K idempotent matrices pdf

Bimatrices are an advanced tool which can handle over one linear models at a time. This means that there is an index k such that bk o. A matrix a is idempotent if and only if all its eigenvalues are either 0 or 1. Pdf some basic properties of idempotent matrices researchgate. We also determine the maximum number of nonzero entries in kidempotent 01 matrices of a given order as well as the kidempotent 01 matrices attaining this maximum. In mathematics, a matrix plural matrices is a rectangular array see irregular matrix of numbers, symbols, or expressions, arranged in rows and columns. In this paper, the authors explored the k potent matrices using the definition in 9 or the case 1, k in 8 and generalize the work done by wu and slowik in 2 and 3 respectively and show. If u, v is an edge of a digraph da corresponding to an idempotent matrix a, then there exists a path u, w1, w2. Idempotency of linear combinations of an idempotent matrix.

A necessary and sufficient condition for the equation axb c to have a solution is aa k idempotent matrices. Radhakrishnan and others published on kidempotent circulant matrices find, read and cite all the research you need on researchgate. Idempotent matrix, idempotent bimatrix, sk idempotent, sk idempotent bimatrix, eigen value of sk idempotent bimatrix. Spectral and spectral theory of idempotent matrices. The concept of kidempotent matrices was introduced in 4. Sums of idempotent matrices sums of ldempotent matrices pei yuan wu department of applied mathematics national chiao tung university hsinchu, taiwan, peoples republic of china. There is a positive integer k such that v has pure tropical dimension k, generator dimension k and dual dimension k if and only if v is the column space of an idempotent if and only if v is projective as an ftmodule.

In this paper, we introduce and study the concept of. We prove if ataa, then a is a symmetric idempotent matrix. If a square 01 matrix a satisfies aka, then a is said to be kidempotent. Banerjee 1 has recently given proofs of various properties of real symmetric idempotent matrices. From the similarity theory of matrices, any idempotent matrix a. Idempotency of linear combinations of two idempotent matrices. In other words, a matrix is orthogonal if it is its own inverse. When nonnegative matrices are product of nonnegative. We also determine the maximum number of nonzero entries in k idempotent 01 matrices of a given order as well as the k idempotent 01 matrices attaining this maximum number. Some basic results in skidempotent matrices and their properties are give. A primitive idempotent is an idempotent a such that ar is directly indecomposable. If a square 01 matrix a satisfies aka, then a is said to be k idempotent. If a square 01 matrix a satisfies a k a, then a is said to be k idempotent.

If e is an idempotent in m nft, we say that e has rank k if the dimension in any sense of ce is k. Rank equalities for idempotent matrices with applications core. The category of commutative associative kalgebras with 1 will be denoted by kalg. Chapter 3 spectral and spectral theory of idempotent. In this research work, the concept of idempotent matrices is introduced and. On the equality of rank of a fifthidempotent matrix. Other functions of idempotent matrices p1 and p2 studied quite intensively in the literature are the products p 1 p 2 and p 2 p 1.

Note that this result implies the trace of an idempotent matrix is equal. The idempotency of necessarily implies that it is a null matrix. The concept of skidempotent and sidempotent are introduced. A ring r is called strongly separative if if for all. The individual values in the matrix are called entries. If a square 01 matrix a satisfies a k a, then a is said to be kidempotent. Pdf idempotency of linear combinations of an idempotent matrix. On spectral theory of kn arithmetic mean idempotent. Let a be the pola of matrices described in the example above. We also determine the maximum number of nonzero entries in kidempotent 01 matrices of a given order as well as the kidempotent 01 matrices attaining this maximum number.

The aim of this note is to give shorter proofs of some of these. A matrix is idempotent if it is both symmetric and aa a. Suppose a is an n n matrix such that aa ka for some k 2r. If t is a linear transformation whose matrix is an idempotent matrix, then we prove that rn is the direct sum of the image of t and the kernel of t.

Provided that they have the same size each matrix has the same number of rows and the same number. Some of our proofs are suggested by the fact that idempotent matrices represent projection operators, and by the known properties of such operators see e. Example the identity matrix is idempotent, because i2 i i i. Viewed this way, idempotent matrices are idempotent elements of matrix rings. This video lecture on idempotent, involutory and nilpotent matrix will help students to understand following concepts of gate engineering mathematics. Let k be a xed product of disjoint transposition in s. The relationship between idempotent matrices and kgroup periodic. Introduction a kidempotent matrix is defined and some of its basic characterization are derived, 2. A is idempotent matrix, at is called transpose of a. Idempotent linear transformation and direct sum of image. In this paper, the authors explored the kpotent matrices using the definition in 9 or the case 1, k in 8 and generalize the work done by wu and slowik in 2 and 3 respectively and show. The key notion of the paper is that of an elementary idempotent. Contributions to the study on k idempotent matrices. A trivial idempotent refers to either of the elements 0 and 1, which are always idempotent.

Then ais a product of nonnegative idempotent matrices. Proof for any two k idempotent matrices and we have commutes with the permutation matrix by theorem 2. An idempotent a in r is called a central idempotent if ax xa for all x in r. In linear algebra, an idempotent matrix is a matrix which, when multiplied by itself, yields itself.

In this paper, we give a characterization of k idempotent 01 matrices. I came across a pdf file on the internet today about an idempotent matrix. Algebra consider the algebra m 2k of 2 2 matrices over a eld k. Proof for any two kidempotent matrices and we have commutes with the permutation matrix by theorem 2.