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Periodic Matrix

There are various types of matrices such as square matrix, row matrix, column matrix, null matrix and some special matrices such as periodic matrix etc. in this article we will be discussing one of the sub-topic of a matrix i.e., the periodic matrix.

The article begins with the introduction of the matrix and then the introduction of the periodic matrix and explains the periodic matrix using an example.

Matrix refers to a rectangular arrangement of numbers irrespective of whether they are real or complex enclosed by {} or [].

Matrix has a wide variety of applications, especially in the field of engineering, physics, economics and various other branches of mathematics.

Definition

When a square matrix where the number of rows and the number of columns are equal, satisfies a relation A k +1 = A for some positive integer k, then it is called a periodic matrix.

Therefore, the period of the matrix is known to be the least value of k for which A k +1 = A satisfies.

In the relation of A k +1 = A, A is periodic with k as the period. However,

•In cases where k = 1, this is called idempotent.

A periodic matrix is one of the sub-topics of a matrix. To have a better understanding of the periodic matrix, let us make it clear with the help of some examples of a periodic matrix.

Square Matrix

Before understanding the periodic matrix it is important to understand the square matrix.
A= [{4,5,3}, {1,6,4}, {3,9,5}]
Is a square matrix. A matrix is a set of numbers that are arranged in rows and columns to form a rectangular array.

The name of the matrix originates from its shape where the number of rows (m) and the number of columns (n) are equal i.e., which is 3 x 3. Therefore the above matrix is an example of a 3 x 3 matrix.

Example

One of the illustrations of the periodic matrix
A = [{1,-2,-6}, {-3,2,9}, {2,0,-3}]
When we further solve it,
A² = A.A = [{5,-6,-6}, {9,10,9}, {-4,-4,-3}]
A³ = A.A.A = [{1,-2,-6}, {-3,2,9}, {2,0,-3}]

Therefore, with the above example of a periodic matrix

A³ = A satisfying the relation of Ak+1 = A, where the value of k = 2
Hence, the period of the above matrix is 2.

Properties

Some of the properties of a periodic matrix that directly comes from the definition are
•If A is a non-singular m-periodic matrix then, Am =1
•If A is a m x m circulant matrix, then A is an m-periodic matrix
•If A is a non-singular m-periodic matrix, then A-1 is a non-singular m-periodic matrix
•The companion matrix of a polynomial p(r) = rn+1 then r is a periodic matrix of period n.

Conclusion

As we know the matrix has always been an integral part of mathematics. There are different types of matrices such as the column matrix, row matrix, square matrix etc who have their important role in the topics related to vector and linear algebras. However, the above article talks about the periodic matrix which has a relation with the square matrix and therefore, after reading the whole article and reading the example, you will have a clear understanding and for better results do practice more.