Dinetai o pinakas A

A=(({{1, 2, 3, 4}, {2, 3, 4, 5}, {3, 4, 5, 6}}))//N;

Oi 4 ypoxwroi me apaloifh

Ston (R^4) brisketai o upoxwros twn seirwn, Row(A) h R(A) kai exei diastash 2 opws blepoume apo ta 2 mh mhdenika dianusmata

RowReduce[A]

Mia mh or8okanonikh bash tou upoxwrou Row(A) h R(A) einai ta 2 mh mhdenika dianusmata

RowSpaceBasis = Delete[RowReduce[A], -1]

RowSpaceBasis[[1]] // Norm

RowSpaceBasis[[1]] . RowSpaceBasis[[2]]

O upoxwros tou (R^4) Nullspace(A) exei  bash ta 2 mh mhdenika dianusmata

NullSpace[A]

A . NullSpace[A][[1]] // Chop

Prosexte pws  oi 2 upoxwroi tou (R^(4 ))einai or8ogwnioi meta3u tous.

RowSpaceBasis[[1]] . NullSpace[A][[1]] // Chop

Ston (R^3) brisketai o upoxwros twn sthlwn, Col(A) or R((A^T)) kai exei diastash 2 opws blepoume apo ta 2 mh mhdenika dianusmata (pou apoteloun mia mh or8okanonikh bash)

RowReduce[Transpose[A]]

ColumnSpaceBasis = 
      Delete[RowReduce[Transpose[A]], {{3}, {4}}]

ColumnSpaceBasis[[1]] // Norm

ColumnSpaceBasis[[1]] . ColumnSpaceBasis[[2]]

O upoxwros tou (R^3)  Nullspace((A^T)) exei diastash 1 kai h bash tou apoteleitai apo 1 dianusma

NullSpace[Transpose[A]]

Transpose[A] . NullSpace[Transpose[A]][[1]] // Chop

Kai edw  oi 2 upoxwroi tou (R^(3 ))einai or8ogwnioi meta3u tous.

ColumnSpaceBasis[[1]] . NullSpace[Transpose[A]][[1]] // 
      Chop

Oi 4 upoxwroi me SVD analush tou pinaka A

{h, s, a} = SingularValues[A]

Ston (R^4) brisketai o upoxwros  pou kaluptetai (spanned) apo ton a kai o opoios einai o upoxwros twn seirwn, Row(A) h R(A). H diafora einai oti o a mas dinei mia or8okanonikh bash.

a // MatrixForm

a[[1]] // Norm

a[[1]] . a[[2]] // Chop

Oi 2 upoxwroi tou (R^(4 ))einai or8ogwnioi meta3u tous.

a[[1]] . NullSpace[A][[1]] // Chop

Ston (R^3) brisketai o upoxwros pou kaluptetai (spanned) apo ton h kai o opoios einai o upoxwros twn sthlwn, Col(A) h R((A^T)). Kai edw h diafora einai oti o h mas dinei mia or8okanonikh bash.

Transpose[h] // MatrixForm

h[[1]] // Norm

h[[1]] . h[[2]] // Chop

Oi 2 upoxwroi tou (R^(3 ))einai or8ogwnioi meta3u tous.

h[[1]] . NullSpace[Transpose[A]][[1]] // Chop

Pinakes ta3hs 1

Me thn boh8eia ths SVD analushs, o pinakas A mporei na graftei kai san a8roisma pinakwn ta3hs 1 ws e3hs: A = (Subscript[A, 1]) + 
            (Subscript[A, 2]) = (Subscript[s, 1]) (Subscript[h, 1]) ((Subscript[a, 1])^T)  +  (Subscript[s, 2]) (Subscript[h, 2]) 
              ((Subscript[a, 2])^T), opou

(A1 = s[[1]] Outer[Times, h[[1]], a[[1]]]) // 
      MatrixForm

(A2 = s[[2]] Outer[Times, h[[2]], a[[2]]]) // 
      MatrixForm

A1 + A2 // MatrixForm

Oi pinakes (Subscript[A, 1]) kai (Subscript[A, 2]) einai ta3hs 1 opws prokuptei apo svd analush tous h apo thn sunarthsh

MatrixRank[A1] == MatrixRank[A2] == 1