39

CHAPTER 3

Spectral and -spectral theory of

-idempotent matrices

In this chapter, eigen and -eigen values of a -idempotent matrix are found.

Relations between -eigen values of a matrix and eigen values of the matrix are

analyzed (compare [29]). -diagonalizability and -similarity of -idempotent matrices

are discussed. The spectral and -spectral resolution of a -idempotent matrix (cf.[32]) is

found and all of its oblique projectors are determined as a polynomial in . Relations

among the multiplicity of eigen values of a -idempotent matrix and the matrix

functions such as tr , det and rank are discussed. A set of necessary conditions is

listed for a -idempotent matrix to be reduced to an idempotent matrix. A list comprising

a set of conditions when a -idempotent matrix becomes null, is also determined. An

equivalent condition for -idempotent matrix to be normal is derived.

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3.1 -eigen value of a matrix

In this section, we shall define a -eigen value of a matrix as a special case of

generalized eigen value problem ( cf.[40] ) for some matrices and . For

that, first we define a permutation function K on the unitary space ℂ .

Let

ℂ then K is defined by,

K

ℂ ,where is the fixed disjoint product of transpositions in .

If is the associated permutation matrix of then it can be easily seen that K .

It is also clear that K K .

i.e., K K

ℂ

Example 3.1.1

Let

ℂ and let

Then K

Also K K

Definition 3.1.2

A -eigen value of a matrix is defined to be a zero of the polynomial

det . This polynomial is known as -characteristic polynomial.

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Definition 3.1.3

A non-zero vector in ℂ is said to be a -eigen vector of a complex

matrix associated with a -eigen value if it satisfies K where K is as

defined before. This is equivalent to .

Example 3.1.4

is a -idempotent matrix. The -eigen values of are

1,1 and -1. A -eigen vector corresponding to the -eigen value 1 is . It can

be verified that .

i.e.,

Theorem 3.1.5

If is a complex matrix inℂ then

(i) is a ( -eigen value, -eigen vector) pair for if and only if it is an (eigen

value, eigen vector) pair for .

(ii) Every matrix satisfies the - characteristic equation of .

(iii) Any set of -eigen vectors corresponding to distinct -eigen values of a matrix

must be linearly independent.

Proof

(i) If is a ( -eigen value, -eigen vector) pair for then

K

Therefore is a (eigen value, eigen vector) pair for . By retracing the

above arguments, we see that the converse is also true.

(ii) By Cayley-Hamilton theorem, every matrix satisfies its characteristic equation.

That is,

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det

det det

det det

by (3.1)

Therefore the matrix satisfies the - characteristic equation of . In a similar

manner, it can be easily proved that the matrix satisfies the -characteristic

equation of .

(iii) Any set of -eigen vectors corresponding to distinct -eigen values of a matrix

is the set of eigen vectors corresponding to distinct eigen values of the matrix

by what we have proved above in (i). Hence they are linearly independent.

Remark 3.1.6

If and denote the spectrum and -spectrum of respectively then it is

true that

by theorem 3.1.5 (i)

Theorem 3.1.7

Let ℂ . Then the quadratic form of is the quadratic form for .

Proof

Let be a -eigen vector of .

Then

If KA is a function defined by,

KA , then it is clearly a -quadratic form

in . Equivalently , we have KA .

Hence it is the quadratic form for .

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Remark 3.1.8

(i) A -hermitian matrix is positive -definite (positive -semi definite) if and only

if all -eigen values of are positive (non-negative).

(ii) It can be seen that, is positive -definite if and only if is positive definite.

(iii) If a -hermitian matrix is positive -definite then is diagonalizable with all

real eigen values of .

Definition 3.1.9

A matrix ℂ is said to be upper [lower] -triangular if whenever

[ ]. This is equivalent to ‘ is upper [lower] -triangular if and only if

is upper [lower] triangular.

Note 3.1.10

The elements are known as -diagonal elements of .

Remark 3.1.11

If a matrix ℂ has linearly independent -eigen vectors, we can form a

matrix in which the columns are linearly independent -eigen vectors of . Hence we

have where is the -diagonal matrix of and it is denoted by

.

The matrix consisting of linearly independent -eigen vectors of in example 3.1.4

is given by

and

Therefore

Definition 3.1.12

Two matrices and in ℂ are said to be -similar if there exists a non-

singular matrix ℂ such that . Equivalently, is -similar to if

and only if is similar to .

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Example 3.1.13

A matrix

is -similar to the matrix in example 3.1.4.

For a non singular matrix

and

, it can be

verified that .

Note that . That is is similar to .

Note 3.1.14

If and are -similar matrices then they have the same -characteristic

polynomial and hence the same -spectrum.

Remark 3.1.15

Let be the associated permutation matrix of a disjoint product of transpositions

‘ ’. Let be a non-singular matrix such that then we have

that is .

Theorem 3.1.16

Let be a -idempotent matrix.

(i) Let be a matrix similar to then is -idempotent if where

.

(ii) Let be a matrix -similar to then is -idempotent if where

.

Proof

(i)

by remark 3.1.15

. Hence is -idempotent.

(ii)

by remark 3.1.15

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Therefore is similar to . Hence is -idempotent by what we have proved

above in (i).

Theorem 3.1.17

Let and be two -idempotent matrices. If is -similar to then is

similar to .

Proof

Since is -similar to , we can find a non-singular matrix such that

Hence is similar to .

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3.2 Spectral characterizations of -idempotent matrices

In this section, the spectral resolution of a -idempotent matrix is determined as

well as the diagonalizability of -idempotent matrices is proved.

Theorem 3.2.1

Let be a -idempotent matrix. Then the eigen values of are zero or cube root

of unity.

Proof

Let be an eigen value of a -idempotent matrix . Then

by

by theorem 2.1.7 (c) and

Since , we have or and where

Example 3.2.2

is a -idempotent.

The eigen values of are .

Theorem 3.2.3

If a matrix ℂ is -idempotent then it is diagonalizable and the spectrum

where

. Moreover, there exist unique disjoint oblique

projectors for such that

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Proof

Since [ by theorem 2.1.7 (c) ], the polynomial is a multiple

of of and then every root of has multiplicity 1 (cf.[13], pp.11) . Hence the

matrix is diagonalizable.

Moreover, it is clear that . by theorem 3.2.1

We define ’s by the following formula,

Using , we have

In the case that for , we see that . Similarly when

0 .

By spectral theorem [see theorem 1.2.4], we see that the non-zero ’s so obtained are

disjoint oblique projectors (i.e., and for ) to satisfy the

decompositions and . The uniqueness of the decompositions can be proved as

follows:

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Suppose if possible, let ’s be non-zero disjoint oblique projectors such that

, for complex numbers and

. We wish to prove that this is

actually identical with and except for notations and order of terms. First, it is

proved that ’s are precisely the eigen values of the matrix .

Since , there exists a non-zero vector in the range of such that and

for .

Therefore is an eigen value of [ i.e., . Conversely, if is an eigen

value of then

Since ’s are disjoint, we can find at least one among the non-zero vectors for

which is linearly independent. Hence, it follows that for some . These

arguments show that the set of ’s equals the set of eigen values of . By suitably

changing the order of terms, we have .

Since the expression for is unique in terms of , we have for .

Hence the decompositions and are unique.

Example 3.2.4

Consider the -idempotent matrix in example 2.1.2. The oblique

projectors of are found to be

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It can be easily verified that the above projectors are disjoint (i.e., for ) and

satisfy the decompositions and .

Remark 3.2.5

By theorem 3.2.3, we observe that every -idempotent matrices are

diagonalizable. Let denote the multiplicity of eigen values ,1 and

respectively of a -idempotent matrix . Note that denotes the multiplicity of eigen

values as well as because ‘imaginary roots occur in pairs’. By spectral theorem, it

can be seen that a -idempotent matrix can be reduced to the following diagonal form (all

the entries off the main diagonal are understood to be zero),

That is we can find a matrix such that

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For example, consider -idempotent matrix in example 2.1.2. The eigen values

of are and . It is an easy exercise to find such that

that is

Theorem 3.2.6

Let denote the multiplicity of eigen values , 1 and

respectively of a -idempotent matrix of order . Then

(i) tr

(ii) det

(iii) rank tr

Proof

By diagonalizing the matrix , we can find a matrix such that satisfying .

It is obvious that

(i) tr

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(ii) If then we have det .

Otherwise, det

(iii) Clearly, rank

by ]

tr

Example 3.2.7

(i) Consider the -idempotent matrix in example 2.1.2. The eigen values

of are and . That is

tr

det

rank

(ii) Consider the -idempotent matrix in example 3.2.2. The eigen values

of are and . That is

tr

det

rank

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3.3 -spectral characterizations of -idempotent matrices

A -eigen value of a -idempotent matrix is found and an equivalent condition for

normality of a -idempotent matrix is also determined in this section.

Theorem 3.3.1

Let be a -idempotent matrix. Then the -eigen values of are and .

Proof

Pre multiplying by , we have

using

Pre multiplying by , we have

by theorem 2.1.7 (c)

using

Since , we have .

Example 3.3.2

The -eigen values of example 3.2.2 are .

Theorem 3.3.3

If a matrix ℂ is -idempotent then . Moreover, there

exist unique disjoint oblique projectors for such that

Proof

By and theorem 3.3.1, it is clear that .

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We define ’s by the following formula

Then

In the case that for , we have . It can be proved that the

non-zero ’s so obtained are unique disjoint oblique projectors such that satisfying the

decompositions and analogous to the proof of theorem 3.2.3.

Example 3.3.4

Consider the -idempotent matrix in example 2.1.2. The oblique

projectors of are found to be

.

It can be easily verified that the above projectors are disjoint and satisfy the

decompositions and .

Remark 3.3.5

The diagonalizability of (where is a -idempotent matrix) implies the

-diagonalizability of . Let denote the multiplicity of -eigen values

of then we can find a matrix such that

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For example, consider the -idempotent matrix in example 2.1.2. The

-eigen values of are and . It can be found that

It can be easily verified that .

That is

.

Theorem 3.3.6

Let be a -idempotent matrix. Then holds if and only if

or .

Proof

Suppose then by theorem 3.3.3

Therefore, by

But using

i.e.,

From and , we have .

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

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Example 3.3.7

Consider the -idempotent matrix in example 2.2.6. It can be easily

determined that the -eigen values of are and . That is . We observe

that . But the -eigen values of the matrix in example 2.1.2 are and .

Clearly .

Theorem 3.3.8

Let be a -idempotent matrix. Then holds [i.e., reduces to idempotent

(cf. lemma 2.2.5)] if and only if any one of the following condition holds.

(i) is idempotent

(ii) is -idempotent

(iii) is square hermitian

(iv) is hermitian

(v) is -hermitian

(vi) is -square hermitian

(vii) is cube hermitian

(viii) is -cube hermitian

(ix) is -cube hermitian

(x) is skew cube hermitian

(xi) is skew -cube hermitian

(xii) is skew square hermitian

(xiii) is skew -square hermitian

Proof

(i) Since is idempotent, we have

Therefore by

Hence

By theorem 3.3.6,

(ii) Since is -idempotent, we have by theorem 3.3.3

Since is -idempotent, we have by theorem 3.2.3

Hence

That is

Therefore

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By theorem 3.3.6, .

(iii)

Pre multiplying by , we have

Post multiplying by , we have

Since [by remark 2.1.4], we have

(iv) Since is hermitian, the eigen values of are real.

Therefore ℝ

That is .

Since is diagonalizable [by theorem 3.2.3], we have is idempotent.

(v)

. Hence is hermitian, which is (iv).

(vi)

. Hence is hermitian, which is (iv).

(vii) Since is cube hermitian

Pre and post multiplying by , we have

From and , we have

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(viii)

. Hence is cube hermitian, which is (vii).

(ix)

by theorem 2.1.7 (e)

. Hence is hermitian, which is (iv).

(x) It is true that

Since is skew cube hermitian, we have

(xi)

. Hence is skew cube hermitian, which is (x).

(xii)

Pre multiplying by , we have

Post multiplying by , we have

Since [by remark 2.1.4], we have

Therefore and hence it can be proved as before (x) that .

(xiii)

As before (xii), it can be proved that .

Theorem 3.3.9

Let be a -idempotent matrix. Then becomes null (i.e., if and only if any

one of the following condition holds.

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(a) is skew hermitian

(b) is skew hermitian

(c) is skew -hermitian

(d) is skew -hermitian

(e) is skew square hermitian

(f) is skew -square hermitian

(g) is skew cube hermitian

(h) is skew -cube hermitian

Proof

(a) Since is skew hermitian, the eigen values of are pure imaginary.

That is ℝ

It follows that .

Therefore .

(b) Since is skew hermitian, we have

ℝ

It follows that and hence .

(c)

Hence is skew hermitian, which is (b).

(d)

Hence is skew hermitian, which is (a).

(e)

Hence is skew -hermitian, which is (c).

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(f)

Hence is skew hermitian, which is (a).

(g)

by theorem 2.1.7 (e)

Hence is skew hermitian, which is (b).

(h)

Hence is skew hermitian, which is (a).

Theorem 3.3.10

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

(1) is normal

(2) is square hermitian

(3) is normal

(4) is hermitian

Proof

Suppose if is null then we observe that the theorem is trivially true. Assume that

.

(1) (2) :

By theorem 3.2.3, the matrix can be written as

, not all

Since is normal, ’s are orthogonal projectors (i.e., )

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Hence becomes

But it is an easy exercise to prove that

Thus we have and hence is square hermitian.

(2) (3) :

by

by

From and , we have

Hence is normal.

(3) (4) :

By theorem 3.3.3, the matrix can be written as

, not both

Since is normal, we have

Hence is hermitian.

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(4) (1) :

Therefore is square hermitian and it implies that is normal.