Study of Some Useful Properties of a Matrix and its Characteristics

The importance of matrices in linear algebra and their applications is so great that they can be considered the soul of linear algebra. Matrix applications include not only mathematics but also other fields of science and real life, such as probability theory and statistics, electronics, engineering, computer science, cryptography, wireless communication [1], and so on. From ancient times to the present, mathematicians all over the world have used the matrix. There are some matrices reviews in this section.

A matrix is a m x n array of numbers, where m represents the number of rows and n represents the number of columns. A matrix is a rectangular array that contains the real constants of a linear equation or a system of linear equations. All of the elements of a matrix are referred to as entries [2,3]. Let the following system of linear equations x + 4y + 5z = 0 6x + 2y + 3z = 0 2x - 7z = 0 Then the matrix of the above system of linear equations is Order of Matrix A matrix of m numbers of rows and n numbers of columns is said to be a matrix of order m * n [3]. Let A be a matrix of the following forms

Here, matrix has 3 rows and 4 columns. So, matrix A is of 3*4 order . Square Matrix An m*n matrix A is called a square matrix if the numbers of rows is m equal to the number of the columns n(m= n). Example of a square matrix is as following

A diagonal matrix is a square matrix, where all the elements other than the diagonal are zero. Example of a diagonal matrix is as following: Identity Matrix A diagonal matrix is said to be an identity matrix if all the diagonal elements are 1 [4]. The following is an example of an identity matrix Zero Matrix The matrix in which all the elements are zero is said to be a zero matrix [5]. As an example of a zero matrix we can consider the following matrix A. A square matrixA = (aij) of order N , is said to be  Symmetric if aij = ajifor all i and j , i.e. AT = A , where AT is the transpose of A Skew symmetric if ,aij = - aji , i.e. A = AT . Every square matrix can be expressed as the sum of a symmetric and skew symmetric matrix as:

A = 1/2 (A+ AT) + 1/2 (A- AT)

Where 1/2 (A+ AT) is a symmetric and 1/2 (A- AT) is a skew symmetric matrix.  Singular if det(A)=0 and Nonsingular if det(A) ≠0 i.e. if there exists a unique matrix B such that AB=I=BA , where B = A -1 is the inverse of A and I is the Identity Matrix.  Null matrix if its every element is zero and is denoted by O.  Diagonal Matrix when aij=0for i≠j and an≠ 0 for all i. Matrix. If the transposition matrix, say P1, is multiplied with a square matrix , then the product P1*A will be the matrix but with the same two rows (or columns) interchanged.  Permutation matrix if it has exactly one non-zero element, namely, unity, in each row and each column and all other entries are zero. e.g. is a permutation matrix.  Sparse matrix if a large percentage of the entries of the matrix are zero.  Dense if relatively large number of elements of the matrix are non-zero.  Triangular matrix: If all the elements above the diagonal of a square matrix are zero, then the matrix is called a Lower-Triangular Matrix. If all the elements below the diagonal of the matrix are zero, then it is called an Upper-Triangular Matrix.

 Tridiagonal matrix if 𝐴hasnon-zero elements only on the diagonal and inpositionsadjacentto the diagonal.

e.g. is a tridiagonal of order 4*4.  where Bi‘s , i = 1,2,3,……,s are sqaure sub-matrices . not necessarily all of the order.  Non- Negative ( positive) if A≥O(>0) , where o is null matrix.  Hermitian if AH = A and Skew - Hermitian if AH = - A, where AH denoted the conjugate transpose of A.  Positive Define if vH Av > 0 where vH = (v complememt )T , and Positive Semi-definite if vH Av ≥ 0 for any vector v≠0.

If is non-singular and positive definite, then is Hermitian and positiveB = AHA definite. If A is symmetric and positive definite, then its eigenvalues are all positive. A All leading minors of A are positive. A  Similar to a square matrix of the same order if a non-singular matrix can be determined such that

B = S-1AS

Similar matrices have same rank and same eignvalues.  Monotone matrix if det(A) ≠ 0 and A-1≥ 0  Orthogonal ifA-1 = AT i.e. ATA = I= A AT  Unitary if A-1 = AH i.e.AHA = I = AAH  Normal if AHA = AAH

In this section, we will define a matrix that is distinct from the matrices defined previously in numerous books and papers.

one 1 or m, are zero; construct a new type of matrix. Consider the following matrix of order 4*4

Here, the matrix A 4*4 is a square matrix where , i = 1,….,4 and j = 1,…,4. All the elements of A 4*4 which does not have i=1 or 4 or j = 1 or 4 are

a22, a23, a32, a33

And a22 = a23 = a32 = a33=0 This matrix resembles the perimeter of a square. So, for the sake of convenience, we can call it a Perimeter matrix.

 A square matrix of the form Am*m must be used.  The smallest value of m is 3.  All elements in the first and last rows, as well as the first and last columns, must have a value other than 0.  Except for the third property, all of the elements are 0.

When all other elements of a Perimeter matrix of order m*m where m = 1, 2, …., n except the zero element are equal, the determinant of the matrix is zero. Example: Consider the following matrix

A =

Then , determinant of A , det(A) = 0 2.4. Theorem det(A) - det(I) ≠det(A - I) 2.5 Theorem Eigenvalues of a tridiagonal matrix: This class of matrices arises commonly in the study of stability of the finite difference processes, and a knowledge of its eigenvalues leads immediately into useful stability conditions. Let , Then its eigenvalues are given by ; where and may be real or complex λs = a + 2√bc cos(sΠ / N+1) , s = 1(1)N ; where a,b and c may be real or complex. Let A be N*N cyclic tridiagonal matrix given by Then the eigenvalues of A are given by λs = a + 2√bc cos(2sΠ / N+1) , s =0,1,2 ,……..,N-1. 2.6 Theorem A square matrix A = (aij) of order N is irreducible if N = 1 or if in case N>1 , then given any two non-empty disjoint subsets S and T and of w , the set of first N positive integers i.e. w = {1,2,3,….,N such that , there exists and such that . ( ) ij a  A N 1N 1N S T w{1,2,...,} w NS T w iS jT 0 ij a 

Matrix theory has numerous applications in mathematics and real-world sciences. This research paper begins by providing a brief overview of various types of matrices. The paper then introduces a new type of matrix in addition to the known matrices. A few properties and theorems are included here.

REFERENCES

[2]. A. K. Kaw. Introduction to MATRIX ALGEBRA. Florida, 2002. [3]. S. Kaur. ―Applications of Matrices,‖ International Journal of Engineering Technology Science and Research, IJETSR, Vol. 4, Issue 11, pp. 284-288, 2017. [4]. S. Lipschutz & M. L. Lipson. Linear Algebra. Schaum‘s Outline Series, The McGraw-Hill Companies, Inc., 2009. [5]. N. H. Hau et el., ―Review of Matrix Theory with Applications in Educations and Decision Sciences,‖ Advances in Decision Sciences, Vol. 24, Issue 1, 2020.

Gantmacher (Author)

ISBN-10 ‏ : 0486788539 ISBN-13 ‏ : 978-0486788531

Research Scholar, Sunrise University, Alwar, Rajasthan