Theorima2:Esto ||A||oo,||A||1,||A||2 oi 
fisikes normes dianismatos ||x||oo,||x||1,
||x||2.Isxuei:
1.||A||oo=||A||g       2.||A||1=||A||s
3.||A||2=[r(ATA)]**1/2
        =r(A) an A simmetrikos

Apod: 1.Gia kathe u me ||u||oo=1 exoume:
||Au||oo=max | sum(j)(aij*uj) |
         i
        <=max ( sum(j)(|aij||uj|) )
         i
        <=max|uj|*max sum(j)(|aij|)
          j       i
         =max sum(j)(|aij|)=||A||g
          i
Esto tora m tetoio oste:||A||g=sum(j)(|amj|)
Orizoume to dianisma V me   1 an amj>=0
                        vj=-1 an amj<0
Profanos ||v||oo=1 kai ||Av||oo>=
||(Av)||m=|sum(j)(amjvj)|=sum(j)(|amj|)
=||A||g
2.Gia kathe u,me ||u||1=1 exoume
||Au||1=sum(i)(|sum(j)(aijuj)|)
<=sum(i,j)(|aij||uj|)=sum(j)(|uj|sum(i)(|aij|)
<=(sum(j)|uj|)max sum(i)(|aij|)=||A||s
Esto tora m tetoio oste: ||A||s=sum(i)|aim|
Tote gia em=(0,0,...,1,0,0,..,0) exoume
||em||=1 kai:        m
||Aem||1=sum(i)(|aim|)=||A||s
3.O pinakas ATA einai simmetrikos kai 
exei pragmatikes,profanos mi arnhtikes 
idiotimes l'1,..l'n kai antistoixa 
idiodianismata u1,..,un ta opoia apote
loun mia orthokanoniki basi sto Rn.E
xoume tote gia x anikei sto Rn:
                      T   T
x=sum(j)(cjuj) , ||x||2 =x *x=
               T
=(sum(i)(ciui)) (sum(j)(cjuj))
=sum(i,j)(cicjuiTuj)=sum(i)(ci**2)
            T       T   T
||Ax||2=(Ax) *(Ax)=x *(A Ax)=
              T
(sum(i)(ciui)) (sum(j)(l'jcjuj))=
=sum(i)(l'ici**2)
Epomenos (||Ax||2 **2)/(||x||2 **2)
=sum(i)(l'ici**2)/sum(i)(ci**2)
<=maxl'i=r(ATA)
   i
Eksalou,an l'k=maxl'i=r(ATA) tote
                i
              T  T        T
||Auk||2 **2 =ukA Auk=l'kukuk=l'k=
r(ATA)||uk||2 **2
Sinepos,||A||2 **2=sup||Ax||2 **2/ 
||x||2 **2=r(ATA)  x<>0
