104

CHAPTER 6

Generalized inverses of -idempotent matrix

Various generalized inverses [11] of a -idempotent matrix are studied and

the corresponding inverses for the elements in group

are determined (cf. theorem 2.1.9). A condition for the Moore Penrose inverse (see

definition 1.2.6) of a -idempotent matrix to be -idempotent is derived (cf.[34]). A

column and row inverse [41] of a -idempotent matrix is found in section 2 and then

it is shown that the group inverse (see definition 1.2.7) of a -idempotent matrix is

. A commuting pseudo inverse of the corresponding elements in group is also

found. The -idempotency of EP matrices [20,38,43] is analyzed in this chapter. An

equivalent condition for a -idempotent matrix to be EP is also determined.

105

6.1 Moore Penrose inverse of -idempotent matrix

In this section ( -inverses, (2 -inverses, ( -inverses and ( -inverses of a

-idempotent matrix is found. Various generalized inverses for the matrices

in group (cf. theorem 2.1.9) are also found. A necessary condition

for the Moore Penrose inverse of a -idempotent matrix to be -idempotent is also

derived.

Theorem 6.1.1

If is a -inverse (generalized inverse) of a -idempotent matrix then

(i)

(ii)

(iii)

(iv) or

(v) or

Proof

(i)

(ii)

(iii) A similar proof may be given to show that as in (ii).

(iv)

by theorem 2.1.7 (c)

106

Therefore .

In a similar manner, it can be proved that

( i.e., ).

(v)

by theorem 2.1.7 (c)

Therefore . In a similar manner, it can be proved that

( i.e., ).

Theorem 6.1.2

If is a -inverse (reflexive generalized inverse) of a -idempotent matrix

then

(i)

(ii)

(iii)

(iv) or

(v) or

Proof

The proof is analogous to the proof of theorem 6.1.1.

Theorem 6.1.3

If is a -idempotent matrix then .

Proof

(i)

by theorem 2.1.7 (c)

107

(ii)

From (i) and (ii), we have .

Note 6.1.4

Using in theorems 6.1.1 and 6.1.2 we have ;

; ; ; .

It is also observed that each matrices commutes with their

respective -inverses.

Remark 6.1.5

The set of all -inverses of is given by (cf. corollary 1.2.15)

for some .

Since for a -idempotent matrix , we have

Theorem 6.1.6

If is a -inverse of a -idempotent matrix then

(i)

(ii)

(iii)

(iv)

(v)

Proof

By theorem 6.1.1, the -inverses of of the corresponding matrices is settled.

108

(i)

Therefore

(ii)

Therefore

(iii)

Therefore .

(iv)

Therefore

(v)

109

Therefore .

Theorem 6.1.7

If is a -inverse of a -idempotent matrix then

(i)

(ii)

(iii)

(iv)

(v)

Proof

The proof of (i) to (iii) are analogous to that of (i) to (iii) proved in theorem 6.1.6.

(iv)

Therefore .

(v)

Therefore .

Theorem 6.1.8

If is the Moore Penrose inverse of a -idempotent matrix then

(i)

(ii)

(iii)

110

(iv)

(v)

Proof

Combining theorems 6.1.1, 6.1.2, 6.1.6 and 6.1.7 we see that the results (i) to (iii)

can be easily proved.

(iv) Since is the Moore Penrose inverse of a -idempotent matrix , we have is

a -inverse of by theorem 6.1.6. Similarly is a -inverse of by

theorem 6.1.7.

But we have by theorem 1.2.16

Therefore

by theorem 2.1.7 (c)

(v) By theorem 1.2.18, we have

Theorem 6.1.9

Let be a -idempotent matrix. If is hermitian then .

Proof

It is true that if is -idempotent then . by theorem 6.1.3

Also

Hence we have .

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Theorem 6.1.10

Let be a -idempotent matrix. Then commutes with the associated

permutation matrix .

Proof

If is a -idempotent matrix then by remark 2.1.4 , we see that

Therefore .

Theorem 6.1.11

Let be a -idempotent matrix. If is square hermitian then .

Proof

Since is square hermitian then .

Clearly . by theorem 6.1.3

It is true that

and

Therefore we have .

Theorem 6.1.12

Let be a -idempotent matrix. If is EP (range hermitian) then is also

-idempotent.

Proof

Assume that is EP .By theorem 1.2.13 , we have

112

(1)

by

(2)

by

(3)

by

by

by

(4)

by

by

by

From (1) to (4), we see that

113

Hence

by theorem 1.2.8

by

Therefore is -idempotent.

Corollary 6.1.13

Let be a -idempotent matrix. If is normal then is also -idempotent.

Proof

If is normal then clearly is EP

By theorem 6.1.12, is also -idempotent.

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6.2 Group inverse of -idempotent matrix

In this section, a generalized inverse of a -idempotent matrix whose column

space is identical with the column space of is to be found. This is known as column

inverse or -inverse and it is denoted by

. Similarly, a generalized inverse of the

matrix whose row space is identical with the row space of is to be found. This is

known as row inverse or -inverse and it is denoted by

. The following lemma

establishes the existence of such inverses (cf.[41]).

Lemma 6.2.1

Given matrices and of order , a necessary and sufficient condition for the

matrix to have a generalized inverse such that and

[where denotes the column space of ], is that rank rank .

An inverse with the required property is unique such that if

rank rank rank .

Proof

Let be a generalized inverse of .

Then

But rank rank

rank rank

Therefore we have

rank rank

Hence, rank rank

But rank rank

rank

rank rank by

115

From and , we have rank rank

By lemma 1.2.20 , it follows that is a generalized inverse of .

Now,

This implies that

rank rank and rank rank

and for some matrices and

Hence

Therefore and ( i.e.,

implies that

implies that

Theorem 6.2.2

Let be a -idempotent matrix and let be any generalized inverse

of whose column space is identical with the column space of (column inverse

)

is expressible in the form and a generalized inverse of whose

row space is identical with the row space of [row inverse

] is in the form

.

Proof

We see that rank rank rank .

Taking and in lemma 6.2.1, we have rank rank .

Hence the condition is satisfied.

116

Therefore . It follows that

by theorem 6.1.1

In a similar manner, taking and in lemma 6.2.1, we have

rank rank . Hence the condition is satisfied.

Therefore . It follows that

by theorem 6.1.1

Hence the theorem is proved.

Theorem 6.2.3

Let be a -idempotent matrix. If

is a -inverse (column inverse) of then

(i)

(ii)

(iii)

(iv)

(v)

Proof

By theorem 6.2.2, we see that see

(i)

117

(ii)

(iii)

(iv)

(v)

118

Hence the theorem is proved.

Theorem 6.2.4

Let be a -idempotent matrix. If

is a -inverse (row inverse) of then

(i)

(ii)

(iii)

(iv)

(v)

Proof

By theorem 6.2.2, we see that see

(i)

(ii)

119

(iii)

(iv)

(v)

Hence the theorem is proved.

Remark 6.2.5

Taking in lemma 6.2.1, we have is the

unique generalized inverse with row and column space coincident with that of a

-idempotent matrix .

By lemma 1.2.17,we have

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A reflexive generalized inverse satisfying the property is known as group inverse

and it is denoted by . It is true that a group inverse when exists is unique.

Theorem 6.2.6

If is the group inverse of a -idempotent matrix then

(i)

(ii)

(iii)

(iv)

(v)

(vi)

Proof

By theorem 6.1.3, we have .

Since , we have .

By the note 6.1.4, we see that (ii) to (vi) can be easily proved.

Remark 6.2.7

For a square matrix of order , Drazin defined a pseudo inverse satisfying the

following conditions

i.

ii. , for some positive integer .

iii.

-is known as commuting pseudo inverse of (cf.[41]) and it is denoted by . The

smallest positive integer for which the above condition is satisfied, is called the index of

. It is true that exists and is unique (cf.[41]).

121

Theorem 6.2.8

If is the commuting pseudo inverse of a -idempotent matrix then

(i)

(ii)

(iii)

(iv)

(v)

(vi)

Proof

(i)

. Hence the index of is one.

Therefore .

The proof of (ii) is similar to (i).

(iii) Clearly commutes with itself.

Since is a tripotent matrix [by theorem 2.1.7 (e)], we have

It follows that .

The proof of (iv) is similar to (iii).

(v) Since is idempotent [by theorem 2.1.7 (d)], it is obvious that

(vi) commutes with itself.

Since is tripotent ,we have

122

Hence the proof of the theorem.

Theorem 6.2.9

If is -idempotent matrix such that where is a Jordan normal form

of then .

Proof

If is -idempotent matrix, we have rank rank .

By theorem 1.2.18,

Since is -idempotent,

by theorem 6.2.6

Since , we have

From and ,

Therefore .

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6.3 -idempotency of EP matrix

In this section, an equivalent condition for a -idempotent matrix to be EP is

derived.

Theorem 6.3.1

Let be a -idempotent matrix. Then and are complementary

subspaces of the unitary space.

Proof

It is true that and are clearly subspaces of the unitary space.

Let . Hence there exists a vector such that

and

It follows that .

Therefore .

Remark 6.3.2

A basis for is useful in a number of applications such as, for example

numerical computation of Moore Penrose inverse and group inverse. The general solution

of the linear non-homogenous equation is the sum of any particular solution and

the general solution of the homogenous equation . This general solution consists

of all linear combinations of the elements of any basis . A basis for the range space

and null space of a -idempotent matrix as in example 2.1.2 are given by

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Basis of

Basis of

Theorem 6.3.3

Let be a -idempotent matrix. Then the following are equivalent.

(1) is square hermitian

(2) is hermitian

(3)

(4) is EP

Proof

(1)⟹(2):

Since is square hermitian,

Hence is hermitian.

(2)⟹(3):

By theorem 6.1.9 , we have

by theorem 6.2.6

By theorem 1.2.13, we see that

is EP if and only if . That is (3) is equivalent to (4).

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Corollary 6.3.4

Let be a -idempotent matrix. If is cube hermitian then reduces to an

idempotent matrix.

Proof

Since is cube hermitian, we have

by theorem 2.1.7 (c)

by

This implies that is hermitian. By theorem 6.3.3 , the matrix is square hermitian.

From and , we have .

Thus and hence is idempotent.

Theorem 6.3.5

Let be a -idempotent matrix. If is EP then and are also EP.

Proof

If is EP then . by theorem 6.3.3

by theorem 6.2.6

by theorem 6.2.6

Therefore is EP.

From ,

126

Therefore is EP.

Theorem 6.3.6

Let be a -idempotent matrix. If is a EP matrix then

Proof

By theorem 1.2.13 , if is a EP matrix then

By theorem 6.1.8, we have

by