Hermitian matrix

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a Hermitian matrix (or self-adjoint matrix) is a <a href="/facts/Complex_number/2w7mqI7e">complex</a> <a href="/facts/Square_matrix/5TP9mNNn">square matrix</a> that is equal to its own <a href="/facts/Conjugate_transpose/7eBWtqVG">conjugate transpose</a>—that is, the element in the i-th row and j-th column is equal to the <a href="/facts/Complex_conjugate/7TEaoQDe">complex conjugate</a> of the element in the j-th row and i-th column, for all indices i and j:

A
        
           is Hermitian
        
        
        
        ⟺
        
        
        
          a
          
            i
            j
          
        
        =
        
          
            
              a
              
                j
                i
              
            
            ¯
          
        
      
    
    {\displaystyle A{\text{ is Hermitian}}\quad \iff \quad a_{ij}={\overline {a_{ji}}}}

or in matrix form:

A
        
           is Hermitian
        
        
        
        ⟺
        
        
        A
        =
        
          
            
              A
              
                
                  T
                
              
            
            ¯
          
        
        .
      
    
    {\displaystyle A{\text{ is Hermitian}}\quad \iff \quad A={\overline {A^{\mathsf {T}}}}.}

Hermitian matrices can be understood as the complex extension of real <a href="/facts/Symmetric_matrix/08CGwjNl">symmetric matrices</a>.
If the <a href="/facts/Conjugate_transpose/7eBWtqVG">conjugate transpose</a> of a matrix 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is denoted by 
 
 
 
 
 A
 
 
 H
 
 
 
 ,
 
 
 {\displaystyle A^{\mathsf {H}},}
 
 then the Hermitian property can be written concisely as

 
 
 
 A
 
  is Hermitian
 
 
 
 ⟺
 
 
 A
 =
 
 A
 
 
 H
 
 
 
 
 
 {\displaystyle A{\text{ is Hermitian}}\quad \iff \quad A=A^{\mathsf {H}}}

Hermitian matrices are named after <a href="/facts/Charles_Hermite/kj1bGlMr">Charles Hermite</a>, who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of always having real <a href="/facts/Eigenvalues_and_eigenvectors/8TjEoT8u">eigenvalues</a>. Other, equivalent notations in common use are 
 
 
 
 
 A
 
 
 H
 
 
 
 =
 
 A
 
 †
 
 
 =
 
 A
 
 ∗
 
 
 ,
 
 
 {\displaystyle A^{\mathsf {H}}=A^{\dagger }=A^{\ast },}
 
 although in <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a>, 
 
 
 
 
 A
 
 ∗
 
 
 
 
 {\displaystyle A^{\ast }}
 
 typically means the <a href="/facts/Complex_conjugate/7TEaoQDe">complex conjugate</a> only, and not the <a href="/facts/Conjugate_transpose/7eBWtqVG">conjugate transpose</a>.

Hermitian matrix open-in-new

Hermitian matrix