h p://www.aimsp ess.com/jou nal/Ma h
AIMS Ma hema ics, 8(1): 1937–1958.
DOI:10.3934/ma h.2023100
Recei ed: 10 July 2022
Re ised: 29 Sep embe 2022
Accep ed: 08 Oc obe 2022
Published: 26 Oc obe 2022
Resea ch a icle
Suppo ing ec o s s. p incipal componen s
Almudena P. M´
a quez1,*, F ancisco Ja ie Ga c´
ıa-Pacheco1, M´
ı iam Mengiba -Rod ´
ıguez2and
Albe o S´
anchez-Alzola3
1Depa men o Ma hema ics, College o Enginee ing, Uni e si y o Cadiz, 11519 Pue o Real, Spain
2Depa men o Compu e Science and A i icial In elligence, Uni e si y o G anada, 18071
G anada, Spain
3Depa men o S a is ics and Ope a ion Resea ch, Uni e si y o Cadiz, 11519 Pue o Real, Spain
*Co espondence: Email: [email p o ec ed].
Abs ac : Le T:X→Ybe a bounded linea ope a o be ween Banach spaces X,Y. A ec o x0∈SX
in he uni sphe e SXo Xis called a suppo ing ec o o Tp o ided ha kT(x0)k=sup{kT(x)k:kxk=
1}=kTk. Since ma ices induce linea ope a o s be ween ini e-dimensional Hilbe spaces, we can
conside hei suppo ing ec o s. In his manusc ip , we un eil he ela ionship be ween he p incipal
componen s o a ma ix and i s suppo ing ec o s. Applica ions o ou esul s o eal-li e p oblems a e
p o ided.
Keywo ds: bounded linea ope a o ; Hilbe space; mean ope a o ; p incipal componen s; suppo ing
ec o
Ma hema ics Subjec Classi ica ion: 51F30, 54E35, 54E45
1. In oduc ion
Suppo ing Vec o Analysis (SVA) is a ela i ely ecen echnique ha allows one o sol e
analy ically many eal-li e p oblems ha used o be ackled by means o Heu is ic me hods. The lack
o ma hema ical o malism o Heu is ic me hods esul ed many imes in unp edic able solu ions, ha
is, ma hema ical solu ions whose eal-li e in e p e a ions make no sense. Suppo ing ec o s came
in o play o o e come his issue. This way, suppo ing ec o s we e used in a success ul way o
sol e mul iobjec i e op imiza ion p oblems coming om di e en disciplines, such as Bioenginee ing,
Physics, and S a is ics [4, 6–9, 15, 22], imp o ing conside ably he esul s achie ed by o he me hods
like, o ins ance, Heu is ic echniques [10,11,20, 21].
In [4, 6, 15], i was p o en ha Singula Value Decomposi ion (SVD) can be seen as a pa icula
case o SVA. This ac igge ed he new end o es a ing S a is ical no ions om he pe spec i e o
1938
Func ional Analysis and Ope a o Theo y. The main objec i e o his manusc ip is o s udy P incipal
Componen Analysis (PCA) by means o SVA.
2. Ma e ials and me hods
We will e iew se e al basic no ions om Ope a o Theo y ha will u n ou o be c ucial o he
de elopmen o his manusc ip .
2.1. Cen e ing and s anda dizing
I x=(x1,...,xn)∈Rn, hen he mean o xis de ined as x:=1
nPn
i=1xi, and i s s anda d de ia ion is
gi en by sx:=q1
nPn
i=1(xi−x)2. No ice ha
√nsx=x−x2,(2.1.1)
whe e x:=(x,n
. . ., x)deno es he cons an ec o o e m x(in gene al, i a∈R, hen a:=(a,n
. . ., a)
deno es he cons an ec o o e m a).
We say ha x∈Rnis cen e ed p o ided ha x=0, and i is s anda dized p o ided ha x=0 and
sx=1. In he la e si ua ion, kxk2=√n, in iew o (2.1.1). The subse o cen e ed ec o s o Rnis
usually deno ed by cen(Rn), ha is, cen(Rn) :={x∈Rn:x=0}.The subse o s anda dized ec o s o
Rnis usually deno ed by s an(Rn), ha is,
s an(Rn) :={x∈Rn:x=0 and sx=1}.
Acco ding o (2.1.1), s an (Rn)⊆√nS`n
2, whe e S`n
2s ands o he uni sphe e o `n
2:=(Rn,k·k2). In
Topology, S`n
2is deno ed as Sn−1.
2.2. P incipal componen analysis
The co a iance o wo ec o s x,y∈Rnis de ined as
sx,y:=1
n
n
X
i=1
(xi−x) (yi−y).
No ice ha sx,x=s2
x, ha is, he a iance o x. The co a iance ma ix o a gi en ma ix A∈Mm×nis
de ined by sa1,...,an:=sai,aji,j=1,...,n, whe e a1,...,ans and o he column ec o s o A.
Conside a ma ix A∈Mm×n. The p incipal componen s o Aa e de ined as Ax1,...,Axn, whe e
{x1,...,xn}is an o de ed o hono mal basis o eigen ec o s o sa1,...,an, so ing he eigen alues o sa1,...,an
dec easingly.
We e e he eade o [24] o a wide pe spec i e on PCA. In e es ing applica ions o PCA o
ce ain Enginee ing ields, such as ideo p ocessing and Big Da a, ha e been p o ided in [3,12].
2.3. Suppo ing ec o analysis
Le X,Ybe Banach spaces. Le T:X→Ybe a bounded linea ope a o . The ope a o no m o Tis
gi en by
kTk:=sup{kT(x)k:kxk=1}.(2.3.1)
AIMS Ma hema ics Volume 8, Issue 1, 1937–1958.
1939
The ec o space CL(X,Y) o con inuous linea ope a o s om X o Ybecomes a Banach space when
endowed wi h he ope a o no m (2.3.1). In he case X=Y,CL(X,Y) is simply deno ed as CL(X).
I Y=K(Ro C), hen CL(X,Y) is deno ed as X∗, ha is, he dual space o X. I is also common o
deno e CL(X,Y) by B(X,Y) and CL(X) by B(X).
The suppo ing ec o no ion was o mally posed o he i s ime in [5]. Howe e , his concep can
be ound implici ly and sca e ed h oughou he li e a u e o Banach Space Theo y [1,2, 18,19].
The se o suppo ing ec o s o a bounded linea ope a o T:X→Ybe ween Banach spaces X,Y
is de ined by
supp (T) :={x∈SX:kT(x)k=kTk} =a g max
kxk=1kT(x)k.(2.3.2)
He e, SXs ands o he uni sphe e o X, and BXdeno es he (closed) uni ball o X. In he in ini e-
dimensional se ing, i may occu ha (2.3.2) is emp y. No e ha supp (T)=supp (λT) o all λ∈
K {0}, and supp (T)=SKsupp (T), whe e K=Ro C. Fo a opological and geome ical analysis
o he abo e se , we s ongly e e he eade o [13,14,23].
Fo linea unc ionals, a special subse o suppo ing ec o s is wo h ega ding. Conside a
con inuous linea unc ional ∈X∗in he dual X∗o a Banach space X. We de ine he se o 1-
suppo ing ec o s o by
supp 1( ) :={x∈SX: (x)=k k}.(2.3.3)
No ice ha 1-suppo ing ec o s a e pa icula cases o suppo ing ec o s; in o he wo ds, supp 1( )⊆
supp ( ). In he upcoming sec ions, 1-suppo ing ec o s will be e y much elied on. The ollowing
ema k highligh s a s anda d geome ical p ope y sa is ied by 1-suppo ing ec o s.
Rema k 2.1. Conside a Banach space X and a nonze o linea unc ional ∈X∗ {0}. Fo e e y
x,y∈supp 1( )and e e y λ∈[0,1], we ha e ha λx+(1 −λ)y∈supp 1( ), ha is, supp 1( )is a
con ex subse o he uni sphe e SXo X.
A di ec consequence o Rema k 2.1 is ha supp 1( ) is ei he emp y o a single on in s ic ly
con ex Banach spaces, like, o ins ance, Hilbe spaces.
2.4. Hilbe space heo y
Rep esen a ion Theo y is one o he mos impo an heo ies in Ma hema ics. A majo esul in
Rep esen a ion Theo y is undoub edly he Riesz Rep esen a ion Theo em. This is a key esul in
Func ional Analysis and is c ucial o wo king wi h sel -adjoin ope a o s on Hilbe spaces.
Riesz Rep esen a ion Theo em. In a Hilbe space H, o e e y h∗∈H∗ he e exis s a unique h ∈H
sa is ying h∗=(·|h). This assignmen be ween H and H∗is a su jec i e linea isome y.
In iew o Rema k 2.1 and unde he se ings o he Riesz Rep esen a ion Theo em, o e e y h∈
H {0}, we ha e ha supp 1(h∗)=nh
khko, ha is, h
khkis he only 1-suppo ing ec o o h∗.
On he o he hand, he o hogonal subspace o a closed subspace Vo a Hilbe space His deno ed
by V⊥. The o hogonal p ojec ion o Hon o Vis usually deno ed by pV. Obse e ha H=V⊕2V⊥,
ha is, o each h∈H,
h=pV(h)+pV⊥(h),and khk2=kpV(h)k2+kpV⊥(h)k2.(2.4.1)
AIMS Ma hema ics Volume 8, Issue 1, 1937–1958.
1940
The adjoin o a bounded linea ope a o T:H→Kbe ween Hilbe spaces H,Kis de ined as he
unique bounded linea ope a o T∗:K→Hsa is ying (T(h)|k)=(h|T∗(k)) o each h∈Hand each
k∈K. Basic p ope ies sa is ied by he adjoin ope a o a e kT∗k=kTk,(T∗)∗=T, (T+S)∗=T∗+S∗,
(T◦S)∗=S∗◦T∗and (λT)∗=λT∗.
Lemma 2.2. E e y bounded linea ope a o T :H→K be ween Hilbe spaces H,K e i ies ha
cl(T(H)) =ke (T∗)⊥and T(H)⊥=cl(T(H))⊥=ke (T∗).
P oo . Fi s o , i is a i ial obse a ion ha T(H)⊥=cl(T(H))⊥. Fix an a bi a y h∈H. Fo e e y
k∈ke (T∗), (T(h)|k)=(h|T∗(k))=(h|0) =0. This shows ha T(h)∈ke (T∗)⊥. The a bi a iness
o h∈Hmeans ha T(H)⊆ke (T∗)⊥. By aking o hogonal complemen s, we ob ain ha
ke (T∗)⊆T(H)⊥. Con e sely, ix an a bi a y k∈T(H)⊥. Fo e e y h∈H, 0 =(T(h)|k)=(h|T∗(k)).
This shows ha T∗(k)=0, and hence k∈ke (T∗). The a bi a iness o k∈T(H)⊥means ha
T(H)⊥⊆ke (T∗). By aking o hogonal complemen s, we inally ob ain ha ke (T∗)⊥⊆cl(T(H)). 
A bounded ope a o T∈B(H) is said o be sel -adjoin p o ided ha T∗=T. I His complex,
hen T∈B(H) is sel -adjoin i and only i (T(h)|h)∈R o all h∈H. A sel -adjoin ope a o is called
posi i e p o ided ha (T(h)|h)≥0 o all h∈H.
Fo e e y T∈B(H), σ(T) :={λ∈C:λI−T<U(B(H))}is he spec um o T, whe e U(B(H))is
he mul iplica i e g oup o in e ible ope a o s on H. Among o he spec al p ope ies, he spec um
is compac and nonemp y, and kTk ≥ max |σ(T)|. A special subse o he spec um, called he poin
spec um, σp(T) :={λ∈C: ke (λI−T),{0}},whose elemen s a e called he eigen alues o T, will
be e y much employed. No e ha σp(T)⊆σ(T). Fu he mo e, o each λ∈σp(T), VT(λ) :=
{h∈H:T(h)=λh}s ands o he subspace o eigen ec o s associa ed wi h λ. In case he e is no
con usion wi h T, we will simply deno e VT(λ) by V(λ).
Suppose nex ha kTkis an eigen alue o T, ha is, kTk ∈ σp(T). In his si ua ion, since kTk ≥
max |σ(T)|, we conclude ha kTkis he maximum o |σ(T)|; in o he wo ds, kTk=max |σ(T)|. In his
case, we w i e kTk=λmax(T). Obse e also ha V(kTk)∩SX⊆supp (T). Indeed, i x∈V(kTk)∩SX,
hen T(x)=kTkx, so kT(x)k=kTk, and hence x∈supp (T).
Ne e heless, in gene al, kTk<σp(T), unless, o ins ance, Tis compac , sel -adjoin and posi i e.
This is why we ha e o ely on he adjoin T∗and on he s ongly posi i e ope a o T∗◦T. I is
s aigh o wa d o check ha he eigen alues o a sel -adjoin ope a o a e eal, and he eigen alues
o a sel -adjoin posi i e ope a o a e posi i e. When Tis compac , i holds ha T∗◦Tis compac ,
sel -adjoin and posi i e.
The ollowing esul , on which we will s ongly ely la e on, can be ound in [6, Theo em 4], which
is i sel a e inemen o [14, Theo em 9].
Theo em 2.3. Le H,K be Hilbe spaces. Le T ∈B(H,K). Then, kTk2=kT∗◦Tk, and supp (T)⊆
supp (T∗◦T). Fu he mo e, supp (T),∅i and only i kT∗◦Tk ∈ σp(T∗◦T). In his si ua ion,
kTk=√λmax (T∗◦T), and supp (T)=V(λmax(T∗◦T))∩SH.
3. Resul s
In his sec ion, we will s a e and p o e all he no el heo ems o his wo k. This sec ion is di ided
in o ou subsec ions, in which we will deal wi h suppo ing ec o s, p incipal componen s and he
AIMS Ma hema ics Volume 8, Issue 1, 1937–1958.
1941
opological s uc u e o he subse s o cen e ed and s anda dized ec o s.
3.1. Topological s uc u e o he subse s o cen e ed and s anda dized ec o s
This subsec ion begins un eiling he opological s uc u e o he subse s cen(Rn) and s an(Rn) o
cen e ed and s anda dized ec o s, espec i ely. No ice ha cen(R)={0}and s an(R)=∅.
Theo em 3.1. I n ≥1, hen cen(Rn)is linea ly isome ic, and hence homeomo phic, o `n−1
2. I n ≥2,
hen s an(Rn)is linea ly isome ic, and hence homeomo phic, o Sn−2.
P oo . No ice ha x+y=x+y, and x = x o all x∈Rnand all ∈R. As a consequence,
·:Rn→R
x7→ x=1
nPn
i=1xi
(3.1.1)
is a linea unc ional (usually called he mean unc ional). Nex , simply obse e ha cen(Rn)=ke (·).
Thus, by bea ing in mind (2.1.1), we immedia ely ob ain ha s an(Rn)=ke (·)∩√nS`n
2=√nSke (·).
In o he wo ds, s an(Rn) is p ecisely a mul iple o he uni sphe e o he Hilbe subspace ke (·)o `n
2.
Since dim (ke (·)) =n−1, we ha e ha ke (·)is linea ly isome ic o `n−1
2. As a consequence, he uni
sphe e o ke (·),1
√ns an(Rn), is linea ly isome ic o he uni sphe e o `n−1
2,S`n−1
2=Sn−2.
The ollowing lemma aims a compu ing he no m o he mean unc ional as an elemen o he dual
space o `n
2as well as i s only 1-suppo ing ec o .
Lemma 3.2. In `n
2∗,k·k=1
√nand supp 1(·)=n1
√no.
P oo . H¨
olde ’s Inequali y ensu es ha
|x|=
1
n
n
X
i=1
xi≤
n
X
i=1
1
n|xi| ≤ 
n
X
i=1
1
n2
1
2
n
X
i=1
x2
i
1
2
=1
√nkxk2
o e e y x∈`n
2. This shows ha k·k≤1
√n. On he o he hand, k1k2=√n, ha is, 1
√n∈S`n
2. Finally,
1
√n=1
n
n
X
i=1
1
√n=1
√n.
As a consequence, k·k=1
√nand supp 1(·)=n1
√no.
As a di ec consequence o Lemma 3.2, he s anda d de ia ion o a ec o x∈Rncan be ew i en
as he ollowing (2.1.1):
sx=1
√nx−x2=k·kx−x2.(3.1.2)
By bea ing in mind he Riesz Rep esen a ion Theo em, Lemma 3.2 assu es ha i h:=1
n, hen
h∗=·in H:=`n
2. No ice ha h:=1
n=1
n,n
. . ., 1
ncan be seen as a ( ini e) con ex se ies. This ac will
help us gene alize hese concep s o an in ini e-dimensional sepa able Hilbe space se ing la e on in
he Discussion.
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1942
De ini ion 3.3. Fo n ≥1, he mean ope a o is de ined by
•:Rn→Rn
x7→ x=(x,n
. . ., x),(3.1.3)
and he cen e ing ope a o is de ined as
cen : Rn→Rn
x7→ cen(x) :=x−x=(x1−x,n
. . ., xn−x).(3.1.4)
Fo n ≥2, he s anda dizing ope a o is de ined as
s an : Rn Rn1→Rn
x7→ s an(x) :=√nx−x
kx−xk2.(3.1.5)
I is a i ial obse a ion ha
s an(x)=√ncen(x)
kcen(x)k2
=cen(x)
k·kkcen(x)k2
(3.1.6)
o all x∈Rn.
Recall ha a p ojec ion on a Banach space Xis a con inuous, linea , and idempo en map P:X→X.
I s dual ope a o P∗:X∗→X∗is also a p ojec ion. The complemen a y p ojec ion o Pis
de ined as IX−P, which is also a p ojec ion. E e y non-ze o p ojec ion has no m g ea e han o
equal o 1. A 1-p ojec ion is a p ojec ion o no m 1 (also called a con ac i e p ojec ion), and a
(1,1)-p ojec ion is a 1-p ojec ion whose complemen a y p ojec ion is also a 1-p ojec ion (also called
bicon ac i e). O hogonal p ojec ions in Hilbe spaces a e he mos ep esen a i e examples o
bicon ac i e p ojec ions.
The inal esul o his i s subsec ion se es o show ha bo h he mean ope a o and he cen e ing
ope a o a e complemen a y p ojec ions o each o he o no m 1, ha is, bicon ac i e, o he Euclidean
no m. E en mo e, he mean ope a o and he cen e ing ope a o a e o hogonal p ojec ions o each
o he . This will allow us o di ec ly ob ain he K¨
onig-Huygens Theo em (3.1.7) as a di ec consequence
o he Py hago ean Theo em in Hilbe spaces. The K¨
onig-Huygens Theo em p o ides he classical
decomposi ion o he 2-no m o a ec o o Rnin e ms o he mean and he s anda d de ia ion.
Theo em 3.4. The cen e ing ope a o is a linea p ojec ion on Rnwhose ke nel is ke (cen) =Rn1,
whose ange is cen(Rn)=ke (·), and whose complemen a y p ojec ion is he mean ope a o .
Fu he mo e, i we conside Rnendowed wi h he Euclidean no m, hen k•k =kcenk=1, and •
and cen a e complemen a y o hogonal p ojec ions. As a consequence, o e e y x ∈Rn,
kxk2
2=kxk2
2+kx−xk2
2=n|x|2+s2
x.(3.1.7)
P oo . In he i s place, no ice ha •is clea ly linea , since •=·1; in o he wo ds, x=(x,n
. . ., x)=
x(1,n
. . ., 1)=x1 o all x∈Rn. As a consequence, cen is linea as well because cen =IRn−•, ha is, he
di e ence o wo linea ope a o s on Rn. On he o he hand, obse e ha i x∈Rnis a cons an ec o ,
ha is, x= 1= o some ∈R, hen cen(x)=0. Con e sely, i cen(x)=0, hen x=x=x1, and
hence x∈Rn1is a cons an ec o . This shows ha ke (cen) =Rn1. F om Theo em 3.1, we al eady
AIMS Ma hema ics Volume 8, Issue 1, 1937–1958.
1943
know ha cen(Rn)=ke (·). Nex , le us p o e ha cen is a p ojec ion. Fix an a bi a y x∈Rn. No ice
ha
cen (cen(x))=cen x−x=cen (x)−cen x=x−x−0=x−x=cen (x).
Since •=IRn−cen, we conclude ha he mean ope a o is he complemen a y p ojec ion o he
cen e ing ope a o . Finally, le us compu e k•k and kcenk. In acco dance wi h Lemma 3.2, o e e y
x∈Rnwe ha e ha
kxk2=√n|x| ≤ √n1
√nkxk2=kxk2,
meaning ha k•k ≤ 1. Nex ,

1
√n2
= 1
√n,n
. . ., 1
√n!2
=√n1
√n=1.
This shows ha k•k =1. In o de o p o e ha •and cen a e o hogonal p ojec ions, i only su ices o
ealize ha R1and cen(Rn) a e o hogonal subspaces. Indeed, o e e y ∈Rand e e y x∈Rnwi h
x=0, we ha e ha ( 1|x)= (1|x)= Pn
i=1xi= nx =0, and hence R1⊆cen(Rn)⊥, o equi alen ly,
cen(Rn)⊆(R1)⊥. Fu he mo e, he Py hago ean Theo em in `n
2allows ha
k 1+xk2
2=k 1k2
2+kxk2
2
o e e y ∈Rand e e y x∈Rnwi h x=0. Nex , i y∈(R1)⊥, hen 0 =1
n(1|y)=1
nPn
i=1yi=y,
meaning ha y∈ke (·)=cen(Rn). As a consequence, cen(Rn)⊥=R1, esul ing in
kxk2
2=kxk2
2+kx−xk2
2=n|x|2+s2
x
o e e y x∈Rn. I only emains o show ha kcenk=1, bu his is a di ec consequence o he ac
ha cen is an o hogonal p ojec ion. 
3.2. Suppo ing ec o s and i s p incipal componen
In his subsec ion, we will p o ide su icien condi ions o he suppo ing ec o s o coincide wi h
he i s p incipal componen . Fi s , we will need some de ini ions.
De ini ion 3.5. A ma ix is said o be cen e ed (s anda dized) p o ided ha all o i s column ec o s
a e cen e ed (s anda dized) ec o s.
The ollowing lemma displays a simple cha ac e iza ion o cen e ed ma ices.
Lemma 3.6. Le A ∈Mm×n(R). The ollowing condi ions a e equi alen :
1) A is cen e ed.
2) Ax is cen e ed o all x ∈Rn.
P oo . Le {e1,...,en}deno e he canonical basis o Rn. No ice ha he columns o Aa e p ecisely Aej
o j=1,...,n. Suppose i s ha Ais cen e ed. Fix an a bi a y x∈Rn. The linea i y o he mean
unc ional allows ha
Ax =
n
X
j=1
xjAej=
n
X
j=1
xjAej=0.
AIMS Ma hema ics Volume 8, Issue 1, 1937–1958.
1944
As a consequence, Ax is cen e ed o all x∈Rn. Con e sely, suppose now ha Ax is cen e ed o all
x∈Rn. In pa icula , Aej=0 o j=1,...,n, meaning ha he columns o Aa e cen e ed ec o s, ha
is, Ais cen e ed.

The ollowing cha ac e iza ion o cen e ed ma ices is a bi mo e sophis ica ed. Fi s , a echnical
lemma is needed.
Lemma 3.7. Conside x,y∈Rm. Then,
1) msx,y=x•y i and only i ei he x o y is cen e ed.
2) x is s anda dized i and only i x is cen e ed and x •x=m.
P oo .
1) Le us obse e ha
msx,y=
m
X
i=1
(xi−x) (yi−y)
=
m
X
i=1
xiyi−
m
X
i=1
xyi−
m
X
i=1
xiy+
m
X
i=1
x y
=
m
X
i=1
xiyi−x
m
X
i=1
yi−y
m
X
i=1
xi+x y
m
X
i=1
1
=
m
X
i=1
xiyi−xmy −ymx +x y
=x•y−mx y.
As a consequence, msx,y=x•yi and only i ei he xo yis cen e ed.
2) By de ini ion, xis s anda dized i and only i xis cen e ed and sx=1. We know ha hen
kx−xk2=√msx. In he con ex o cen e ed ec o s, he p e ious exp ession becomes √x•x=
kxk2=√msx. The e o e, xis s anda dized i and only i xis cen e ed and x•x=m.

Recall ha he co a iance ma ix o a gi en ma ix A∈Mm×nis de ined by sa1,...,an:=sai,aji,j=1,...,n,
whe e a1,...,ans and o he column ec o s o A. Also, ecall ha i Bis a squa e ma ix, hen
diag(B) s ands o he diagonal o B.
P oposi ion 3.8. Le A ∈Mm×n(R). The ollowing condi ions a e equi alen :
1) A is cen e ed.
2) A A=msa1,...,an.
3) diag A A=diag msa1,...,an.
AIMS Ma hema ics Volume 8, Issue 1, 1937–1958.
1945
P oo . No ice ha A A=ai•aji,j=1,...,n. Suppose i s ha Ais cen e ed. In iew o Lemma 3.7,
msai,aj=ai•aj o all i,j∈ {1,...,n}, meaning ha A A=msa1,...,an. Nex , i A A=msa1,...,an, hen
we i ially ha e ha diag A A=diag msa1,...,an. Finally, assume ha diag A A=diag msa1,...,an.
Then, msai,ai=ai•ai o all i=1,...,n. In acco dance wi h Lemma 3.7, aiis cen e ed o all
i=1,...,n, meaning ha Ais cen e ed. 
As an immedia e consequence o Lemma 3.7 and P oposi ion 3.8, we ob ain he ollowing co olla y.
Co olla y 3.9. Le A ∈Mm×n(R). The ollowing condi ions a e equi alen :
1) A is s anda dized.
2) A A=msa1,...,anand diag A A=(m,n
. . ., m).
Finally, we ha e ga he ed all he necessa y ools o p o e ou main esul s o his subsec ion. Simply
keep in mind he small obse a ion ha i B∈Mn(R), hen σp(αB)=ασp(B) and VαB(αλ)=VB(λ) o
all α∈R {0}and all λ∈σp(B).
Theo em 3.10. Le A ∈Mm×n(R). I A is cen e ed, hen
supp (A)=nx∈S`n
2:Ax is he i s p incipal componen o Ao,
whe e A is seen as a linea ope a o
A:`n
2→`m
2
x7→ Ax.(3.2.1)
P oo . Acco ding o Theo em 2.3, supp (A)=V(λmax(A∗◦A))∩S`n
2. No ice ha he adjoin o A,
A∗, coincides wi h i s anspose, A . On he o he hand, since Ais cen e ed, P oposi ion 3.8 allows ha
A A=msa1,...,an. Finally,
supp (A)=VA∗◦A(λmax(A∗◦A))∩S`n
2
=VA Aλmax(A A)∩S`n
2
=Vmsa1,...,anλmax(msa1,...,an)∩S`n
2
=Vmsa1,...,anmλmax(sa1,...,an)∩S`n
2
=Vsa1,...,anλmax(sa1,...,an)∩S`n
2
=nx∈S`n
2:Ax is he i s p incipal componen o Ao.

We will conclude his subsec ion wi h an example o a cen e ed ma ix Awhose las p incipal
componen has a leas wo dimensions.
Rema k 3.11. Le T :H→K be a bounded linea ope a o be ween Hilbe spaces H,K. Then,
ke (T)⊆ke (T∗◦T). As a consequence, i 0∈σp(T), hen 0∈σp(T∗◦T).
No ice ha i T∈B(H) is a sel -adjoin posi i e ope a o on a Hilbe space Hsuch ha ke (T),
{0}, hen 0 is he minimum o σp(T) since all he eigen alues o Ta e posi i e.
AIMS Ma hema ics Volume 8, Issue 1, 1937–1958.
1952
4. Discussion
We will discuss how o anspo he mean unc ional, he mean ope a o and he cen e ing ope a o
o abs ac se ings, such as Hilbe spaces, Banach algeb as and p obabili y spaces.
4.1. Gene aliza ion o Hilbe spaces
As men ioned in Lemma 3.2, in iew o he Riesz Rep esen a ion Theo em, i h0:=1
n=1
n,n
. . ., 1
n,
hen h∗
0=·is p ecisely he mean unc ional in H:=`n
2. Ac ually, kh0k2=1
√n=kh∗
0k. A his s age,
he key is o ealize ha h0can be seen as a ( ini e) con ex se ies. Recall ha a con ex se ies is a
con e gen se ies P∞
n=1 nsuch ha P∞
n=1 n=1 and n≥0 o all n∈N. No ice ha i P∞
n=1 nis a
con ex se ies, hen P∞
n=1 2
n≤P∞
n=1 n=1, and hence ( n)n∈N∈`2. Now, we a e in he igh posi ion o
de ine he no ions o mean and s anda d de ia ion on sepa able Hilbe spaces.
Le Hbe a sepa able Hilbe space wi h an o hono mal basis (en)n∈N. Fix a con ex se ies P∞
n=1 n.
Le h∈Hand w i e h=P∞
n=1(h|en)en. The mean o h, wi h espec o ( n)n∈N, is de ined as
h:=∞
X
n=1
n(h|en).(4.1.1)
The mean unc ional, wi h espec o ( n)n∈N, is gi en by
·:H→K
h7→ h:=P∞
n=1 n(h|en).(4.1.2)
By elying on H¨
olde ’s Inequali y, i is no ha d o check ha he mean unc ional is an elemen o H∗
whose no m is p ecisely pP∞
n=1 2
n=k( n)n∈Nk2. In acco dance wi h he Riesz Rep esen a ion Theo em,
he e exis s h0∈Hsuch ha (h|h0)=h o all h∈H. I we le h=P∞
n=1(h|en)en, hen we ob ain ha
∞
X
n=1
(h|en) (en|h0)=
∞
X
n=1
(h|en)en
h0
=∞
X
n=1
n(h|en).(4.1.3)
By aking h=en o e e y n∈Nin (4.1.3), we conclude ha (en|h0)= n o e e y n∈N. In pa icula ,
h0=P∞
n=1 nen. As expec ed,
kh0k=k( n)n∈Nk2=
∞
X
n=1
2
n=k·k.
In iew o he Riesz Rep esen a ion Theo em, h∗
0=(·|h0)=·. No ice ha , in o de o conclude ha
h∗
0=(·|h0)=·, he Riesz Rep esen a ion Theo em is eally no needed. Le us ge back o a second o
`n
2. Fo e e y x∈`n
2,
x=(x,n
. . ., x)=x1=x
1
√n
1
n
1
√n
=·
1
√n
(x)
1
n
1
√n
.
Then, going back o a gene al sepa able Hilbe space H, he mean ope a o is de ined as
H→H
h7→ h:=h∗
0
kh∗
0k(h)h0
kh0k.(4.1.4)
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1953
I is no ha d o check ha he mean ope a o is an o hogonal p ojec ion on Hwhose complemen a y
p ojec ion is, p ecisely, he cen e ing ope a o :
H→H
h7→ cen(h) :=h−h.(4.1.5)
4.2. Gene aliza ion o Banach algeb as
A Banach algeb a is a eal o complex algeb a Aendowed wi h a comple e ec o no m ha is also
a ing no m, ha is, kabk≤kakkbk o all a,b∈A. We say ha Ais uni al i i is uni a y, ha is, i has
a uni y 1∈A, and k1k=1. In his si ua ion, acco ding o he Hahn-Banach Theo em, he e exis s a
con inuous linea unc ional, which we will deno e by 1∗∈A∗, such ha k1∗k=1 and 1∗(1)=1. Then,
he mean unc ional is p ecisely 1∗, ha is, de ined by
A→K
a7→ a:=1∗(a).(4.2.1)
The mean ope a o is de ined as
A→A
a7→ a:=1∗(a)1.(4.2.2)
Finally, he cen e ing ope a o is de ined as
A→A
a7→ cen(a) :=a−a.(4.2.3)
Following (3.1.2), he a iance o an elemen a∈Acan be de ined by
s2
a:=1∗a−a2=1∗(a2)−1∗(a)2.(4.2.4)
No ice ha his gene aliza ion o uni al Banach algeb as p esen s a weakness: The exis ence o he
unc ional 1∗∈A∗is gua an eed by he Hahn-Banach Theo em, bu i s uniqueness is no gua an eed. In
ac , in many uni al Banach algeb as, such as `∞(Λ) o ins ance, 1is no a smoo h poin o B`∞(Λ)[16,
Theo em 2.9], and hus he e a e in ini ely many unc ionals o no m 1 a aining hei no m a 1. I
seems no i ial o o e come his issue. Maybe an op ion is o y o eno m equi alen ly he uni al
Banach algeb a in such a way ha 1becomes a smoo h poin o he new uni ball o he algeb a o ,
a leas , o ind ano he smoo h poin in he uni ball. Acco ding o [16, Theo em 2.1], he canonical
uni ec o eλis a smoo h poin o B`∞(Λ) o each λ∈Λ. Ano he possibili y may ely on cons uc ing
he mean unc ional in a C∗-algeb a. Recall ha a C∗-algeb a is a Banach algeb a Aendowed wi h an
an imul iplica i e and an ilinea in olu ion ∗:A→Asa is ying ha kaa∗k=kak2 o all a∈A.
4.3. Gene aliza ion o p obabili y spaces
A p obabili y space is a 3- uple (Ω,Σ,P) whe e (Ω,Σ) is a measu able space and P:Σ→[0,1] is a
p obabili y measu e, ha is, a coun ably addi i e posi i e measu e such ha P(Ω)=1. I Xis a Banach
space, by L1((Ω,Σ,P),X) we deno e he Banach space o all absolu ely in eg able unc ions, ha is,
L1((Ω,Σ,P),X) :=( ∈XΩ: is measu able and ZΩk kdP<∞),
AIMS Ma hema ics Volume 8, Issue 1, 1937–1958.
1954
endowed wi h he no m
k k1:=ZΩk kdP.
Fo each ∈L1((Ω,Σ,P),X), he mean o is de ined as
µ( ) :=ZΩ
dP.(4.3.1)
The mean unc ional is gi en by
µ:L1((Ω,Σ,P),X)→X
7→ µ( ) :=RΩ dP.(4.3.2)
No ice ha he mean unc ional is ac ually an ope a o , bu we keep calling i unc ional no o mis ake
i wi h he mean ope a o , which will be de ined nex . I can be easily shown ha he mean unc ional
has no m equal o 1. Indeed,
kµ( )k=ZΩ
dP≤ZΩk kdP=k k1
o e e y ∈L1((Ω,Σ,P),X), meaning ha kµk ≤ 1. Now, i we choose any x∈SX, hen xχΩ∈
SL1((Ω,Σ,P),X), and
µ(xχΩ)=ZΩ
xχΩdP=xP(Ω)=x.
Hence,
kµ(xχΩ)k=ZΩ
xχΩdP
=kxkP(Ω)=1=kxχΩk1.
This p o es ha kµk=1 and xχΩ∈supp (µ). The mean ope a o is hen de ined as
L1((Ω,Σ,P),X)→L1((Ω,Σ,P),X)
7→ µ( )χΩ,(4.3.3)
and he cen e ing ope a o is
L1((Ω,Σ,P),X)→L1((Ω,Σ,P),X)
7→ −µ( )χΩ.(4.3.4)
Finally, i Xis a uni al Banach algeb a, hen he na u al way o de ining he a iance o ∈
L1((Ω,Σ,P),X) is
σ( ) :=µ( −µ( )χΩ)2=ZΩ
( −µ( )χΩ)2dP.(4.3.5)
5. Conclusions
I is well known in he li e a u e o he Geome y o Banach Spaces ha Hilbe spaces a e ansi i e
Banach spaces, meaning ha e e y wo elemen s o he uni sphe e o a Hilbe space can be anspo ed
one in o ano he by means o a su jec i e linea isome y. This ac con e s Hilbe spaces wi h a
AIMS Ma hema ics Volume 8, Issue 1, 1937–1958.
1955
ce ain eedom when i comes o choosing he con ex se ies ha de ines he mean unc ional, he mean
ope a o , he cen e ing ope a o and he s anda d de ia ion. Theo em 3.1, Lemma 3.2 and Theo em 3.4
can be anspo ed o he mo e gene al scope p o ided by sepa able Hilbe spaces discussed in he
p e ious sec ion. In ac , in he Discussion we un eiled how o ex end he mean unc ional, he mean
ope a o , he cen e ing ope a o and he a iance o spaces o absolu ely in eg able unc ions de ined
on a p obabili y space and alued on a uni al Banach algeb a.
On he o he hand, he s udy o P incipal Componen Analysis h ough Suppo ing Vec o Analysis
is a e olu iona y end ha allows one o look a hese S a is ical concep s om a Func ional Analysis
iewpoin , which is mo e gene al and wo ks in in ini e dimensional en i onmen s, making possible
applica ions in e y speci ic se ings such as, o ins ance, Quan um Mechanical Sys ems.
Acknowledgmen s
This wo k has been pa ially suppo ed by he Resea ch G an PGC-101514-B-I00 awa ded by he
Minis y o Science, Inno a ion and Uni e si ies o Spain and co- inanced by he 2014-2020 ERDF
Ope a ional P og amme and by he Depa men o Economy, Knowledge, Business and Uni e si y o
he Regional Go e nmen o Andalusia unde P ojec e e ence FEDER-UCA18-105867. The APCs
ha e been paid by he Depa men o Ma hema ics o he Uni e si y o Cadiz.
Con lic o in e es
The au ho s decla e ha hey ha e no con lic o in e es .
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Supplemen a y: Py hon GUI code
Le A∈Mm×n(R) be a gene al ma ix. The algo i hm o compu e he suppo ing ec o wi h
i s co esponding i s p incipal componen , he second eigen ec o wi h i s co esponding second
p incipal componen and he nex eigen ec o s and p incipal componen s o Ais he ollowing:
de PCA(X, num_componen s):
X_meaned = X - np.mean(X, axis=0)
co _ma = np.co (X_meaned, ow a =False)
eigen_ alues, eigen_ ec o s = np.linalg.eigh(co _ma )
so ed_index = np.a gso (eigen_ alues)[::-1]
so ed_eigen ec o s = eigen_ ec o s[:, so ed_index]
eigen ec o _subse = so ed_eigen ec o s[:, 0:num_componen s]
X_new_ alues = np.do (eigen ec o _subse . anspose(),
X_meaned. anspose()). anspose()
e u n X_new_ alues, eigen ec o _subse
One o he no el ies o his wo k is o apply he PCA me hod wi h a di e en p ocedu e speci ically,
using an algo i hm based on he ma hema ical idea ha he second eigen ec o , wi h i s associa ed
second p incipal componen , is he suppo ing ec o o he o iginal poin s p ojec ed o he o hogonal
complemen o he o iginal suppo ing ec o . Consequen ly, all he p incipal componen s can be
compu ed in an i e a i e p ocess ia a suppo ing ec o .
Le x∈Rnbe a ec o . Fi s , we show he ollowing unc ion o calcula e he o hogonal
complemen o x:
de calcula e_o hogonal_complemen (x, no malize=T ue, h eshold=1e-15):
x = np.asa ay(x)
, c = x.shape
i < c:
impo wa nings
wa nings.wa n(’ ewe ows han columns’, Use Wa ning)
s, , d = np.linalg.s d(x)
ank = ( > h eshold).sum()
AIMS Ma hema ics Volume 8, Issue 1, 1937–1958.

1958
oc = s[:, ank:]
i no malize:
k_oc = oc.shape[1]
oc = oc.do (np.linalg.in (oc[:k_oc, :]))
e u n oc
Nex , by using he p e ious unc ion, we calcula e he o hogonal complemen o he suppo ing
ec o , ep esen ing a hype plane wi h dimension n−1. A e wa ds, he poin s o he o iginal ma ix,
used o he ini ial suppo ing ec o , a e p ojec ed o his hype plane. Hence, a new ma ix is de i ed,
whose i s p incipal componen co esponds o he second p incipal componen o he o iginal ma ix.
Thus, by applying his i e a i e algo i hm, we can calcula e all he p incipal componen s ia he
o hogonal complemen o a suppo ing ec o .
I is easie o unde s and his algo i hm conside ing a special case. Le A∈M3×3(R) be a ma ix.
In an in ui i e manne , each ow o he ma ix ep esen s poin s o R3. Now, we p esen a p ocess o
compu ing he i s eigen ec o (suppo ing ec o ) and i s co esponding i s p incipal componen as
be o e, bu wi h a di e en p ocedu e o ob ain he ollowings. As men ioned, we no ice ha he i s
p incipal componen ia a suppo ing ec o o a de i ed ma ix om he o iginal one coincides wi h
he second p incipal componen o he o iginal ma ix. This de i ed ma ix is cons uc ed p ojec ing he
o iginal poin s o he o iginal ma ix o a plane o med by he o hogonal complemen ( wo ec o s) o
he ini ial suppo ing ec o (one ec o ) and he o igin poin . The e o e, we ob ain he same esul by
applying he PCA p esen ed be o e and his algo i hm de e mining he nex eigen ec o as a suppo ing
ec o o he o iginal suppo ing ec o .
suppo ing_ ec o = PCA(inpu _ma ix, 1)
o hogonal_complemen = calcula e_o hogonal_complemen (suppo ing_ ec o )
1 = Vec o (o hogonal_complemen .T[0, :])
2 = Vec o (o hogonal_complemen .T[1, :])
poin = Poin ([0, 0, 0])
plane = Plane. om_ ec o s(poin , 1, 2)
p ojec ed_poin s = np.a ay([plane.p ojec _poin (x) o x in inpu _ma ix])
suppo ing_ ec o _p ojec ed_poin s = PCA(p ojec ed_poin s, 1)
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AIMS Ma hema ics Volume 8, Issue 1, 1937–1958.