# Inner Product over Fuzzy Matrices

Authors: A. Nagoor Gani; K. Kannan; A. R. Manikandan
Journal: Journal of Mathematics
 (2016)

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## Abstract

The purpose of this study was to introduce the inner product over fuzzy matrices. By virtue of this definition,α-norm is defined and the parallelogram law is proved. Again the relative fuzzy norm with respect to the inner product over fuzzy matrices is defined. Moreover Cauchy Schwarz inequality, Pythagoras, and Fundamental Minimum Principle are established. Some equivalent conditions are also proved.


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## Body

## 1. Introduction

The concept of fuzzy set was introduced by Zadeh [1] in 1965. Biswas [2] and El-Abyad and El-Hamouly [3] first tried to give a meaningful definition of fuzzy inner product space and associated fuzzy norm function. Also those definitions are restricted to the real linear space only. Felbin [4], Ragab, and Emam [5] introduced fuzzy 2-normed linear spaces. Meenakshi and Cokilavany [6], Bag and Samanta [7], and so forth have given different definitions of fuzzy normed linear spaces. Gani and Kalyani [8] introduced the concept of binormed sequences in fuzzy matrices. Gani and Kalyani [9] introduced the properties of fuzzy m-norm matrices. Gani and Kalyani Introduced the definition of fuzzy equivalence relation. Recently Majumdar and Samanta [10] and Hasankhani et al. [11] have introduced a definition of fuzzy inner product space whose associated norm is of Felbin type. Somasundaram and Beaula [12] defined the 2-fuzzy 2-normed linear space. Some standard results in fuzzy 2-normed linear spaces were extended. The organization of the paper is as follows. Section 2 provides some preliminary results which are used in this paper. In Section 3, we study some results on inner product over fuzzy matrices and the parallelogram law is proved. Section 4 is devoted to establishing Cauchy Schwarz inequality, Pythagoras, and Fundamental Minimum Principle theorem in fuzzy setting.In this paper, we have introduced the new concept of inner product over fuzzy matrices onMnF, the set of all fuzzy sets of P∗MnF. We introduce the notion of α-norm on an inner product over fuzzy matrices. With the help of it, the standard parallelogram law is proved. Some important results on inner product over fuzzy matrices are proved. Moreover Cauchy Schwarz inequality, Pythagoras, and Fundamental Minimum Principle are established. Some equivalent conditions are also proved.


## 2. Preliminaries

We considerF=[0,1] the fuzzy algebra with operation [+,·] and the standard order “≤” where a+b=max⁡{a,b} and a·b=min⁡{a,b} for all a, b in F. F is a commutative semiring with additive and multiplicative identities 0 and 1, respectively. Let MMN(F) denote the set of all m×n fuzzy matrices over F. In short, Mn(F) is the set of all fuzzy matrices of order n. Define “+” and scalar multiplication in MnF as A+B=[aij+bij], where A=[aij] and B=[bij], and cA=[caij], where c is in [0,1]; with these operations Mn(F) forms a linear space.


## 3.m-Norm and Binorm Fuzzy Matrices

Definition 1.
LetMn(F) be the set of all (n×n) fuzzy matrices over F=[0,1]. For every A in Mn(F) define m-norm of A denoted by Am as(1)Am=max⁡aij,whereA=aijorAm=max⁡a11,a12,…,aij,…,annorAm=∑i=1n∑j=1naij.Definition 2.
LetA be in MnF and let α be in [0,1] such that ‖A‖m=α. Then the pair (A,α) is called a fuzzy point in MnF and it is denoted by PAα. The dual fuzzy point for PAα is the point with α-norm (1-α) denoted by P∗=P1-α.Definition 3.
The set of all fuzzy points inMnF is given by P∗MnF={PAα/A∈MnF,α∈[0,1]}.InF we follow the usual ≤ order relation; correspondingly we define an order relation in P∗MnF as follows.Definition 4.
We definePAα≤PBβ if and only if α<β and PAα=PBβ if and only if A=B (then automatically α=β).Definition 5.
A binorm with respect to them-norm in MnF is a real valued function θ defined on P∗MnF×P∗MnF to [0,1] satisfying the following conditions: (i)
θP1,P2=0 if and only if P1=P2.(ii)
θP1,P2=θP2∗,P1∗.(iii)
θαP1,P2=αθP1,P2,∀α∈[0,1].(iv)
θP1+P2,P3≤θP1,P3+θP2,P3.Then (MnF,θ) is called a fuzzy binormed linear space with respect to the m-norm.
[θP1,P2 is defined for P1=P2 as 0 (by (i)). Therefore θP1,P2≠0 is defined for P1<P2].(i)
WhenP1=P2 both the fuzzy points coincide.ThereforeθP1,P2=0.ConverselyθP1,P2=0⇒ both fuzzy points should coincide P1=P2.(ii)
P1<P2⇒α1<α2⇒1-α2<1-α1⇒P2∗<P1∗.ThereforeθP1,P2=θP2∗,P1∗.(iii)
Case (i): letα<α1; (2)θαP1,P2⟹αP1<P2⟹αα1<α2⟹α<α2sinceα<α1<α2,(3)αθP1,P2⟹αα1<αα2⟹α<α⟹α<α2sinceα<α1<α2.From (2) and (3),(4)θαP1,P2⟹αθP1,P2.Case (ii): letα1<α;(5)θαP1,P2⟹αP1<P2⟹αα1<α2⟹α<α2sinceα<α1<α2,(6)αθP1,P2⟹αα1<αα2⟹α1<α2ifα2<αsinceα<α1<α2.From (5) and (3),(7)θαP1,P2⟹αθP1,P2.(iv)
Case (i): letα2<α1;(8)P1+P2<P3⟹α1+α2<α3⟹α1<α3⟹P1<P3,(9)θP1+P2,P3=θP1,P3.Case (ii): letα1<α2;(10)P1+P2<P3⟹α1+α2<α3⟹α2<α3⟹P2<P3,(11)θP1+P2,P3=θP2,P3.From (9) and (11),(12)θP1+P2,P3=eitherθP1,P3orθP2,P3.ThereforeθP1+P2,P3≤θP1,P3+θP2,P3.SinceθP2,P3≥0, [in fact θP1+P2,P3=θP1,P3+θP2,P3].Thus θ is a fuzzy binorm with respect to the m-norm in MnF and hence MnF is a fuzzy binormed linear space with respect to the m-norm.Definition 6.
LetMn(F) be a linear space over K (the field is real or complex number). A fuzzy subset θ of Mn(F)×R is called a fuzzy norm of Mn(F) if and only if, for all PA,PB∈P∗(MnF) and c in K.θ1
For alls∈R with s≤0, θPA,s=0.θ2
For alls∈R with s>0, θPA,s=1.θ3
For alls∈R with s>0, θcPA,s=θ(PA,s/c), if c≠0.θ4
For alls,t∈R, PA,PB∈P∗(MnF), θPA+PB,s+t≥min⁡θPA,s,θPB,t.θ5
θPA,· is a nondecreasing function of R and lim⁡θPA,s=1 as s→∞.The pair (MnF,θ) will be referred to as a fuzzy normed linear space.Theorem 7.
LetMnF be nonempty and P∗(MnF) be the set of all fuzzy sets in MnF. If P∗MnF={PAα/A∈MnF and α∈[0,1]} and PA is a bounded function for PA≤1.
LetK be the space of real numbers; then MnF is a linear space over the field K; where the addition and scalar multiplication are defined by(13)PA+PB=A,α+B,β=A+B,α∧βA,α∈PA,B,β∈PB,kPB=kPA,αA,α∈PA,wherekinK.Proof.
M
n
F is said to be fuzzy normed space if to every PA∈P∗(MnF) there is an associated nonnegative real number θ(PA) called the α-norm of PA in such a way that
(i)(14)θPA=0iffPA=0,θPA=0⟹PAαA∈MnF,α∈0,1=0⟹A=0,αin0,1⟹PA=0;therefore θPA=0 if and only if [PA]=0;
(ii) one has(15)θkPA=kθPA,∀kinK,θkPA=θkA,αA,α∈MnF,k∈K=kθA,αA∈MnF,α∈0,1,θkPA=kθPA;
(iii) one has(16)θPA+PB≤θPA+θPBfor  every,PA,PB∈P∗MnF.For(17)θPA+PB=θA,α+θB,βA,B∈MnF,α,β∈0,1=A+B,α∧βA,B∈MnF,α,β∈0,1≤θA,α∧β+θB,α∧βA,α,B,β∈MnF,θPA+PB≤θPA+θPB.Therefore (MnF,θ) is a fuzzy normed linear space.


## 4. Inner Product over Fuzzy Matrices

Definition 8.
LetMnF be a linear space over the filed C. The fuzzy subset η defined a mapping from P∗(MnF)×P∗(MnF)×C to [0,1] such that, for all PA,PB,Pg∈P∗(MnF), α∈C:(i)
Fors,t∈C, η(PA+PB,Pg,s+t)≥min⁡ηPA,Pg,s,ηPB,Pg,t.(ii)
Fors,t∈C, η(PA+PB,st)≥min⁡ηPA,PA,s2,ηPB,PB,t2.(iii)
Fors,t∈C, ηPA,PB,s=ηPB,PA,s-.(iv)
Fors∈C,ηkPA,PB,s=ηPA,PB,s/k, for all α≠0∈C.(v)
ηPA,PA,s=0 for all s∈C∖R+.(vi)
ηPA,PA,s=1 for all s>0 if and only if PA=0.(vii)
ηPA,PA,·:R→I(=[0;1]) is a monotonic nondecreasing function of R and lim⁡ηPA,PA,s=1 as s→∞.Then η is said to be 2-fuzzy inner product (2-FIP) on MnF and the pair MnF,θ is called a 2-fuzzy inner product space (2-FIPS).Example 9.
If(18)PA=0.80.30.20.60.90.60.10.70.8,PB=0.20.30.70.70.50.60.80.60.7,Pg=0.30.60.70.10.50.20.20.60.3,s=0.70.40.50.40.80.20.10.60.7,t=0.60.40.20.40.50.30.50.30.4,PA=0.80.30.20.60.90.60.10.70.8,PA=0.80.90.60.70.8+0.30.60.60.10.8+0.20.60.90.10.7=0.80.8+0.6+0.30.6+0.1+0.20.6+0.1=0.8+0.3+0.2,PA=0.8,PB=0.20.30.70.70.50.60.80.60.7,PB=0.20.50.60.60.7+0.30.70.60.80.7+0.70.70.50.80.6=0.20.5+0.6+0.30.7+0.6+0.70.6+0.5=0.2+0.3+0.6,PB=0.6,Pg=0.30.60.70.10.50.20.20.60.3,Pg=0.30.50.20.60.3+0.60.10.20.20.3+0.70.10.50.20.6=0.30.3+0.2+0.60.1+0.2+0.70.1+0.2=0.3+0.2+0.2,Pg=0.3,s=0.70.40.50.40.80.20.10.60.7,s=0.7,s-=0.70.40.10.40.80.60.50.20.7,s-=0.70.80.60.20.7+0.40.40.60.50.7+0.10.40.80.50.2=0.70.7+0.2+0.40.4+0.5+0.10.2+0.5=0.7+0.4+0.1,s-=0.7,therefore s=s-=0.7. Consider(19)t=0.60.40.20.40.50.30.50.30.4,t=0.4,t-=0.60.40.50.40.50.30.20.30.4,t-=0.60.50.30.30.4+0.40.40.30.20.4+0.50.40.50.20.3=0.60.4+0.3+0.40.4+0.2+0.50.3+0.2=0.4+0.4+0.3,t-=0.4.Therefore t=t-=0.4. One has the following:(i)
Fors,t∈C,(20)ηPA+PB,Pg,s+t≥min⁡ηPA,Pg,s,ηPB,Pg,t⟹η0.8+0.6,0.3,0.7+0.4≥min⁡η0.8,0.4,0.7,η0.6,0.3,0.4⟹η0.8,0.3,0.7≥min⁡η0.8,0.4,0.7,η0.6,0.3,0.4⟹η0.8,0.3,0.7≥η0.6,0.3,0.4.(ii)
Fors,t∈C,(21)ηPA+PB,st≥min⁡ηPA,PA,s2,ηPB,PB,t2⟹η0.8+0.6,0.70.4≥min⁡η0.80.8,0.70.7,η0.60.6,0.40.4⟹η0.8+0.6,0.4≥min⁡η0.8,0.7,η0.6,0.4⟹η0.8,0.4≥η0.6,0.4.(iii)
Fors,t∈C,(22)ηPA,PB,s=ηPB,PA,s-⟹η0.8,0.6,0.7=η0.6,0.8,0.7⟹η0.8,0.6,0.7=η0.8,0.6,0.7.(iv)
Fors,t∈C,(23)ηkPA,PB,s=ηPA,PB,sk.For(24)k=0.9⟹η0.90.8,0.6,0.7=η0.8,0.6,0.70.9⟹η0.8,0.6,0.7=η0.8,0.6,0.7.Theorem 10.
If a 2-fuzzy inner product (2-FIP) onMnF is strictly convex, and if min⁡{θPA,st,θPB,st}≥min⁡{θPA,s2,θPB,t2}, then θ are 2-fuzzy inner product on MnF.Proof.
(i) Fors,t∈C and PA,PB,Pg∈P∗(MnF), we have(25)η′PA+PB,Pg,s+t=η′PA+PB,Pg,s+t+0=θPA+PB,s+t∀θPg,0=θPA+PB,s+t≥min⁡θPA,s,θPB,t=min⁡η′PA,PB,s,η′PB,Pg,t.Therefore η′PA+PB,Pg,s+t=min⁡{η′PA,PB,s,η′PB,Pg,t}.
(ii) One has(26)η′PA,PB,st=θPA,st=θPB,st=min⁡θPA,st,θPB,st≥min⁡θPA,s2,θPB,t2=min⁡η′PA,PA,s2,η′PB,PB,t2.
(iii) One has(27)η′PA,PB,s=θPA,s=θPA,s-,η′PA,PB,t-=θPB,t-=η′PB,PA,t-,η′PA,PB,s-=η′PB,PA,s-.
(iv) One has(28)η′kPA,PB,s=θkPA,s=θPA,skk≠0,η′kPA,PB,s=η′PA,PB,sk.
(v) One has(29)η′PA,PA,s=0∀tinC.
(vi) One has(30)η′PA,PA,s=0∀t>0⟹θPA,s=1∀t>0⟹PA=0.
(vii) Sinceη′PA,PA,·=θPA,· and θPA,· is a monotonic nondecreasing function of R and lim⁡θPA,s=1, this implies that μ′ has also the property. Thus η′ is a 2-fuzzy inner product on Mn(F).Definition 11.
Let(MnF,θ) be a 2-FIP satisfying the condition {ηPA,PB,s>0,whent>0} implying that f=0. Then, for all α in [0,1], define θPAα=inf⁡{t:ηPA,PA,s2≥α} as a crisp norm on MnF, called the α-norm on MnF generated by η. Now using these definitions let one define fuzzy norm on MnF and verify the conditions as follows.Theorem 12.
Letθ be a 2-FIP on MnF; then θ:MnF×R→[0,1] defined by(31)θPA,s=ηPA,PA,s2whensinR,s>0,θPA,s=0,whensinRands≤0is a fuzzy α-norm on MnF.Proof.
(i)θPA,s=0 for all s in R and s≤0.
(ii) For alls>0, ηPA,PA,s2=1 if and only if PA=0.
It follows thatθPA,s=0 if and only if PA=0.
(iii) For alls>0 and k≠0,(32)θkPA,s=ηkPA,kPA,s2,θkPA,s=ηPA,PA,s2k.
(iv) One has(33)θPA+PB,s+t≥min⁡θPA,s,θPB,tfor  everys,t∈R+,PA,PB∈P∗MnF⟹θPA+PB,s+t=ηPA+PB,PA+PB,s+t2=ηPA+PB,PA+PB,s2+st+st+t2=ηPA,PA,s2+ηPA,PA,st+ηPB,PB,t2≥ηPA,PA,s2+ηPB,PB,s2,θPA+PB,s+t=θPA,s+θPB,t.Theorem 13 (Cauchy Schwarz inequality).
Letθbe a 2-fuzzy inner product on MnF, α∈[0,1], and let θ be a α-norm generated from 2-FIP η on MnF; then ηPA,PB,st≤θPA,sθPB,t, for all PA,PB∈P∗(MnF).Proof.
We can assume thatPB>0∀r∈R+⟹(34)0≤θPA-rPB,s+t2=ηPA-rPB,PA-rPB,s2-st-st+t2≤ηPA,PA,s2+ηPA,-rPB,-st+η-rPB,PA,-st+η-rPB,-rPB,t2=ηPA,PA,s2+rηPA,PB,-st-1+rηPB,PA,-st-1+ηrPB,rPB,t2=ηPA,PA,s2-rηPA,PB,2st+r2ηPB,PB,t2.We can find a fuzzy sequence (35)r⟶ηPA,PB,stηPB,PB,t2.Then, taking limit into the above,(36)0≤ηPA,PA,s2-ηPA,PB,3stηPB,PB,t2+ηPA,PB,2stηPB,PB,t2,ηPA,PB,st≤ηPA,PA,s2ηPB,PB,t2.Therefore ηPA,PB,st≤θPA,sθPB,t.Theorem 14 (parallelogram law).
Letθ be a 2-fuzzy inner product on MnF, α∈[0,1], and let θ be a α-norm generated from 2-FIP η on MnF; then θPA+PBα2+θPA-PBα2=2ηPA,PA,s2+2ηPB,PB,t2.Proof.
Consider(37)θPA+PBα2+θPA-PBα2=inf⁡s2,s∈R+,θPA+PB,s≥α+inf⁡t2,t∈R+,θPA-PB,t≥α,where θ is the fuzzy α-norm induced from η.
One has the following:(38)θPA+PBα2+θPA-PBα2=inf⁡s2+t2,s,t∈R+,θPA+PB,s≥α,θPA-PB,t≥α,θPA+PBα2+θPA-PBα2=inf⁡s2+t2,s,t∈R+,θPA+PB,s∧θPA-PB,t≥α⟹θPA+PBα2+θPA-PBα2=θPA+PB,sθPA+PB,t=ηPA+PB,PA+PB,s+t2+ηPA-PB,PA-PB,s-t2=ηPA+PB,PA+PB,s2+st+st+t2+ηPA-PB,PA-PB,s2-st-st+t2=ηPA,PA,s2+ηPA,PB,st+ηPB,PA,st+ηPB,PB,t2+ηPA,PA,s2+ηPA,-PB,-st+η-PB,PA,st+η-PB,-PB,t2=ηPA,PA,s2+ηPA,PB,st+ηPB,PA,st+ηPB,PB,t2+ηPA,PA,s2+ηPA,PB,-st-1+ηPB,PA,-st-1+ηPB,PB,t2=ηPA,PA,s2+ηPA,PB,st+ηPB,PA,st+ηPB,PB,t2+ηPA,PA,s2-ηPA,PB,st-ηPB,PA,st+ηPB,PB,t2=ηPA,PA,s2+ηPB,PB,t2+ηPA,PA,s2+ηPB,PB,t2,θPA+PBα2+θPA-PBα2=2ηPA,PA,s2+2ηPB,PB,t2.Therefore, θPA+PBα2+θPA-PBα2=2ηPA,PA,s2+2ηPB,PB,t2.Theorem 15 (Pythagoras).
Letθ be a 2-fuzzy inner product on MnF, α∈[0,1], and let θ be a α-norm generated from 2-FIP η on MnF; then θPA+PBα2=θPAα2+θPBα2.Proof.
Consider the following:(39)θPAα=inf⁡s2,s∈R+,θPA,s≥α,θPBα=inf⁡t2,t∈R+,θPB,t≥α,θPA+PBα2=inf⁡s2+t2,s,t∈R+,θPA,s+θPB,t≥α,(40)θPA+PBα2=inf⁡s2+t2,s,t∈R+,θPA+PB,s+t≥α,(41)θPAα2+θPBα2=inf⁡s2,s∈R+,θPA,s≥α+inf⁡t2,t∈R+,θPB,t≥α=inf⁡s2,s∈R+,ηPA,PA,s2≥α+inf⁡t2,t∈R+,ηPB,PB,t2≥α=inf⁡s2+t2,s,t∈R+,θPA+PB,s+t≥α≤inf⁡s2+t2,s,t∈R+,θPA+PB,s+t≥α,∗θPAα2+θPBα2≤θPA+PBα2From40.Also(42)θPAα2+θPBα2=ηPA,PA,s2+ηPB,PB,t2=ηPA,PA,s2+ηPB,PB,t2+ηPA,PB,st+ηPB,PA,st≥inf⁡ηPA+PB,PA+PB,s+t2≥inf⁡s2+t2,s,t∈R+,θPA+PB,s+t≥α,∗∗θPAα2+θPBα2≥θPA+PBα2From40.From ∗ and ∗∗, (43)θPA+PBα2=θPAα2+θPBα2.Definition 16.
A sequence{PAn} in a fuzzy α-normed linear space (MnF,θ) is called a Cauchy sequence with respect to 2-norm if lim⁡θPAn,PBmα=0 as n, m→∞.Definition 17.
A sequence{PAn} in a fuzzy α-norm linear space (MnF,θ) is called a convergent sequence with respect to 2-norm if there exists PAn∈P∗(MnF), such that lim⁡θPAn,PBmα=0.Definition 18.
A fuzzy 2-normed linear space(MnF,θ) is said to be complete if every Cauchy sequence converges.Definition 19.
A complete fuzzy 2-normed linear space(MnF,θ) is called 2-fuzzy Banach space.Definition 20.
A complex 2-fuzzy Banach space(MnF,θ) is said to be 2-fuzzy Hilbert space if its norm is induced by the 2-fuzzy inner product.Theorem 21 (Fundamental Minimum Principle).
A nonempty closed convex fuzzy subsetc of a fuzzy Hilbert space MnF has a unique element of smallest α-norm.Proof.
Ifc is a 2-fuzzy convex set.
It contains(PA+PB)/2 whenever it contains PA and PB; we know that (44)cλPA+1-λPB≥min⁡cPA,cPB.And so,(45)cPA+PB2≥min⁡cPA,cPB.Let d=inf⁡{θPAα:PA∈C} where θPAα=inf⁡{s>0:ηPA,PA,s2≥α}.
LetPA, PB be a closed convex fuzzy set C, inside a fuzzy Hilbert space, so that both PA and PB are within ε>0 of the inf⁡d of the α-norm in C. Then there exists a sequence {PAn} in C such that θPAnα→d+ε.
By parallelogram law,(46)θPA+PBα2+θPA-PBα2=2θPAα2+θPBα2.Since c is convex,(47)cPAn+PBm2≥min⁡cPAn,cPBm,soPAn+PBm2∈C,θPAn+PBm2≥d+ε⟹θPAn+PBm≥2d+ε,θPA-PBα2=2θPAα2+θPBα2-θPA+PBα2from46≤2θPAα2+θPBα2-4d+ε2=2d+ε2+d+ε2-4d+ε2,θPA-PBα2=4d+ε2-4d+ε2=0.{PAn} is a 2-fuzzy Cauchy sequence in C, since MnF is complete and c is closed convex. Thus c such that PAn→PA; also θPAnα=θlim⁡PAn=lim⁡θPAnα=d, which implies that PA∈C with smallest α-norm. Now let us prove the uniqueness part. Suppose that there exists a PB∈C other than PA with the same α-norm d.
Again, by parallelogram law,(48)θPAn+PBm2α2=θPAα22+θPBα22-θPA-PBα22≤d+ε22+d+ε22≤d+ε2.Which is a contradiction to the definition of d+ε:(49)θPAn+PBm≥2d+ε.Hence there exists a unique element PA∈P∗(MnF) of smallest α-norm.Theorem 22.
The 2-FIPPA,PB∈P∗(MnF) is a continuous function on P∗(Mn1F)×P∗(Mn2F).Proof.
LetMnF be a 2-FIP and let {[PAn]} and {[PBn]} be sequence in P∗(MnF) such that θ(PAn)α=θ(PA)α and lim⁡θPBnα=θ(PB)α; then,(50)θPAn,PBn-PA,PB=θPAn,PBn-PAn,PB+PAnPB-PA,PB≤θPAn,PBn-PAn,PB+θPAnPB-PA,PB=θPAnPBn-PB+θPAn-PA,PB≤θPAnθPBn-PB+θPAn-PAθPB.Since {[PAn]} and {[PBn]} converge,(51)lim⁡θPAn-PAα=0,lim⁡θPBn-PBα=0.Theorem 23.
LetMnF be a 2-FIP and θ is indeed a α-norm on P∗(MnF). It also satisfies (i)
θPA>0 for every nonzero PA∈P∗(MnF),(ii)
θ(Px,Py,PA)≤θPx,PyθPA, for all PA∈P∗(MnF) and Px,Py∈P∗(MnF).Proof.
Consider(52)θtPA2=θtPA,tPA=tθPA,tPA≤tθPAθtPA,θtPA≤tθPA.For λ>0,(53)θPA=θ1ttPA≤1tθtPA,θtPA≤tθPA∀PA∈P∗MnF,tinR+.Also, (54)θPA+PBα2=θPA+PB,PA+PB≤θPAPA+PB+θPBPA+PB≤θPAθPA+PB+θPBθPA+PB≤θPA+θPBθPA+PB,θPA+PB≤θPA+θPB.(i)
LetPA∈P∗(MnF) be a nonzero element in MnF; then PA·PA>0 and hence θPA=θPA·PA1/2>0 in P∗(MnF).(ii)
LetPx,Py∈P∗(MnF);(55)θPx,Py,PA2=θPx,PyPA,θPx,PyPA=θPx,Py2PA,PA.Since(MnF,θ) is a fuzzy norm,(56)θPx,Py2PA,PA≤θPx,Py2θPA,PA=θPx,Py2θPA2.
The square roots exist and are unique inP∗(MnF).
Thereforeθ(Px,Py,PA)≤θPx,Pyθ(PA).Theorem 24 (polarisation identity).
IfPA,PB are element of 2-FIPS (MnF,η), then(57)4ηPA,PB,st=θPA+PBα2-θPA-PBα2+iθPA+iPBα2-iθPA+iPBα2.Proof.
Consider(58)
θ
P
A
+
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α
2
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2
+
i
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4
η
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.Therefore, (59)4ηPA,PB,st=θPA+PBα2-θPA-PBα2+iθPA+iPBα2-iθPA+iPBα2.


## 5. Concluding Remarks

In this paper the concept of inner product over fuzzy matrices has been discussed. We plan to extend our research work to (1) orthogonal of fuzzy matrices and (2) the Moore Penrose inverse and spectral inverse of fuzzy matrices.



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*Source: 6521893-2016-02-21.xml*