The key notion of the paper is that of an elementary idempotent. 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. Note that this result implies the trace of an idempotent matrix is equal. Proof for any two kidempotent matrices and we have commutes with the permutation matrix by theorem 2. 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. Baksalary linear algebra and its applications 321 2000 37.
If a square 01 matrix a satisfies aka, then a is said to be k idempotent. The idempotency of necessarily implies that it is a null matrix. Then ais a product of nonnegative idempotent matrices. A is idempotent matrix, at is called transpose of a. The concept of kidempotent matrices was introduced in 4. Algebra consider the algebra m 2k of 2 2 matrices over a eld k. In this paper, we give a characterization of kidempotent 01 matrices. On spectral theory of kn arithmetic mean idempotent. This means that there is an index k such that bk o. The category of commutative associative kalgebras with 1 will be denoted by kalg. Let a be the pola of matrices described in the example above. Rank equalities for idempotent matrices with applications core. Example the identity matrix is idempotent, because i2 i i i. Idempotency of linear combinations of two idempotent matrices.
The aim of this note is to give shorter proofs of some of these. Contributions to the study on k idempotent matrices. Provided that they have the same size each matrix has the same number of rows and the same number. In linear algebra, an idempotent matrix is a matrix which, when multiplied by itself, yields itself.
In this paper, we introduce and study the concept of. Introduction and preliminaries matrices provide a very powerful tool for dealing with linear models. Recall that an idempotent in an algebra is an element esuch that e2 e. If a square 01 matrix a satisfies a k a, then a is said to be kidempotent. The individual values in the matrix are called entries. A trivial idempotent refers to either of the elements 0 and 1, which are always idempotent. 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 concept of skidempotent and sidempotent are introduced. 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. 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. A matrix is idempotent if it is both symmetric and aa a. Introduction a kidempotent matrix is defined and some of its basic characterization are derived, 2.
Bimatrices are an advanced tool which can handle over one linear models at a time. This video lecture on idempotent, involutory and nilpotent matrix will help students to understand following concepts of gate engineering mathematics. In this paper, we give a characterization of k idem. A primitive idempotent is an idempotent a such that ar is directly indecomposable. Suppose a is an n n matrix such that aa ka for some k 2r. 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. Let k be a xed product of disjoint transposition in s. Matrix algebra, matrix relations, matrix identities, derivative of determinant, derivative of inverse matrix, di erentiate a. A local idempotent is an idempotent a such that ara is a local ring. If u, v is an edge of a digraph da corresponding to an idempotent matrix a, then there exists a path u, w1, w2. A ring r is called strongly separative if if for all.
A necessary and sufficient condition for the equation axb c to have a solution is aa k idempotent matrices. If a square 01 matrix a satisfies a k a, then a is said to be k idempotent. Viewed this way, idempotent matrices are idempotent elements of matrix rings. 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. A matrix a is idempotent if and only if all its eigenvalues are either 0 or 1. Pdf idempotency of linear combinations of an idempotent matrix. On the equality of rank of a fifthidempotent matrix. The relationship between idempotent matrices and kgroup periodic.
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. We prove if ataa, then a is a symmetric idempotent matrix. 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. Idempotency of linear combinations of an idempotent matrix. From the similarity theory of matrices, any idempotent matrix a. In other words, a matrix is orthogonal if it is its own inverse. In mathematics, a matrix plural matrices is a rectangular array see irregular matrix of numbers, symbols, or expressions, arranged in rows and columns. When nonnegative matrices are product of nonnegative. Proof for any two k idempotent matrices and we have commutes with the permutation matrix by theorem 2.
In this paper, we give a characterization of k idempotent 01 matrices. An idempotent a in r is called a central idempotent if ax xa for all x in r. Banerjee 1 has recently given proofs of various properties of real symmetric idempotent matrices. Spectral and spectral theory of idempotent matrices.
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