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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., ).
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2.1 Characterizations of k-idempotent matrices
A -idempotent matrix is defined and its characterizations are discussed in this
section.
Definition 2.1.1
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 2.1.2
If
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 2.1.3
In particular if then the associated permutation matrix reduces to
identity matrix and -idempotent matrix reduces to idempotent matrix.
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Remark 2.1.4
implies that . The following relations can also be obtained
which would be useful in computational aspects
or
or
or
Theorem 2.1.5
Let be a -idempotent matrix. Then is -idempotent if and only if is
idempotent.
Proof
Hence ,which implies that is idempotent.
Conversely, if is idempotent then commutes with the permutation matrix
(cf. lemma 2.2.5)
Hence is -idempotent.
Remark 2.1.6
Consider . Then the associated permutation matrix
. With
reference to this ,
(i)
is -idempotent as well as idempotent.
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is -idempotent.
(ii)
is -idempotent but not idempotent.
is not -idempotent.
Theorem 2.1.7
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, If is non-singular then
and .
(d) is idempotent.
(e) and are tripotent matrices.
Proof
(a)
A similar proof may be given for the remaining matrices.
(b)
- times
(c)
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Since , we have is quadripotent.
If is non-singular then and are immediate consequences of
.
(d)
(e)
The proof is similar for .
Example 2.1.8
Consider the matrix in (ii) of remark 2.1.6. The following observations can be
made.
(i) is -idempotent.
(ii)
is also -idempotent.
(iii) It can be seen that .
(iv) Clearly, is idempotent.
(v)
and
are tripotent
matrices.
Theorem 2.1.9
If is a -idempotent matrix then is a group
under matrix multiplication.
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Proof
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
Hence is a group under matrix multiplication.
Remark 2.1.10
(i) If then is a cyclic subgroup of .
.
(ii) If is non-singular then by theorem 2.1.7 (c) , the group becomes
.
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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 2.2.1
Let and be two -idempotent matrices. Then is -idempotent if and
only if .
Proof
iff
Generalization:
Let be -idempotent matrices. Then is -idempotent if and
only if for and in .
Proof
Since ’s are -idempotent matrices, we have
Here
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If is -idempotent then from ,
Hence, it follows that .
Conversely, if we assume that then from
Hence is -idempotent.
Remark 2.2.2
Theorem 2.2.1 fails when we relax the condition that and anty commute. For
example,
Let
and
Clearly, and are -idempotent matrices.
But
and
i.e., .
Also, is not a -idempotent matrix.
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Theorem 2.2.3
Let and be -idempotent matrices. If then is also a
-idempotent matrix.
Proof
Hence the matrix is -idempotent.
Generalization:
If be -idempotent matrices belonging to a commuting family of
matrices then is a -idempotent matrix.
Proof
Hence the matrix is -idempotent.
Remark 2.2.4
If we relax the condition of commutability of matrices and in theorem 2.2.3
then the product need not be k-idempotent. For example the matrices and in remark
2.2.2 can be considered.
It can be seen that . Also the product is not a -idempotent matrix.
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If denotes the commutator of the matrices and (see definition 1.2.19),
theorems 2.2.1 and 2.2.3 can be restated as
If and are two -idempotent matrices then
is -idempotent if and only if .
is -idempotent if .
By theorem 1.2.1, the generalization of theorem 2.2.3 can also be restated as
‘For -idempotent matrices , if there is a unique matrix such that
then the product is -idempotent’.
Lemma 2.2.5
Let be a -idempotent matrix. Then is idempotent if and only if ,
where is the associated permutation matrix of ‘ ’.
Proof
Assume that .
Pre multiplying by , we have
But is -idempotent
Hence is idempotent. By retracing the above steps the converse follows.
Example 2.2.6
is a -idempotent matrix and it also commutes with the
associated permutation matrix
, that is . We see that is
idempotent. Lemma 2.2.5 fails if we relax the condition of commutability of matrices
and . Examples 2.1.2 and 2.1.8 are not idempotents. Note that in such cases.
Theorem 2.2.7
If and are k-idempotent matrices then commutes with the
permutation matrix .
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Proof
Hence
Theorem 2.2.8
Let and are two commuting k-idempotent matrices. The -idempotency of
necessarily implies that it is a null matrix.
Proof
For any two k-idempotent matrices and we have commutes with
the permutation matrix by theorem 2.2.7.
If 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
2.1.7 (c). Substituting in ,
Since and are commuting -idempotent matrices, is also -idempotent by
theorem 2.2.3. Hence . by theorem 2.1.7 (c)
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Pre multiplying by , we have .
It follows that .
Remark 2.2.9
For example the matrices and in remark 2.2.2 can be considered.
.Clearly the matrix commutes
with the permutation matrix . It can be easily verified that is not a
-idempotent matrix.
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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 2.3.1
Any two of the following imply the other one. If ℂ then
(a) is -idempotent
(b) is -hermitian
(c) is square hermitian
Proof
(a) and (b) (c):
and
. Hence is square hermitian.
(b) and (c) (a):
Substituting in ,
We have . Hence is -idempotent.
(c) and (a) (b):
Substituting in ,
We have . Hence is -hermitian.
Corollary 2.3.2
Let be -hermitian -idempotent matrix. If is non-singular then is unitary.
Proof
Since is -hermitian -idempotent, is square hermitian by theorem 2.3.1
If is non-singular then by theorem 2.1.7 (c)
Therefore and hence is unitary.
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Example 2.3.3
Let
(i) is -idempotent.
(ii) is square hermitian.
(iii) is -hermitian.
(iv) is unitary
Theorem 2.3.4
Let be a -idempotent matrix. If is cube hermitian then it reduces to an
orthogonal projector.
Proof
Since is a -idempotent matrix, we have
by remark 2.1.4
If is cube hermitian then
Pre and post multiplying by , we have
by
by
Substituting in , we have . Hence is an orthogonal projector.
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Corollary 2.3.5
Let be a -idempotent matrix. If is -cube hermitian then it reduces to an
orthogonal projector.
Proof
If is -cube hermitian then
by remark 2.1.4
Hence reduces to cube hermitian matrix. By theorem 2.3.4, the matrix is an
orthogonal projector.
Note 2.3.6
Let be a -idempotent matrix. If then by theorem 2.1.7 (c) we have
(i.e., reduces to hermitian matrix )
Theorem 2.3.7
Let be a -idempotent matrix. Then the following are equivalent.
(i) is cube hermitian.
(ii) is hermitian.
(iii) is square hermitian.
Proof
(i) (ii):
If is cube hermitian then
by theorem 2.1.7 (e)
Hence is hermitian.
(ii) (iii):
If is hermitian then
. Hence is square hermitian.
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(iii) (ii):
If is square hermitian then
But
Hence is cube hermitian.
Theorem 2.3.8
Let be a -idempotent matrix. Then the necessary and sufficient condition for
the matrix to be square hermitian is
(i) is idempotent.
(ii)
Proof
Assume that is square hermitian.
Pre and post multiplying by respectively,
we have and
since , we have
.
Hence is idempotent and substituting this in , we have .
Conversely, assume that is idempotent with .
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Hence is square hermitian.
Corollary 2.3.9
Let be a -idempotent matrix. Then the necessary and sufficient condition for
to be -square hermitian is
(i) is idempotent
(ii)
Proof
Assume that is -square hermitian.
is square hermitian.
By theorem 2.3.8, this is equivalent to is idempotent and
Theorem 2.3.10
Let be a -idempotent matrix. Then the following are equivalent.
(i) is -cube hermitian
(ii) is hermitian
(iii) is -square hermitian
(iv) is -hermitian
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Proof
(i) (ii):
If is -cube hermitian then
by theorem 2.1.7 (e)
.
Hence is hermitian.
(ii) (iii):
If is hermitian then
.
Hence is -square hermitian.
(iii) (iv):
If is -square hermitian then
Pre and post multiplying by , we have
.
Hence is -hermitian.
(iv) (i):
If is -hermitian then
But by theorem 2.1.7 (e)
.
Hence is -cube hermitian.
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Note 2.3.11
The following theorem can be considered to be a converse of the above theorem
2.3.10.
Theorem 2.3.12
If is hermitian and -square hermitian then is -idempotent.
Proof
Assume that and
Combining the above two relations, we have .
Hence is -idempotent.