20

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.