Properties of Unitary transforms :
1) Energy conservation :
In the unitary transformation,
 =  ,
 = 
Proof
  =  =    =  .
This implies, unitary transformation preserves signal energy or equivalently the length of vector
 in
 dimensional vector space. That is , every unitary transformation is simply a rotation of in
 dimensional vector space. Alternatively , a unitary transform is a rotation of basis coordinates and components of
 are projections of on the new basis. Similarly , for 2D unitary transformations, it can be proved that
 = 
Example: Consider the vector  =  and  = 
This transformation
 =  can be written as  =  ;  =  where
 ,  new basis vectors,denote the columns of  and , are projections of in the new coordinate system.
 =  =   = 
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