Importance of matrix algebra

• It is an essential tool in expressing maps that are formed between the “finite-dimensional vector spaces”.
• It is useful in manipulating and representing linear equations.
• It can also assist in representing quadratic forms that are essential in studying the matrices of hessian.
• It is a basic framework in quantum mechanics.
• Metrics are useful in studying finite groups. Each finite group has a form of representation as a set of various invertible matrices.
• It is also used in 3D geometry to describe various transformations such as perspective projection, scaling, rotation, and translation.
• Markov chains can be represented in matrices. It provides a process of providing tabular presentation of collected data.
• It is useful in solving various systems of equations and also in physics and engineering.
• It is used in calculating algorithms that are used in software of data mining.

Conclusion

One of the most common tools in computer science and electrical engineering is the matrix which represents numbers in rectangular grids. The number represents the data and is also used for presenting mathematical equations. The multiplication of matrices can give a good estimation of complex calculations in a smaller time frame. It is also used in genetic analysis and image processing. Solving the problems of linear equations can be an essential tool. In order to solve linear equations of different variables, matrix multiplication is used and also in reducing complex dynamics.

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