CHAPTER 2 -idempotent matrices

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1 CHAPTER 2 -idempotent matrices A -idempotent matrix is defined and some of its basic characterizations are derived (see [33]) in this chapter. It is shown that if is a -idempotent matrix then it is quadripotent (i.e.,. Necessary and sufficient condition for the sum of two -idempotent matrices to be -idempotent, is determined and then it is generalized for the sum of -idempotent matrices. A condition for the product of two -idempotent matrices to be -idempotent is also determined and then it is generalized for the product of -idempotent matrices. Relations between power hermitian matrices and -idempotent matrices are investigated (cf. [32]). It is proved that a -idempotent matrix reduces to an idempotent matrix when it commutes with the associated permutation matrix (i.e., ). 20

2 2.1 Characterizations of k-idempotent matrices A -idempotent matrix is defined and its characterizations are discussed in this section. Definition For a fixed product of disjoint transpositions, a matrix in C is said to be -idempotent if. This is equivalent to, where is the associated permutation matrix of. Example Then Here is a -idempotent matrix with. The associated permutation matrix is a matrix with ones on its southwest northeast diagonal and zeros everywhere else. That is, It can be easily verified that. Note In particular if then the associated permutation matrix reduces to identity matrix and -idempotent matrix reduces to idempotent matrix. 21

3 Remark implies that. The following relations can also be obtained which would be useful in computational aspects or or or Theorem Let be a -idempotent matrix. Then is -idempotent if and only if is idempotent.,which implies that is idempotent. Conversely, if is idempotent then commutes with the permutation matrix (cf. lemma 2.2.5) is -idempotent. Remark Consider. Then the associated permutation matrix. With reference to this, (i) is -idempotent as well as idempotent. 22

4 is -idempotent. (ii) is -idempotent but not idempotent. is not -idempotent. Theorem Let be a -idempotent matrix. Then (a) and (when it exists) are also -idempotent. (b) is -idempotent for all positive integers. (c) is quadripotent when is not an idempotent. Further, is non-singular then and. (d) is idempotent. (e) and are tripotent matrices. (a) A similar proof may be given for the remaining matrices. (b) - times (c) 23

5 Since, we have is quadripotent. is non-singular then and are immediate consequences of. (d) (e) The proof is similar for. Example made. Consider the matrix in (ii) of remark The following observations can be (i) is -idempotent. (ii) is also -idempotent. (iii) It can be seen that. (iv) Clearly, is idempotent. (v) and are tripotent matrices. Theorem is a -idempotent matrix then is a group under matrix multiplication. 24

6 Using the remark 2.1.4, the following matrix multiplication table can be found.. From the above table, we see that is closed under matrix multiplication. It is obvious that matrix multiplication is associative. We observe that acts as an identity element in. Besides, it is the only element in having this property. The inverse for each elements are given by, and is a group under matrix multiplication. Remark (i) then is a cyclic subgroup of.. (ii) is non-singular then by theorem (c), the group becomes. 25

7 2.2 Sums and Products of -idempotent matrices In this section, the sums and products of -idempotent matrices are discussed and some related results are obtained. Theorem Let and be two -idempotent matrices. Then is -idempotent if and only if. iff Generalization: Let be -idempotent matrices. Then is -idempotent if and only if for and in. Since s are -idempotent matrices, we have Here 26

8 is -idempotent then from,, it follows that. Conversely, if we assume that then from is -idempotent. Remark example, Theorem fails when we relax the condition that and anty commute. For Let and Clearly, and are -idempotent matrices. But and i.e.,. Also, is not a -idempotent matrix. 27

9 Theorem Let and be -idempotent matrices. then is also a -idempotent matrix. the matrix is -idempotent. Generalization: matrices then be -idempotent matrices belonging to a commuting family of is a -idempotent matrix. the matrix is -idempotent. Remark we relax the condition of commutability of matrices and in theorem then the product need not be k-idempotent. For example the matrices and in remark can be considered. It can be seen that. Also the product is not a -idempotent matrix. 28

10 denotes the commutator of the matrices and (see definition ), theorems and can be restated as and are two -idempotent matrices then is -idempotent if and only if. is -idempotent if. By theorem 1.2.1, the generalization of theorem can also be restated as For -idempotent matrices, if there is a unique matrix such that then the product is -idempotent. Lemma Let be a -idempotent matrix. Then is idempotent if and only if, where is the associated permutation matrix of. Assume that. Pre multiplying by, we have But is -idempotent is idempotent. By retracing the above steps the converse follows. Example is a -idempotent matrix and it also commutes with the associated permutation matrix, that is. We see that is idempotent. Lemma fails if we relax the condition of commutability of matrices and. Examples and are not idempotents. Note that in such cases. Theorem and are k-idempotent matrices then commutes with the permutation matrix. 29

11 Theorem Let and are two commuting k-idempotent matrices. The -idempotency of necessarily implies that it is a null matrix. For any two k-idempotent matrices and we have commutes with the permutation matrix by theorem is k-idempotent then by lemma 2.2.5, it reduces to an idempotent matrix. i.e., Since and are k-idempotent matrices, we have and by theorem (c). Substituting in, Since and are commuting -idempotent matrices, is also -idempotent by theorem by theorem (c) 30

12 Pre multiplying by, we have. It follows that. Remark For example the matrices and in remark can be considered..clearly the matrix commutes with the permutation matrix. It can be easily verified that is not a -idempotent matrix. 31

13 2.3 -idempotency of power hermitian matrices In this section, conditions for power hermitian matrices to be -idempotent are derived and some related results are given. Theorem Any two of the following imply the other one. C then (a) (b) (c) is -idempotent is -hermitian is square hermitian (a) and (b) (c): and. is square hermitian. (b) and (c) (a): Substituting in, We have. is -idempotent. (c) and (a) (b): Substituting in, We have. is -hermitian. Corollary Let be -hermitian -idempotent matrix. is non-singular then is unitary. Since is -hermitian -idempotent, is square hermitian by theorem is non-singular then by theorem (c) Therefore and hence is unitary. 32

14 Example Let (i) is -idempotent. (ii) is square hermitian. (iii) is -hermitian. (iv) is unitary Theorem Let be a -idempotent matrix. is cube hermitian then it reduces to an orthogonal projector. Since is a -idempotent matrix, we have by remark is cube hermitian then Pre and post multiplying by, we have by by Substituting in, we have. is an orthogonal projector. 33

15 Corollary Let be a -idempotent matrix. is -cube hermitian then it reduces to an orthogonal projector. is -cube hermitian then by remark reduces to cube hermitian matrix. By theorem 2.3.4, the matrix is an orthogonal projector. Note Let be a -idempotent matrix. then by theorem (c) we have (i.e., reduces to hermitian matrix ) Theorem Let be a -idempotent matrix. Then the following are equivalent. (i) (ii) (iii) is cube hermitian. is hermitian. is square hermitian. (i) (ii): is cube hermitian then by theorem (e) is hermitian. (ii) (iii): is hermitian then. is square hermitian. 34

16 (iii) (ii): is square hermitian then But is cube hermitian. Theorem Let be a -idempotent matrix. Then the necessary and sufficient condition for the matrix to be square hermitian is (i) (ii) is idempotent. Assume that is square hermitian. Pre and post multiplying by respectively, we have since and, we have is idempotent and substituting this in, we have. Conversely, assume that is idempotent with.. 35

17 is square hermitian. Corollary Let be a -idempotent matrix. Then the necessary and sufficient condition for to be -square hermitian is (i) (ii) is idempotent Assume that is -square hermitian. is square hermitian. By theorem 2.3.8, this is equivalent to is idempotent and Theorem Let be a -idempotent matrix. Then the following are equivalent. (i) (ii) (iii) (iv) is -cube hermitian is hermitian is -square hermitian is -hermitian 36

18 (i) (ii): is -cube hermitian then by theorem (e). is hermitian. (ii) (iii): is hermitian then (iii) (iv): is -square hermitian.. is -square hermitian then Pre and post multiplying by, we have is -hermitian.. (iv) (i): is -hermitian then But by theorem (e). is -cube hermitian. 37

19 Note The following theorem can be considered to be a converse of the above theorem Theorem is hermitian and -square hermitian then is -idempotent. Assume that and Combining the above two relations, we have. is -idempotent. 38