Conjugate transpose

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the conjugate transpose, also known as the Hermitian transpose, of an 
  
    
      
        m
        ×
        n
      
    
    {\displaystyle m\times n}
  
 <a href="/facts/Complex_number/2w7mqI7e">complex</a> <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> 
  
    
      
        
          A
        
      
    
    {\displaystyle \mathbf {A} }
  
 is an 
  
    
      
        n
        ×
        m
      
    
    {\displaystyle n\times m}
  
 matrix obtained by <a href="/facts/Transpose/8wmsagGS">transposing</a> 
  
    
      
        
          A
        
      
    
    {\displaystyle \mathbf {A} }
  
 and applying <a href="/facts/Complex_conjugate/7TEaoQDe">complex conjugation</a> to each entry (the complex conjugate of 
  
    
      
        a
        +
        i
        b
      
    
    {\displaystyle a+ib}
  
 being 
  
    
      
        a
        −
        i
        b
      
    
    {\displaystyle a-ib}
  
, for real numbers 
  
    
      
        a
      
    
    {\displaystyle a}
  
 and 
  
    
      
        b
      
    
    {\displaystyle b}
  
). There are several notations, such as 
  
    
      
        
          
            A
          
          
            
              H
            
          
        
      
    
    {\displaystyle \mathbf {A} ^{\mathrm {H} }}
  
 or 
  
    
      
        
          
            A
          
          
            ∗
          
        
      
    
    {\displaystyle \mathbf {A} ^{*}}
  
, 
  
    
      
        
          
            A
          
          ′
        
      
    
    {\displaystyle \mathbf {A} '}
  
, or (often in physics) 
  
    
      
        
          
            A
          
          
            †
          
        
      
    
    {\displaystyle \mathbf {A} ^{\dagger }}
  
.
</p><p>For <a href="/facts/Real_number/R02gw5Pb">real</a> matrices, the conjugate transpose is just the transpose, 
  
    
      
        
          
            A
          
          
            
              H
            
          
        
        =
        
          
            A
          
          
            T
          
        
      
    
    {\displaystyle \mathbf {A} ^{\mathrm {H} }=\mathbf {A} ^{\operatorname {T} }}
  
.
</p>

Conjugate transpose open-in-new

Conjugate transpose