CHAPTER 4
Inequalities on -idempotent matrices
In this chapter, it is shown that all the standard partial orderings [2,7,24] such as
Lowener, star and rank subtractivity are preserved under the fixed product of disjoint
transpositions . That is all the partial orderings are preserved under -unitary similarity.
Relation between -hermitian matrix and -idempotent matrix is derived here by means
of Lowener partial order. It is proved that all the partial orderings are preserved for
-idempotent matrices when they are squared.
62
4.1
-invariant partial orderings on matrices
In this section, it is to be proved that all the standard partial orderings are
preserved under -unitary similarity. The Lowener partial order, the star order and the
minus order (rank subtractivity order) denoted by
,
and
respectively are defined
as follows:
Definition 4.1.1
ℂ
For

iff

iff

iff rank
,
.
and
rank
rank
.
Remark 4.1.2
The following useful result (cf.[25]) explains the relation
partial order and the spectral radius of the transformation. For
and
where
max
between Lowener
ℂ
,
,
-an eigen value of
is the spectral radius of .
Lemma 4.1.3
If
is the associated permutation matrix of , then for
ℂ
,
Proof
by remark 4.1.2
and
and
by theorem 1.2.9
and
and
by theorem 1.2.9
63
by remark 4.1.2
Similarly,
and
by remark 4.1.2
and
by theorem 1.2.9
and
and
by theorem 1.2.9
by remark 4.1.2
Hence the proof.
Result 4.1.4
It can be easily verified that Lowener ordering is preserved under unitary
similarity. That is
Theorem 4.1.5
Lowener ordering is preserved for -unitary similarity.
Proof
by lemma 4.1.3
by result 4.1.4
by lemma 4.1.3
If
then
is unitarily -similar to .
If
then
is unitarily -similar to .
64
Therefore
and hence Lowener ordering is preserved for -unitary similarity.
Note 4.1.6
If
in theorem 4.1.5, it reduces to result 4.1.4.
Remark 4.1.7
The following result (see[24]) establishes a relation between star ordering and
ℂ
rank subtractivity. For
,
and
Lemma 4.1.8
If
is the associated permutation matrix of , then for
ℂ
,
Proof
and
and
and
Similarly,
and
and
and
65
Result 4.1.9
It can be easily verified that star ordering is preserved under unitary similarity.
That is
Theorem 4.1.10
Star ordering is preserved for -unitary similarity.
Proof
by lemma 4.1.8
by result 4.1.9
by lemma 4.1.8
If
then
is unitarily -similar to .
If
then
is unitarily -similar to .
Therefore
and hence star ordering is preserved for -unitary similarity.
Note 4.1.11
If
in theorem 4.1.10, it reduces to result 4.1.9.
Remark 4.1.12
The following result exhibits an equivalent condition for minus ordering of two
matrices
and . For
ℂ
,
Lemma 4.1.13
If
is the associated permutation matrix of , then for
ℂ
,
66
Proof
rank
rank
rank
rank
rank
rank
rank
rank
rank
Similarly,
rank
rank
rank
rank
rank
rank
rank
rank
rank
Result 4.1.14
It can be easily verified that rank subtractivity ordering is preserved under unitary
similarity. That is
Theorem 4.1.15
Rank subtractivity ordering is preserved for -unitary similarity.
Proof
by lemma 4.1.13
by result 4.1.14
by lemma 4.1.13
If
then
is unitarily -similar to .
If
then
is unitarily -similar to .
67
Therefore
and hence rank subtractivity ordering is preserved for
-unitary
similarity.
Note 4.1.16
If
in theorem 4.1.15, it reduces to result 4.1.14.
68
4.2 Partial orderings on -idempotent matrices
In this section, partial orderings on -idempotent matrices is discussed and a
relation between a square hermitian matrix and a -idempotent matrix through Lowener
partial order is discovered.
Lemma 4.2.1
If
for matrices
and
then
is -hermitian.
Proof
implies that
Hence
by lemma 4.1.3
is -hermitian positive semi definite.
That is
is -hermitian.
Theorem 4.2.2
Let
and
. If
is a -idempotent matrix then
.
Proof
implies that
Therefore
Since
Hence from
, we have
by lemma 4.2.1.
, we have
69
That is
Therefore
Theorem 4.2.3
Let
. If
is -hermitian -idempotent then
is -hermitian and vice
versa.
Proof
Since
, we have
is -hermitian.
by lemma 4.2.1
That is
If
is -hermitian -idempotent then
is square hermitian by theorem 2.3.1.
becomes
Hence
is -hermitian.
On the other hand, if
is -hermitian -idempotent then
is square hermitian by
theorem 2.3.1.
becomes
70
Hence
is -hermitian.
Theorem 4.2.4
Let
and
are -idempotent matrices. Then
if and only if
Proof
Assuming that
Therefore
Conversely, if we assume
, then we have
by what we have proved
above and theorem 2.1.7 (b).
That is
.
by theorem 2.1.7 (c)
Hence the theorem is proved.
Theorem 4.2.5
Let
and
are -idempotent matrices. Then
if and only if
Proof
Assuming that
then we have
(i)
From (i),
by theorem 2.1.7 (a)
71
From (ii),
From
and
, we have
Conversely, if we assume that
then we have
by what we have proved
above and theorem 2.1.7 (b).
That is
by theorem 2.1.7 (c)
Hence the theorem is proved.
Theorem 4.2.6
Let
and
are -idempotent matrices. Then
if and only if
Proof
Assuming that
by remark 4.1.12
Therefore
Conversely, if we assume that
by remark 4.1.12
then we have
by what we have proved
above and theorem 2.1.7 (b).
That is
by theorem 2.1.7 (c)
Hence the theorem is proved.
72
Theorem 4.2.7
If
and
such that
are two disjoint square hermitian and skew square hermitian matrices
then
is -idempotent.
Proof
Since
, we have
is -hermitian.
by lemma 4.2.1
That is
becomes,
Hence
is -idempotent.
Note 4.2.8
The following theorem can be considered to be somehow a reverse of the above
theorem 4.2.7
Theorem 4.2.9
If
and
are two disjoint -idempotent matrices such that
then
and
are square hermitian matrices.
Proof
implies that
is -hermitian
by lemma 4.2.1
73
Squaring
From
That is
, we have
and
and
, solving for
and
we have
and
.
are square hermitian matrices.
74