Symmetric And Skew Symmetric Matrices

Matrix or matrices (plural) are a simple array of numbers organised in the form of rows and columns. Matrices are one of the most helpful tools in mathematics. This tool alone significantly reduces the workload to a great extent as compared to other straightforward tools of mathematics.

The evolution of the concept of matrices results from an attempt to get compact and simple methods of solving a system of linear equations. Matrices are not only used as a representation of the coefficients in a system of linear equations, but the utility of matrices far exceeds that used.

The applications of matrices are of great significance as they cover a wide field of topics. We use them in electronic spreadsheet programs for personal computers, which in turn are used in different areas of business and science like budgeting, sales projection, cost estimation, analysing the results of an experiment, etc. 

What are Matrices?

Theoretically, matrices are an ordered rectangular array of numbers or functions. These numbers or functions are called entries or the elements of a matrix.

Suppose we wish to express that Rob has 10 notebooks. We can express it as [ 10 ]. Here the number inside  [ ] denotes the number of notebooks Rob has. Now if we wish to express that Rob has 10 notebooks and 5 pencils. We can express it as [10 5]. Here the first number represents the number of notebooks and the second number represents the number of pencils owned by Rob.

This information was easy to express since we took into consideration only one person. Now suppose we want to express this information:

We can express this information in the following way using matrices:

This method of organising numbers or functions into an array of rows and columns forms a matrix and plural of the matrix is matrices. The order of the matrices is determined by the number of rows and columns it has. The matrix above has 3 rows and 2 columns. We show the rows and columns as m X n. With the above matrix, we can represent it as a 3 X 2 matrix.

Types of Matrices

A major part of linear algebra lies in the concepts of vectors and matrices. There are a few types of matrices that are quite common in these concepts. We have discussed below, the 6 main types of matrices that are extremely important in order to understand the topic.

Square Matrix

We defined square matrices as the matrices which have the same number of rows (m) and the same number of columns (n). The name comes from the shape this matrix forms in comparison with the more traditional rectangular matrix.

Here, the number of rows (m) is equal to the number of columns (n) which is 3 X 3. This is an example of an order 3 matrix.

Symmetric Matrix

A symmetric matrix is a matrix in which the top left triangle is identical to the bottom right triangle of the matrix. It is one of the most important types of matrices and is crucial for solving the questions of linear algebra and matrices.

Given above, is an example of a matrix with 4 rows (m) and 4 columns (n) where the top left triangle is identical to the bottom right triangle of the matrix. Hence, this is called a square symmetric matrix of order 4.

Triangle Matrix

A triangle matrix is one that has all of its values in the top right or the bottom left triangle. The rest of the values or elements it contains are 0. If the values in a triangle matrix are in its upper part, it is called an upper right triangular matrix. However, if the values in the matrix are in the bottom-left part of the matrix, it is called a bottom left triangular matrix.

Given above is an example of an upper triangular matrix of order 3 since it has 3 rows and 3 columns.

Skew-Symmetric Matrix

In mathematical terms, a skew-symmetric matrix is defined as a matrix the square matrix that is equal to its transpose matrix when multiplied by -1. For a matrix Z, we can write it as Z = -ZT. 

Example of a Skew-Symmetric Matrix: Z

ZT = 

Properties of Skew-Symmetric Matrix

For a matrix to be skew-symmetric, it must abide by 2 conditions:

• It should be a square matrix

• It should be equal to the negative of its transpose.

If it follows these two conditions, it gains these ‌properties given below:

• The summation of two skew-symmetric matrices will always be a skew-symmetric matrix.

• Since the diagonal elements of a skew-symmetric matrix are zero, its trace also becomes equal to zero.

• Invertible: If a matrix is skew-symmetric, it becomes invertible when added to an identity matrix.

Conclusion

After going through this article, one can understand what a matrix is. Some of the major types of matrices include symmetric and skew-symmetric matrices. The applications of matrices in the field of linear algebra and various other fields as well are immense. The applicability offered by matrices makes them such a crucial tool to be aware of.