Ortho-normalization of A Set of Column Vectors

By

Gram Schmidt Process

&

QR Decomposition

 

             
  Description name value Name value  
  R2 Euclidean Space -> column vectors

 

 V1 V2  
         
  Find  Ortho-normal Vectors (set 1) U1=V1 U2  
         
  < U1,U2 > (set 1)        
  Normalization Factor nu1 nu2  
  Find  Normalized Ortho-normal Vectors (set 1) e1 e2  
         
  Decomposition of V into Q & R Matrix          
  Q matrix ( orthogonal) Q    
         
  R matrix (upper triangular matrix) R    
         
  QR= V1V2 QR    
         
             
  Find  Ortho-normal Vectors (set 2) U1 U2=V2  
         
  < U1,U2 > (set 2)        
  Normalization Factor nu3 nu4  
  Find  Normalized Ortho-normal Vectors (set 2) e1 e2  
         
  Decomposition of V into Q & R Matrix          
  Q matrix (orthogonal) Q    
         
  R matrix (lower triangular matrix) R    
         
  QR QR    
         
             
     
     
  * If a & b are a pair of vectors , then vector projections are (a.b)/ (|a|2)  and (a.b)/ (|b|2)

* For theory, click here.

* For QR decomposition, click here.