𝑘-Kernel Symmetric Matrices

Abstract

In this paper we present equivalent characterizations of 𝑘-Kernel symmetric Matrices. Necessary and sufficient conditions are determined for a matrix to be 𝑘-Kernel Symmetric. We give some basic results of kernel symmetric matrices. It is shown that k-symmetric implies 𝑘-Kernel symmetric but the converse need not be true. We derive some basic properties of 𝑘-Kernel symmetric fuzzy matrices. We obtain k-similar and scalar product of a fuzzy matrix.

1. Introduction

Throughout we deal with fuzzy matrices that is, matrices over a fuzzy algebra ℱ with support [0,1] under max-min operations. For 𝑎,𝑏∈ℱ, 𝑎+𝑏=max{𝑎,𝑏}, 𝑎⋅𝑏=min{𝑎,𝑏}, let ℱ_𝑚𝑛 be the set of all 𝑚×𝑛 matrices over ℱ, in short ℱ_𝑛𝑛 is denoted as ℱ_𝑛. For 𝐴∈ℱ_𝑛, let 𝐴^𝑇, 𝐴⁺, 𝑅(𝐴), 𝐶(𝐴), 𝑁(𝐴), and 𝜌(𝐴) denote the transpose, Moore-Penrose inverse, Row space, Column space, Null space, and rank of 𝐴, respectively. 𝐴 is said to be regular if 𝐴𝑋𝐴=𝐴 has a solution. We denote a solution 𝑋 of the equation 𝐴𝑋𝐴=𝐴 by 𝐴⁻ and is called a generalized inverse, in short, g-inverse of 𝐴. However 𝐴{1} denotes the set of all g-inverses of a regular fuzzy matrix A. For a fuzzy matrix A, if 𝐴⁺ exists, then it coincides with 𝐴^𝑇 [1, Theorem 3.16]. A fuzzy matrix A is range symmetric if 𝑅(𝐴)=𝑅(𝐴^𝑇) and Kernel symmetric if 𝑁(𝐴)=𝑁(𝐴^𝑇)={𝑥∶𝑥𝐴=0}. It is well known that for complex matrices, the concept of range symmetric and kernel symmetric is identical. For fuzzy matrix 𝐴∈ℱ_𝑛, 𝐴 is range symmetric, that is, 𝑅(𝐴)=𝑅(𝐴^𝑇) implies 𝑁(𝐴)=𝑁(𝐴^𝑇) but converse needs not be true [2, page 217]. Throughout, let 𝑘-be a fixed product of disjoint transpositions in 𝑆_𝑛=1,2,…,𝑛 and, 𝐾 be the associated permutation matrix. A matrix 𝐴=(𝑎_𝑖𝑗)∈ℱ_𝑛 is 𝑘-Symmetric if 𝑎_𝑖𝑗=𝑎_{𝑘(𝑗)𝑘(𝑖)} for 𝑖,𝑗=1 to 𝑛. A theory for 𝑘-hermitian matrices over the complex field is developed in [3] and the concept of 𝑘-EP matrices as a generalization of 𝑘-hermitian and EP (or) equivalently kernel symmetric matrices over the complex field is studied in [4–6]. Further, many of the basic results on 𝑘-hermitian and EP matrices are obtained for the 𝑘-EP matrices. In this paper we extend the concept of 𝑘-Kernel symmetric matrices for fuzzy matrices and characterizations of a 𝑘-Kernel symmetric matrix is obtained which includes the result found in [2] as a particular case analogous to that of the results on complex matrices found in [5].

2. Preliminaries

For 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)∈ℱ_1×𝑛, let us define the function 𝜅(𝑥)=(𝑥_𝑘(1),𝑥_𝑘(2),…,𝑥_𝑘(𝑛))^𝑇∈ℱ_𝑛×1. Since 𝐾 is involutory, it can be verified that the associated permutation matrix satisfy the following properties.

Since 𝐾 is a permutation matrix, 𝐾𝐾^𝑇=𝐾^𝑇𝐾=𝐼_𝑛 and 𝐾 is an involution, that is, 𝐾²=𝐼, we have 𝐾^𝑇=𝐾.

(P.1)𝐾=𝐾^𝑇, 𝐾²=𝐼, and 𝜅(𝑥)=𝐾𝑥 For 𝐴∈ℱ_𝑛 ,(P.2)𝑁(𝐴)=𝑁(𝐴𝐾),(P.3)if 𝐴⁺ exists, then (𝐾𝐴)⁺=𝐴⁺𝐾 and (𝐴𝐾)⁺=𝐾𝐴⁺(P.4)𝐴⁺ exist if and only if 𝐴^𝑇 is a g-inverse of 𝐴.

Definition 2.1 (see [2, page 119]). For 𝐴∈ℱ_𝑛 is kernel symmetric if 𝑁(𝐴)=𝑁(𝐴^𝑇), where 𝑁(𝐴)={𝑥/𝑥𝐴=0and𝑥∈ℱ_1×𝑛}, we will make use of the following results.

Lemma 2.2 (see [2, page 125]). For 𝐴,𝐵∈ℱ_𝑛 and 𝑃 being a permutation matrix, 𝑁(𝐴)=𝑁(𝐵)⇔𝑁(𝑃𝐴𝑃^𝑇)=𝑁(𝑃𝐵𝑃^𝑇)

Theorem 2.3 (see [2, page 127]). For 𝐴∈ℱ_𝑛, the following statements are equivalent:(1)𝐴 is Kernel symmetric,(2)𝑃𝐴𝑃^𝑇 is Kernel symmetric for some permutation matrix 𝑃,(3)there exists a permutation matrix 𝑃 such that 𝑃𝐴𝑃^𝑇=[^𝐷0₀₀] with det𝐷>0.

3. 𝑘-Kernel Symmetric Matrices

Definition 3.1. A matrix 𝐴∈ℱ_𝑛 is said to be 𝑘-Kernel symmetric if 𝑁(𝐴)=𝑁(𝐾𝐴^𝑇𝐾)

Remark 3.2. In particular, when 𝜅(𝑖)=𝑖 for each 𝑖=1 to 𝑛, the associated permutation matrix 𝐾 reduces to the identity matrix and Definition 3.1 reduces to 𝑁(𝐴)=𝑁(𝐴^𝑇), that is, 𝐴 is Kernel symmetric. If 𝐴 is symmetric, then 𝐴 is 𝑘-Kernel symmetric for all transpositions 𝑘 in 𝑆_𝑛.
Further, 𝐴 is 𝑘-Symmetric implies it is 𝑘-kernel symmetric, for 𝐴=𝐾𝐴^𝑇𝐾 automatically implies 𝑁(𝐴)=𝑁(𝐾𝐴^𝑇𝐾). However, converse needs not be true. This is, illustrated in the following example.

Example 3.3. Let 𝐴=000.60.5100.50.30,𝐾=001010100𝐾𝐴^𝑇𝐾=.000.60.3100.50.50 (3.1)
Therefore, 𝐴 is not 𝑘-symmetric.
For this 𝐴, 𝑁(𝐴)={0}, since 𝐴 has no zero rows and no zero columns.
𝑁(𝐾𝐴^𝑇𝐾)={0}. Hence 𝐴 is 𝑘-Kernel symmetric, but 𝐴 is not 𝑘-symmetric.

Lemma 3.4. For 𝐴∈ℱ_𝑛, 𝐴⁺ exists if and only if (𝐾𝐴)⁺ exists.

Proof. By [1, Theorem 3.16], For 𝐴∈ℱ_𝑚𝑛 if 𝐴⁺ exists then 𝐴⁺=𝐴^𝑇 which implies 𝐴^𝑇 is a g-inverse of 𝐴. Conversely if 𝐴^𝑇 is a g-inverse of 𝐴, then 𝐴𝐴^𝑇𝐴=𝐴⇒𝐴^𝑇𝐴𝐴^𝑇=𝐴^𝑇. Hence 𝐴^𝑇 is a 2 inverse of 𝐴. Both 𝐴𝐴^𝑇 and 𝐴^𝑇𝐴 are symmetric. Hence 𝐴^𝑇=𝐴⁺: 𝐴⁺exists⟺𝐴𝐴^𝑇𝐴=𝐴⟺𝐾𝐴𝐴^𝑇𝐴=𝐾𝐴⟺(𝐾𝐴)(𝐾𝐴)^𝑇(𝐾𝐴)=𝐾𝐴⟺(𝐾𝐴)^𝑇∈(𝐾𝐴){1}⟺(𝐾𝐴)⁺,exists(By,P.4). (3.2)

For sake of completeness we will state the characterization of 𝑘-kernel symmetric fuzzy matrices in the following. The proof directly follows from Definition 3.1 and by (P.2).

Theorem 3.5. For 𝐴∈ℱ_𝑛, the following statements are equivalent:(1)𝐴 is 𝑘-Kernel symmetric,(2)𝐾𝐴 is Kernel symmetric,(3)𝐴𝐾 is Kernel symmetric,(4)𝑁(𝐴^𝑇)=𝑁(𝐾𝐴),(5)𝑁(𝐴)=𝑁((𝐴𝐾)^𝑇),

Lemma 3.6. Let 𝐴∈ℱ_𝑛, then any two of the following conditions imply the other one, (1)𝐴 is Kernel symmetric,(2)𝐴 is 𝑘-Kernel symmetric,(3)𝑁(𝐴^𝑇)=𝑁((𝐴𝐾)^𝑇).

Proof. However, (1) and (2) ⇒ (3): 𝐴is𝑘-Kernelsymmetric⇒𝑁(𝐴)=𝑁(𝐾𝐴^𝑇𝐾)⇒𝑁(𝐴)=𝑁(𝐾𝐴^𝑇)(By,P.2)Hence,(1)and(2)⇒𝑁(𝐴^𝑇)=𝑁(𝐴)=𝑁((𝐴𝐾)^𝑇). (3.3) Thus (3) holds.
Also (1) and (3) ⇒ (2):
𝐴isKernelsymmetric⇒𝑁(𝐴)=𝑁(𝐴^𝑇)Hence,(1)and(3)⇒𝑁(𝐴)=𝑁((𝐴𝐾)^𝑇)⇒𝑁(𝐴𝐾)=𝑁((𝐴𝐾)^𝑇)(By,P.2)⇒𝐴𝐾isKernelsymmetric⇒𝐴is𝑘-Kernelsymmetric(byTheorem(3.5)). (3.4) Thus (2) holds.
However, (2) and (3) ⇒ (1):
𝐴is𝑘-Kernelsymmetric⇒𝑁(𝐴)=𝑁(𝐾𝐴^𝑇𝐾)⇒𝑁(𝐴)=𝑁((𝐴𝐾)^𝑇)(by,P.2)Hence(2)and(3)⇒𝑁(𝐴)=𝑁(𝐴^𝑇). (3.5) Thus, (1) holds.
Hence the theorem.

Toward characterizing a matrix being 𝑘-Kernel symmetric, we first prove the following lemma.

Lemma 3.7. Let 𝐵=[^𝐷0₀₀], where 𝐷 is 𝑟×𝑟 fuzzy matrix with no zero rows and no zero columns, then the following equivalent conditions hold: (1)𝐵 is 𝑘-Kernel symmetric,(2)𝑁(𝐵^𝑇)=𝑁((𝐵𝐾)^𝑇),(3)𝐾=[^𝐾10_0𝐾2] where 𝐾₁ and 𝐾₂ are permutation matrices of order r and 𝑛-𝑟, respectively,(4)𝑘=𝑘₁𝑘₂ where 𝑘₁ is the product of disjoint transpositions on 𝑆_𝑛={1,2,…,𝑛} leaving (𝑟+1,𝑟+2,…,𝑛) fixed and 𝑘₂ is the product of disjoint transposition leaving (1,2,…,𝑟) fixed.

Proof. Since 𝐷 has no zero rows and no zero columns 𝑁(𝐷)=𝑁(𝐷^𝑇)={0}. Therefore 𝑁(𝐵)=𝑁(𝐵^𝑇)≠{0} and 𝐵 is Kernel symmetric.
Now we will prove the equivalence of (1),(2), and (3). 𝐵 is 𝑘-Kernel symmetric ⇔𝑁(𝐵^𝑇)=𝑁((𝐵𝐾)^𝑇) follows from By Lemma (3.6).
Choose 𝑧=[0𝑦] with each component of 𝑦≠0 and partitioned in conformity with that of 𝐵=[^𝐷0₀₀]. Clearly, 𝑧∈𝑁(𝐵)=𝑁((𝐵^𝑇))=𝑁((𝐵𝐾)^𝑇). Let us partition 𝐾 as 𝐾=[^𝐾1𝐾3_{𝐾𝑇3𝐾2}], Then 𝐾𝐵^𝑇=𝐾₁𝐾₃𝐾^𝑇₃𝐾₂𝐷^𝑇0=𝐾00¹𝐷^𝑇0𝐾^𝑇₃𝐷^𝑇0. (3.6) Now
𝑧=[0𝑦]∈𝑁(𝐵)=𝑁(𝐾𝐵^𝑇)⇒[0y]𝐾₁𝐷^𝑇0𝐾^𝑇₃𝐷^𝑇0=0⇒𝑦𝐾^𝑇₃𝐷^𝑇=0 (3.7) Since 𝑁(𝐷^𝑇)=0, it follows that 𝑦𝐾^𝑇₃=0.
Since each component of 𝑦≠0 under max-min composition 𝑦𝐾^𝑇₃=0, this implies 𝐾^𝑇₃=0⇒𝐾₃=0.
Therefore 𝐾𝐾=¹00𝐾₂. (3.8) Thus, (3) holds, Conversely, if (3) holds, then
𝐾𝐵^𝑇=𝐾₁𝐷^𝑇000,𝑁(𝐾𝐵^𝑇)=𝑁(𝐵). (3.9) Thus (1)⇔(2)⇔(3) holds.
However, (3)⇔(4): the equivalence of (3) and (4) is clear from the definition of 𝑘.

Definition 3.8. For 𝐴,𝐵∈ℱ_𝑛, 𝐴 is 𝑘-similar to 𝐵 if there exists a permutation matrix 𝑃 such that 𝐴=(𝐾𝑃^𝑇𝐾)𝐵𝑃.

Theorem 3.9. For 𝐴∈ℱ_𝑛 and 𝑘=𝑘₁𝑘₂ (where 𝑘₁𝑘₂ as defined in Lemma 3.7). Then the following are equivalent:(1)𝐴 is 𝑘-Kernel symmetric of rank r,(2)𝐴 is 𝑘-similar to a diagonal block matrix [^𝐷0₀₀] with det𝐷>0,(3)𝐴=𝐾𝐺𝐿𝐺^𝑇 and 𝐿∈ℱ_𝑟 with det𝐿>0 and 𝐺^𝑇𝐺=𝐼_𝑟.

Proof. (1)⇔(2).
By using Theorem 2.3 and Lemma 3.7 the proof runs as follows. 𝐴is𝑘-Kernelsymmetric⟺𝐾𝐴isKernelsymmetric∶⟺𝑃𝐾𝐴𝑃^𝑇=𝐸000withdet𝐸>0forsomepermutationmatrix𝑃(ByTheorem(2.3))⟺𝐴=𝐾𝑃^𝑇𝑃𝐸000⟺𝐴=(𝐾𝑃^𝑇𝐾)𝐾𝐸000𝑃(ByP.1)⟺𝐴=𝐾𝑃^𝑇𝐾𝐾₁00𝐾₂𝑃𝐸000⟺𝐴=𝐾𝑃^𝑇𝐾𝐾₁𝑃𝐸000⟺𝐴=𝐾𝑃^𝑇𝐾𝐷000𝑃. (3.10) Thus 𝐴 is 𝑘-similar to a diagonal block matrix [^𝐷0₀₀], where 𝐷=𝐾₁𝐸 and det 𝐷>0 .
However, (2)⇔(3): 𝐴=𝐾𝑃^𝑇𝐾𝐾₁𝑃𝑃𝐸000=𝐾^𝑇1𝑃^𝑇₃𝑃^𝑇₂𝑃^𝑇₄𝐾₁00𝐾₂𝐷0𝑃00¹𝑃₂𝑃₃𝑃₄𝑃=𝐾^𝑇1𝑃^𝑇₂𝐾₁𝐷𝑃₁𝑃₂=𝐾𝐺𝐿𝐺^𝑇,where𝑃𝐺=^𝑇1𝑃^𝑇₂,𝐺^𝑇=𝑃₁𝑃₂,𝐿=𝐾₁𝐷∈ℱ_𝑟𝐺^𝑇𝐺=𝑃₁𝑃₂𝑃^𝑇₁𝑃^𝑇₂=𝑃₁𝑃^𝑇₁+𝑃₂𝑃^𝑇₂=𝐼_𝑟,𝐿∈ℱ_𝑟. (3.11) Hence the Proof.

Let 𝑥,𝑦∈ℱ_1×𝑛𝐴̇ scalar product of 𝑥 and 𝑦 is defined by 𝑥𝑦^𝑇=⟨𝑥,𝑦⟩. For any subset 𝑆∈ℱ_1×𝑛,𝑆^⟂={𝑦∶⟨𝑥,𝑦⟩=0,forall𝑥∈𝑆}.

Remark 3.10. In particular, when 𝜅(𝑖)=𝑖,𝐾 reduces to the identity matrix, then Theorem 3.9 reduces to Theorem 2.3. For a complex matrix 𝐴, it is well known that 𝑁(𝐴)^⟂=𝑅(𝐴^∗), where 𝑁(𝐴)^⟂ is the orthogonal complement of 𝑁(𝐴). However, this fails for a fuzzy matrix hence 𝐶^𝑛=𝑁(𝐴)⊕𝑅(𝐴) this decomposition fails for Kernel fuzzy matrix. Here we shall prove the partial inclusion relation in the following.

Theorem 3.11. For 𝐴∈ℱ_𝑛, if 𝑁(𝐴)≠{0}, then 𝑅(𝐴^𝑇)⊆𝑁(𝐴)^⟂ and 𝑅(𝐴^𝑇)≠ℱ_1×𝑛.

Proof. Let 𝑥≠0∈𝑁(𝐴), since 𝑥≠0, 𝑥_𝑖𝑜≠0 for atleast one 𝑖_𝑜. Suppose 𝑥_𝑖≠0 (say) then under the max-min composition 𝑥𝐴=0 implies, the 𝑖th row of 𝐴=0, therefore, the 𝑖th column of 𝐴^𝑇=0. If 𝑥∈𝑅(𝐴^𝑇), then there exists 𝑦∈ℱ_1×𝑛 such that 𝑦𝐴^𝑇=𝑥. Since 𝑖th column of 𝐴^𝑇=0, it follows that, 𝑖th component of 𝑥=0, that is, 𝑥_𝑖=0 which is a contradiction. Hence 𝑥∉𝑅(𝐴^𝑇) and 𝑅(𝐴^𝑇)≠ℱ_1×𝑛.
For any 𝑧∈𝑅(𝐴^𝑇), 𝑧=𝑦𝐴^𝑇 for some 𝑦∈ℱ_1×𝑛. For any 𝑥∈𝑁(𝐴), 𝑥𝐴=0 and ⟨𝑥,𝑧⟩=𝑥𝑧^𝑇=𝑥(𝑦𝐴^𝑇)^𝑇=𝑥𝐴𝑦^𝑇=0. (3.12) Therefore, z ∈𝑁(𝐴)^⟂, 𝑅(𝐴^𝑇)⊆𝑁(𝐴)^⟂.

Remark 3.12. We observe that the converse of Theorem 3.11 needs not be true. That is , if 𝑅(𝐴^𝑇)≠ℱ_1×𝑛, then 𝑁(𝐴)≠{0} and 𝑁(𝐴)^⟂⊆𝑅(𝐴^𝑇) need not be true. These are illustrated in the following Examples.

Example 3.13. Let 𝐴=000.60.5100.50.30 (3.13) since 𝐴 has no zero columns, 𝑁(𝐴)={0}.
For this 𝐴,𝑅(𝐴^𝑇)={(𝑥,𝑦,𝑧)∶0≤𝑥≤0.6,0≤𝑦≤1,0≤𝑧≤0.5}.
Therefore, 𝑅(𝐴^𝑇)≠ℱ_1×3.

Example 3.14. Let 𝐴=.110010000 (3.14) For this 𝐴, 𝑁(A)={(0,0,𝑧)∶𝑧∈ℱ},𝑁(𝐴)^⟂={(𝑥,𝑦,0)∶𝑥,𝑦∈ℱ}, (3.15) Here, 𝑅(𝐴^𝑇)={(𝑥,𝑦,0)∶0≤𝑦≤𝑥≤1}≠ℱ_1×3.
Therefore, for 𝑥>𝑦∈ℱ, (𝑥,𝑦,0)∈𝑁(𝐴)^⟂ but (𝑥,𝑦,0)∉𝑅(𝐴^𝑇).
Therefore, 𝑁(𝐴)^⟂ is not contained in 𝑅(𝐴^𝑇).