Hermitian matrix

<h2 id="alternative-characterizations">Alternative characterizations</h2>
<p>Hermitian matrices can be characterized in a number of equivalent ways, some of which are listed below:
</p>
<h3>Equality with the adjoint</h3>
<p>A square matrix 
  
    
      
        A
      
    
    {\displaystyle A}
  
 is Hermitian if and only if it is equal to its <a href="/facts/Hermitian_adjoint/gN6RHphd">conjugate transpose</a>, that is, it satisfies

⟨
        
          w
        
        ,
        A
        
          v
        
        ⟩
        =
        ⟨
        A
        
          w
        
        ,
        
          v
        
        ⟩
        ,
      
    
    {\displaystyle \langle \mathbf {w} ,A\mathbf {v} \rangle =\langle A\mathbf {w} ,\mathbf {v} \rangle ,}

for any pair of vectors 
  
    
      
        
          v
        
        ,
        
          w
        
        ,
      
    
    {\displaystyle \mathbf {v} ,\mathbf {w} ,}
  
 where 
  
    
      
        ⟨
        ⋅
        ,
        ⋅
        ⟩
      
    
    {\displaystyle \langle \cdot ,\cdot \rangle }
  
 denotes <a href="/facts/Dot_product/tNz8MLjT">the inner product</a> operation.
</p><p>This is also the way that the more general concept of <a href="/facts/Self-adjoint_operator/VrH1nEAK">self-adjoint operator</a> is defined.
</p>
<h3>Real-valuedness of quadratic forms</h3>
<p>An 
  
    
      
        n
        ×

n
      
    
    {\displaystyle n\times {}n}
  
 matrix 
  
    
      
        A
      
    
    {\displaystyle A}
  
 is Hermitian if and only if

⟨
        
          v
        
        ,
        A
        
          v
        
        ⟩
        ∈
        
          R
        
        ,
        
        
          for all 
        
        
          v
        
        ∈
        
          
            C
          
          
            n
          
        
        .
      
    
    {\displaystyle \langle \mathbf {v} ,A\mathbf {v} \rangle \in \mathbb {R} ,\quad {\text{for all }}\mathbf {v} \in \mathbb {C} ^{n}.}

</p>
<h3>Spectral properties</h3>
<p>A square matrix 
  
    
      
        A
      
    
    {\displaystyle A}
  
 is Hermitian if and only if it is unitarily <a href="/facts/Diagonalizable_matrix/y3MkyTCy">diagonalizable</a> with real <a href="/facts/Eigenvalues_and_eigenvectors/8TjEoT8u">eigenvalues</a>.
</p>
<h2 id="applications">Applications</h2>
<p>Hermitian matrices are fundamental to <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a> because they describe operators with necessarily real eigenvalues. An eigenvalue 
  
    
      
        a
      
    
    {\displaystyle a}
  
 of an operator 
  
    
      
        
          
            
              A
              ^
            
          
        
      
    
    {\displaystyle {\hat {A}}}
  
 on some quantum state 
  
    
      
        
          |
        
        ψ
        ⟩
      
    
    {\displaystyle |\psi \rangle }
  
 is one of the possible measurement outcomes of the operator, which requires the operators to have real eigenvalues.
</p><p>In <a href="/facts/Signal_processing/Npaja6zb">signal processing</a>, Hermitian matrices are utilized in tasks like <a href="/facts/Fourier_analysis/sUA8JP6q">Fourier analysis</a> and signal representation.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> The eigenvalues and eigenvectors of Hermitian matrices play a crucial role in analyzing signals and extracting meaningful information.
</p><p>Hermitian matrices are extensively studied in <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a> and <a href="/facts/Numerical_analysis/QT9cYa9z">numerical analysis</a>. They have well-defined spectral properties, and many numerical algorithms, such as the <a href="/facts/Lanczos_algorithm/t3n5RYKr">Lanczos algorithm</a>, exploit these properties for efficient computations. Hermitian matrices also appear in techniques like <a href="/facts/Singular_value_decomposition/8t3v6wk3">singular value decomposition</a> (SVD) and <a href="/facts/Eigenvalue_decomposition/a2nfF7hJ">eigenvalue decomposition</a>.
</p><p>In <a href="/facts/Statistics/DNPsKGYU">statistics</a> and <a href="/facts/Machine_learning/e0w0XJTu">machine learning</a>, Hermitian matrices are used in <a href="/facts/Covariance_matrix/kLhgjHC6">covariance matrices</a>, where they represent the relationships between different variables. The positive definiteness of a Hermitian covariance matrix ensures the well-definedness of multivariate distributions.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p><p>Hermitian matrices are applied in the design and analysis of <a href="/facts/Communications_system/5t3Ys2DI">communications system</a>, especially in the field of <a href="/facts/Multiple-input_multiple-output/1EXPB7Vl">multiple-input multiple-output</a> (MIMO) systems. Channel matrices in MIMO systems often exhibit Hermitian properties.
</p><p>In <a href="/facts/Graph_theory/VVPCd4TX">graph theory</a>, Hermitian matrices are used to study the <a href="/facts/Spectral_graph_theory/tBttgGTh">spectra of graphs</a>. The Hermitian Laplacian matrix is a key tool in this context, as it is used to analyze the spectra of mixed graphs.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> The Hermitian-adjacency matrix of a mixed graph is another important concept, as it is a Hermitian matrix that plays a role in studying the energies of mixed graphs.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>
</p>
<h2 id="examples-and-solutions">Examples and solutions</h2>
<p>In this section, the conjugate transpose of matrix 
  
    
      
        A
      
    
    {\displaystyle A}
  
 is denoted as 
  
    
      
        
          A
          
            
              H
            
          
        
        ,
      
    
    {\displaystyle A^{\mathsf {H}},}
  
 the transpose of matrix 
  
    
      
        A
      
    
    {\displaystyle A}
  
 is denoted as 
  
    
      
        
          A
          
            
              T
            
          
        
      
    
    {\displaystyle A^{\mathsf {T}}}
  
 and conjugate of matrix 
  
    
      
        A
      
    
    {\displaystyle A}
  
 is denoted as 
  
    
      
        
          
            A
            ¯
          
        
        .
      
    
    {\displaystyle {\overline {A}}.}

</p><p>See the following example:
</p><p>
  
    
      
        
          
            [
            
              
                
                  0
                
                
                  a
                  −
                  i
                  b
                
                
                  c
                  −
                  i
                  d
                
              
              
                
                  a
                  +
                  i
                  b
                
                
                  1
                
                
                  m
                  −
                  i
                  n
                
              
              
                
                  c
                  +
                  i
                  d
                
                
                  m
                  +
                  i
                  n
                
                
                  2
                
              
            
            ]
          
        
      
    
    {\displaystyle {\begin{bmatrix}0&a-ib&c-id\\a+ib&1&m-in\\c+id&m+in&2\end{bmatrix}}}

</p><p>The diagonal elements must be <a href="/facts/Real_number/R02gw5Pb">real</a>, as they must be their own complex conjugate.
</p><p>Well-known families of Hermitian matrices include the <a href="/facts/Pauli_matrices/BdeC2wos">Pauli matrices</a>, the <a href="/facts/Gell-Mann_matrices/8WleYNIf">Gell-Mann matrices</a> and their generalizations. In <a href="/facts/Theoretical_physics/Cs2Y12rQ">theoretical physics</a> such Hermitian matrices are often multiplied by <a href="/facts/Imaginary_number/6Xh9W3hq">imaginary</a> coefficients,<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a><a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> which results in <a href="/facts/Skew-Hermitian_matrix/wxjjnvCC">skew-Hermitian matrices</a>.
</p><p>Here, we offer another useful Hermitian matrix using an abstract example. If a square matrix 
  
    
      
        A
      
    
    {\displaystyle A}
  
 equals the <a href="/facts/Matrix_multiplication/lYDB6Ro2">product of a matrix</a> with its conjugate transpose, that is, 
  
    
      
        A
        =
        B
        
          B
          
            
              H
            
          
        
        ,
      
    
    {\displaystyle A=BB^{\mathsf {H}},}
  
 then 
  
    
      
        A
      
    
    {\displaystyle A}
  
 is a Hermitian <a href="/facts/Positive_semi-definite_matrix/Hbr6AuPS">positive semi-definite matrix</a>. Furthermore, if 
  
    
      
        B
      
    
    {\displaystyle B}
  
 is row full-rank, then 
  
    
      
        A
      
    
    {\displaystyle A}
  
 is positive definite.
</p>
<h2 id="properties">Properties</h2>
<h3>Main diagonal values are real</h3>
<p>The entries on the <a href="/facts/Main_diagonal/rkT2F08Q">main diagonal</a> (top left to bottom right) of any Hermitian matrix are <a href="/facts/Real_number/R02gw5Pb">real</a>.
</p>
Proof
<p>By definition of the Hermitian matrix

H
          
            i
            j
          
        
        =
        
          
            
              H
              ¯
            
          
          
            j
            i
          
        
      
    
    {\displaystyle H_{ij}={\overline {H}}_{ji}}

so for <i>i</i> = <i>j</i> the above follows, as a number can equal its complex conjugate only if the imaginary parts are zero.
</p>

<p>Only the <a href="/facts/Main_diagonal/rkT2F08Q">main diagonal</a> entries are necessarily real; Hermitian matrices can have arbitrary complex-valued entries in their <a href="/facts/Off-diagonal_element/39z9sIRb">off-diagonal elements</a>, as long as diagonally-opposite entries are complex conjugates.
</p>
<h3>Symmetric</h3>
<p>A matrix that has only real entries is <a href="/facts/Symmetric_matrix/08CGwjNl">symmetric</a> <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> it is a Hermitian matrix. A real and symmetric matrix is simply a special case of a Hermitian matrix.
</p>
Proof
<p>
  
    
      
        
          H
          
            i
            j
          
        
        =
        
          
            
              H
              ¯
            
          
          
            j
            i
          
        
      
    
    {\displaystyle H_{ij}={\overline {H}}_{ji}}
  
 by definition. Thus  
  
    
      
        
          H
          
            i
            j
          
        
        =
        
          H
          
            j
            i
          
        
      
    
    {\displaystyle H_{ij}=H_{ji}}
  
 (matrix symmetry) if and only if 
  
    
      
        
          H
          
            i
            j
          
        
        =
        
          
            
              H
              ¯
            
          
          
            i
            j
          
        
      
    
    {\displaystyle H_{ij}={\overline {H}}_{ij}}
  
 (
  
    
      
        
          H
          
            i
            j
          
        
      
    
    {\displaystyle H_{ij}}
  
 is real).
</p>

<p>So, if a real anti-symmetric matrix is multiplied by a real multiple of the imaginary unit 
  
    
      
        i
        ,
      
    
    {\displaystyle i,}
  
 then it becomes Hermitian.
</p>
<h3>Normal</h3>
<p>Every Hermitian matrix is a <a href="/facts/Normal_matrix/WVKxrEK9">normal matrix</a>. That is to say, 
  
    
      
        A
        
          A
          
            
              H
            
          
        
        =
        
          A
          
            
              H
            
          
        
        A
        .
      
    
    {\displaystyle AA^{\mathsf {H}}=A^{\mathsf {H}}A.}

</p>
Proof
<p>
  
    
      
        A
        =
        
          A
          
            
              H
            
          
        
        ,
      
    
    {\displaystyle A=A^{\mathsf {H}},}
  
 so 
  
    
      
        A
        
          A
          
            
              H
            
          
        
        =
        A
        A
        =
        
          A
          
            
              H
            
          
        
        A
        .
      
    
    {\displaystyle AA^{\mathsf {H}}=AA=A^{\mathsf {H}}A.}

</p>

<h3>Diagonalizable</h3>
<p>The finite-dimensional <a href="/facts/Spectral_theorem/Vqc04B21">spectral theorem</a> says that any Hermitian matrix can be <a href="/facts/Diagonalizable_matrix/y3MkyTCy">diagonalized</a> by a <a href="/facts/Unitary_matrix/8fsxv3KD">unitary matrix</a>, and that the resulting diagonal matrix has only real entries. This implies that all <a href="/facts/Eigenvectors/8TjEoT8u">eigenvalues</a> of a Hermitian matrix A with dimension n are real, and that A has n linearly independent <a href="/facts/Eigenvector/8TjEoT8u">eigenvectors</a>. Moreover, a Hermitian matrix has <a href="/facts/Orthogonal/JVIjYPog">orthogonal</a> eigenvectors for distinct eigenvalues. Even if there are degenerate eigenvalues, it is always possible to find an <a href="/facts/Orthogonal_basis/H4DgNCgR">orthogonal basis</a> of C<i>n</i> consisting of n eigenvectors of A.
</p>
<h3>Sum of Hermitian matrices</h3>
<p>The sum of any two Hermitian matrices is Hermitian.
</p>
Proof
<p>
  
    
      
        (
        A
        +
        B
        
          )
          
            i
            j
          
        
        =
        
          A
          
            i
            j
          
        
        +
        
          B
          
            i
            j
          
        
        =
        
          
            
              A
              ¯
            
          
          
            j
            i
          
        
        +
        
          
            
              B
              ¯
            
          
          
            j
            i
          
        
        =
        
          
            
              
                (
                A
                +
                B
                )
              
              ¯
            
          
          
            j
            i
          
        
        ,
      
    
    {\displaystyle (A+B)_{ij}=A_{ij}+B_{ij}={\overline {A}}_{ji}+{\overline {B}}_{ji}={\overline {(A+B)}}_{ji},}
  
 as claimed.
</p>

<h3>Inverse is Hermitian</h3>
<p>The <a href="/facts/Inverse_matrix/fPqXk3V8">inverse</a> of an invertible Hermitian matrix is Hermitian as well.
</p>
Proof
<p>If 
  
    
      
        
          A
          
            −
            1
          
        
        A
        =
        I
        ,
      
    
    {\displaystyle A^{-1}A=I,}
  
 then 
  
    
      
        I
        =
        
          I
          
            
              H
            
          
        
        =
        
          
            (
            
              
                A
                
                  −
                  1
                
              
              A
            
            )
          
          
            
              H
            
          
        
        =
        
          A
          
            
              H
            
          
        
        
          
            (
            
              A
              
                −
                1
              
            
            )
          
          
            
              H
            
          
        
        =
        A
        
          
            (
            
              A
              
                −
                1
              
            
            )
          
          
            
              H
            
          
        
        ,
      
    
    {\displaystyle I=I^{\mathsf {H}}=\left(A^{-1}A\right)^{\mathsf {H}}=A^{\mathsf {H}}\left(A^{-1}\right)^{\mathsf {H}}=A\left(A^{-1}\right)^{\mathsf {H}},}
  
 so 
  
    
      
        
          A
          
            −
            1
          
        
        =
        
          
            (
            
              A
              
                −
                1
              
            
            )
          
          
            
              H
            
          
        
      
    
    {\displaystyle A^{-1}=\left(A^{-1}\right)^{\mathsf {H}}}
  
 as claimed.
</p>

<h3>Associative product of Hermitian matrices</h3>
<p>The <a href="/facts/Matrix_multiplication/lYDB6Ro2">product</a> of two Hermitian matrices A and B is Hermitian if and only if <i>AB</i> = <i>BA</i>.
</p>
Proof
<p>
  
    
      
        (
        A
        B
        
          )
          
            
              H
            
          
        
        =
        
          
            
              (
              A
              B
              
                )
                
                  
                    T
                  
                
              
            
            ¯
          
        
        =
        
          
            
              
                B
                
                  
                    T
                  
                
              
              
                A
                
                  
                    T
                  
                
              
            
            ¯
          
        
        =
        
          
            
              B
              
                
                  T
                
              
            
            ¯
          
        
         
        
          
            
              A
              
                
                  T
                
              
            
            ¯
          
        
        =
        
          B
          
            
              H
            
          
        
        
          A
          
            
              H
            
          
        
        =
        B
        A
        .
      
    
    {\displaystyle (AB)^{\mathsf {H}}={\overline {(AB)^{\mathsf {T}}}}={\overline {B^{\mathsf {T}}A^{\mathsf {T}}}}={\overline {B^{\mathsf {T}}}}\ {\overline {A^{\mathsf {T}}}}=B^{\mathsf {H}}A^{\mathsf {H}}=BA.}
  
 Thus 
  
    
      
        (
        A
        B
        
          )
          
            
              H
            
          
        
        =
        A
        B
      
    
    {\displaystyle (AB)^{\mathsf {H}}=AB}
  
 <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> 
  
    
      
        A
        B
        =
        B
        A
        .
      
    
    {\displaystyle AB=BA.}

</p><p>Thus <i>A</i><i>n</i> is Hermitian if A is Hermitian and n is an integer.
</p>

<h3><i>ABA</i> Hermitian</h3>
<p>If <i>A</i> and <i>B</i> are Hermitian, then <i>ABA</i> is also Hermitian.
</p>
Proof
<p>
  
    
      
        (
        A
        B
        A
        
          )
          
            
              H
            
          
        
        =
        (
        A
        (
        B
        A
        )
        
          )
          
            
              H
            
          
        
        =
        (
        B
        A
        
          )
          
            
              H
            
          
        
        
          A
          
            
              H
            
          
        
        =
        
          A
          
            
              H
            
          
        
        
          B
          
            
              H
            
          
        
        
          A
          
            
              H
            
          
        
        =
        A
        B
        A
      
    
    {\displaystyle (ABA)^{\mathsf {H}}=(A(BA))^{\mathsf {H}}=(BA)^{\mathsf {H}}A^{\mathsf {H}}=A^{\mathsf {H}}B^{\mathsf {H}}A^{\mathsf {H}}=ABA}

</p>

<h3>vH<i>A</i>v is real for complex v</h3>
<p>For an arbitrary complex valued vector v the product 
  
    
      
        
          
            v
          
          
            
              H
            
          
        
        A
        
          v
        
      
    
    {\displaystyle \mathbf {v} ^{\mathsf {H}}A\mathbf {v} }
  
 is real because of 
  
    
      
        
          
            v
          
          
            
              H
            
          
        
        A
        
          v
        
        =
        
          
            (
            
              
                
                  v
                
                
                  
                    H
                  
                
              
              A
              
                v
              
            
            )
          
          
            
              H
            
          
        
        .
      
    
    {\displaystyle \mathbf {v} ^{\mathsf {H}}A\mathbf {v} =\left(\mathbf {v} ^{\mathsf {H}}A\mathbf {v} \right)^{\mathsf {H}}.}
  
 This is especially important in quantum physics where Hermitian matrices are operators that measure properties of a system, e.g. total <a href="/facts/Spin_(physics)/4Jvp7e8P">spin</a>, which have to be real.
</p>
<h3>Complex Hermitian forms vector space over ℝ</h3>
<p>The Hermitian complex n-by-n matrices do not form a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> over the <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a>, ℂ, since the identity matrix <i>I</i><i>n</i> is Hermitian, but <i>i</i> <i>I</i><i>n</i> is not. However the complex Hermitian matrices <i>do</i> form a vector space over the <a href="/facts/Real_numbers/R02gw5Pb">real numbers</a> ℝ. In the 2<i>n</i>2-<a href="/facts/Dimension_of_a_vector_space/z8m02A3W">dimensional</a> vector space of complex <i>n</i> × <i>n</i> matrices over ℝ, the complex Hermitian matrices form a subspace of dimension <i>n</i>2. If <i>E</i><i>jk</i> denotes the n-by-n matrix with a 1 in the <i>j</i>,<i>k</i> position and zeros elsewhere, a basis (orthonormal with respect to the Frobenius inner product) can be described as follows:

E
          
            j
            j
          
        
        
           for 
        
        1
        ≤
        j
        ≤
        n
        
        (
        n
        
           matrices
        
        )
      
    
    {\displaystyle E_{jj}{\text{ for }}1\leq j\leq n\quad (n{\text{ matrices}})}

</p><p>together with the set of matrices of the form

1
            
              2
            
          
        
        
          (
          
            
              E
              
                j
                k
              
            
            +
            
              E
              
                k
                j
              
            
          
          )
        
        
           for 
        
        1
        ≤
        j
        <
        k
        ≤
        n
        
        
          (
          
            
              
                
                  
                    n
                    
                      2
                    
                  
                  −
                  n
                
                2
              
            
            
               matrices
            
          
          )
        
      
    
    {\displaystyle {\frac {1}{\sqrt {2}}}\left(E_{jk}+E_{kj}\right){\text{ for }}1\leq j<k\leq n\quad \left({\frac {n^{2}-n}{2}}{\text{ matrices}}\right)}

</p><p>and the matrices

i
            
              2
            
          
        
        
          (
          
            
              E
              
                j
                k
              
            
            −
            
              E
              
                k
                j
              
            
          
          )
        
        
           for 
        
        1
        ≤
        j
        <
        k
        ≤
        n
        
        
          (
          
            
              
                
                  
                    n
                    
                      2
                    
                  
                  −
                  n
                
                2
              
            
            
               matrices
            
          
          )
        
      
    
    {\displaystyle {\frac {i}{\sqrt {2}}}\left(E_{jk}-E_{kj}\right){\text{ for }}1\leq j<k\leq n\quad \left({\frac {n^{2}-n}{2}}{\text{ matrices}}\right)}

</p><p>where 
  
    
      
        i
      
    
    {\displaystyle i}
  
 denotes the <a href="/facts/Imaginary_unit/1SCllQlU">imaginary unit</a>, 
  
    
      
        i
        =
        
          
            −
            1
          
        
         
        .
      
    
    {\displaystyle i={\sqrt {-1}}~.}

</p><p>An example is that the four <a href="/facts/Pauli_matrices/BdeC2wos">Pauli matrices</a> form a complete basis for the vector space of all complex 2-by-2 Hermitian matrices over ℝ.
</p>
<h3>Eigendecomposition</h3>
<p>If n orthonormal eigenvectors 
  
    
      
        
          
            u
          
          
            1
          
        
        ,
        …
        ,
        
          
            u
          
          
            n
          
        
      
    
    {\displaystyle \mathbf {u} _{1},\dots ,\mathbf {u} _{n}}
  
 of a Hermitian matrix are chosen and written as the columns of the matrix U, then one <a href="/facts/Eigendecomposition_of_a_matrix/a2nfF7hJ">eigendecomposition</a> of A is 
  
    
      
        A
        =
        U
        Λ
        
          U
          
            
              H
            
          
        
      
    
    {\displaystyle A=U\Lambda U^{\mathsf {H}}}
  
 where 
  
    
      
        U
        
          U
          
            
              H
            
          
        
        =
        I
        =
        
          U
          
            
              H
            
          
        
        U
      
    
    {\displaystyle UU^{\mathsf {H}}=I=U^{\mathsf {H}}U}
  
 and therefore

A
        =
        
          ∑
          
            j
          
        
        
          λ
          
            j
          
        
        
          
            u
          
          
            j
          
        
        
          
            u
          
          
            j
          
          
            
              H
            
          
        
        ,
      
    
    {\displaystyle A=\sum _{j}\lambda _{j}\mathbf {u} _{j}\mathbf {u} _{j}^{\mathsf {H}},}

where 
  
    
      
        
          λ
          
            j
          
        
      
    
    {\displaystyle \lambda _{j}}
  
 are the eigenvalues on the diagonal of the diagonal matrix 
  
    
      
        Λ
        .
      
    
    {\displaystyle \Lambda .}

</p>
<h3>Singular values</h3>
<p>The singular values of 
  
    
      
        A
      
    
    {\displaystyle A}
  
 are the absolute values of its eigenvalues:
</p><p>Since 
  
    
      
        A
      
    
    {\displaystyle A}
  
 has an eigendecomposition 
  
    
      
        A
        =
        U
        Λ
        
          U
          
            H
          
        
      
    
    {\displaystyle A=U\Lambda U^{H}}
  
, where 
  
    
      
        U
      
    
    {\displaystyle U}
  
 is a <a href="/facts/Unitary_matrix/8fsxv3KD">unitary matrix</a> (its columns are orthonormal vectors; see above), a <a href="/facts/Singular_value_decomposition/8t3v6wk3">singular value decomposition</a> of 
  
    
      
        A
      
    
    {\displaystyle A}
  
 is 
  
    
      
        A
        =
        U
        
          |
        
        Λ
        
          |
        
        
          sgn
        
        (
        Λ
        )
        
          U
          
            H
          
        
      
    
    {\displaystyle A=U|\Lambda |{\text{sgn}}(\Lambda )U^{H}}
  
, where 
  
    
      
        
          |
        
        Λ
        
          |
        
      
    
    {\displaystyle |\Lambda |}
  
 and 
  
    
      
        
          sgn
        
        (
        Λ
        )
      
    
    {\displaystyle {\text{sgn}}(\Lambda )}
  
 are diagonal matrices containing the absolute values 
  
    
      
        
          |
        
        λ
        
          |
        
      
    
    {\displaystyle |\lambda |}
  
 and signs 
  
    
      
        
          sgn
        
        (
        λ
        )
      
    
    {\displaystyle {\text{sgn}}(\lambda )}
  
 of 
  
    
      
        A
      
    
    {\displaystyle A}
  
's eigenvalues, respectively. 
  
    
      
        sgn
        ⁡
        (
        Λ
        )
        
          U
          
            H
          
        
      
    
    {\displaystyle \operatorname {sgn}(\Lambda )U^{H}}
  
 is unitary, since the columns of 
  
    
      
        
          U
          
            H
          
        
      
    
    {\displaystyle U^{H}}
  
 are only getting multiplied by 
  
    
      
        ±
        1
      
    
    {\displaystyle \pm 1}
  
. 
  
    
      
        
          |
        
        Λ
        
          |
        
      
    
    {\displaystyle |\Lambda |}
  
 contains the singular values of 
  
    
      
        A
      
    
    {\displaystyle A}
  
, namely, the absolute values of its eigenvalues.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a>
</p>
<h3>Real determinant</h3>
<p>The determinant of a Hermitian matrix is real:
</p>
Proof
<p>
  
    
      
        det
        (
        A
        )
        =
        det
        
          (
          
            A
            
              
                T
              
            
          
          )
        
        
        ⇒
        
        det
        
          (
          
            A
            
              
                H
              
            
          
          )
        
        =
        
          
            
              det
              (
              A
              )
            
            ¯
          
        
      
    
    {\displaystyle \det(A)=\det \left(A^{\mathsf {T}}\right)\quad \Rightarrow \quad \det \left(A^{\mathsf {H}}\right)={\overline {\det(A)}}}

Therefore if 
  
    
      
        A
        =
        
          A
          
            
              H
            
          
        
        
        ⇒
        
        det
        (
        A
        )
        =
        
          
            
              det
              (
              A
              )
            
            ¯
          
        
        .
      
    
    {\displaystyle A=A^{\mathsf {H}}\quad \Rightarrow \quad \det(A)={\overline {\det(A)}}.}

</p>

<p>(Alternatively, the determinant is the product of the matrix's eigenvalues, and as mentioned before, the eigenvalues of a Hermitian matrix are real.)
</p>
<h2 id="decomposition-into-hermitian-and-skew-hermitian-matrices">Decomposition into Hermitian and skew-Hermitian matrices</h2>
<p>Additional facts related to Hermitian matrices include:
</p>
<ul><li>The sum of a square matrix and its conjugate transpose 
  
    
      
        
          (
          
            A
            +
            
              A
              
                
                  H
                
              
            
          
          )
        
      
    
    {\displaystyle \left(A+A^{\mathsf {H}}\right)}
  
 is Hermitian.</li>
<li>The difference of a square matrix and its conjugate transpose 
  
    
      
        
          (
          
            A
            −
            
              A
              
                
                  H
                
              
            
          
          )
        
      
    
    {\displaystyle \left(A-A^{\mathsf {H}}\right)}
  
 is <a href="/facts/Skew-Hermitian_matrix/wxjjnvCC">skew-Hermitian</a> (also called antihermitian). This implies that the <a href="/facts/Commutator/eYxzFmFY">commutator</a> of two Hermitian matrices is skew-Hermitian.</li>
<li>An arbitrary square matrix C can be written as the sum of a Hermitian matrix A and a skew-Hermitian matrix B. This is known as the Toeplitz decomposition of C.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a>: 227  
  
    
      
        C
        =
        A
        +
        B
        
        
          with
        
        
        A
        =
        
          
            1
            2
          
        
        
          (
          
            C
            +
            
              C
              
                
                  H
                
              
            
          
          )
        
        
        
          and
        
        
        B
        =
        
          
            1
            2
          
        
        
          (
          
            C
            −
            
              C
              
                
                  H
                
              
            
          
          )
        
      
    
    {\displaystyle C=A+B\quad {\text{with}}\quad A={\frac {1}{2}}\left(C+C^{\mathsf {H}}\right)\quad {\text{and}}\quad B={\frac {1}{2}}\left(C-C^{\mathsf {H}}\right)}
  
</li></ul>
<h2 id="rayleigh-quotient">Rayleigh quotient</h2>
<p class="note">Main article: <a href="/facts/Rayleigh_quotient/yH7sD1K4">Rayleigh quotient</a></p>
<p>In mathematics, for a given complex Hermitian matrix M and nonzero vector x, the Rayleigh quotient<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> 
  
    
      
        R
        (
        M
        ,
        
          x
        
        )
        ,
      
    
    {\displaystyle R(M,\mathbf {x} ),}
  
 is defined as:<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a>: p. 234 <a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a>

R
        (
        M
        ,
        
          x
        
        )
        :=
        
          
            
              
                
                  x
                
                
                  
                    H
                  
                
              
              M
              
                x
              
            
            
              
                
                  x
                
                
                  
                    H
                  
                
              
              
                x
              
            
          
        
        .
      
    
    {\displaystyle R(M,\mathbf {x} ):={\frac {\mathbf {x} ^{\mathsf {H}}M\mathbf {x} }{\mathbf {x} ^{\mathsf {H}}\mathbf {x} }}.}

</p><p>For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the conjugate transpose 
  
    
      
        
          
            x
          
          
            
              H
            
          
        
      
    
    {\displaystyle \mathbf {x} ^{\mathsf {H}}}
  
 to the usual transpose 
  
    
      
        
          
            x
          
          
            
              T
            
          
        
        .
      
    
    {\displaystyle \mathbf {x} ^{\mathsf {T}}.}
  
 
  
    
      
        R
        (
        M
        ,
        c
        
          x
        
        )
        =
        R
        (
        M
        ,
        
          x
        
        )
      
    
    {\displaystyle R(M,c\mathbf {x} )=R(M,\mathbf {x} )}
  
 for any non-zero real scalar 
  
    
      
        c
        .
      
    
    {\displaystyle c.}
  
 Also, recall that a Hermitian (or real symmetric) matrix has real eigenvalues.
</p><p>It can be shown<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> that, for a given matrix, the Rayleigh quotient reaches its minimum value 
  
    
      
        
          λ
          
            min
          
        
      
    
    {\displaystyle \lambda _{\min }}
  
 (the smallest eigenvalue of M) when 
  
    
      
        
          x
        
      
    
    {\displaystyle \mathbf {x} }
  
 is 
  
    
      
        
          
            v
          
          
            min
          
        
      
    
    {\displaystyle \mathbf {v} _{\min }}
  
 (the corresponding eigenvector). Similarly, 
  
    
      
        R
        (
        M
        ,
        
          x
        
        )
        ≤
        
          λ
          
            max
          
        
      
    
    {\displaystyle R(M,\mathbf {x} )\leq \lambda _{\max }}
  
 and 
  
    
      
        R
        (
        M
        ,
        
          
            v
          
          
            max
          
        
        )
        =
        
          λ
          
            max
          
        
        .
      
    
    {\displaystyle R(M,\mathbf {v} _{\max })=\lambda _{\max }.}

</p><p>The Rayleigh quotient is used in the min-max theorem to get exact values of all eigenvalues. It is also used in eigenvalue algorithms to obtain an eigenvalue approximation from an eigenvector approximation. Specifically, this is the basis for Rayleigh quotient iteration.
</p><p>The range of the Rayleigh quotient (for matrix that is not necessarily Hermitian) is called a numerical range (or spectrum in functional analysis). When the matrix is Hermitian, the numerical range is equal to the spectral norm. Still in functional analysis, 
  
    
      
        
          λ
          
            max
          
        
      
    
    {\displaystyle \lambda _{\max }}
  
 is known as the spectral radius. In the context of C*-algebras or algebraic quantum mechanics, the function that to <i>M</i> associates the Rayleigh quotient <i>R</i>(<i>M</i>, <i>x</i>) for a fixed x and <i>M</i> varying through the algebra would be referred to as "vector state" of the algebra.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Complex_symmetric_matrix/08CGwjNl">Complex symmetric matrix</a> – Matrix equal to its transposePages displaying short descriptions of redirect targets</li>
<li><a href="/facts/Haynsworth_inertia_additivity_formula/x71a5V9k">Haynsworth inertia additivity formula</a> – Counts positive, negative, and zero eigenvalues of a block partitioned Hermitian matrix</li>
<li><a href="/facts/Hermitian_form/Kp6qUA8v">Hermitian form</a> – Generalization of a bilinear formPages displaying short descriptions of redirect targets</li>
<li><a href="/facts/Normal_matrix/WVKxrEK9">Normal matrix</a> – Matrix that commutes with its conjugate transpose</li>
<li><a href="/facts/Schur%25E2%2580%2593Horn_theorem/WMQ92oBF">Schur–Horn theorem</a> – Characterizes the diagonal of a Hermitian matrix with given eigenvalues</li>
<li><a href="/facts/Self-adjoint_operator/VrH1nEAK">Self-adjoint operator</a> – Linear operator equal to its own adjoint</li>
<li><a href="/facts/Skew-Hermitian_matrix/wxjjnvCC">Skew-Hermitian matrix</a> – Matrix whose conjugate transpose is its negative (additive inverse) (anti-Hermitian matrix)</li>
<li><a href="/facts/Unitary_matrix/8fsxv3KD">Unitary matrix</a> – Complex matrix whose conjugate transpose equals its inverse</li>
<li><a href="/facts/Vector_space/M1pxMLx2">Vector space</a> – Algebraic structure in linear algebra</li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="https://www.encyclopediaofmath.org/index.php?title=Hermitian_matrix">"Hermitian matrix"</a>, <i><a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a></i>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a>, 2001 [1994]</li>
<li><a href="https://www.cyut.edu.tw/~ckhung/b/la/hermitian.en.php">Visualizing Hermitian Matrix as An Ellipse with Dr. Geo</a> <a href="https://web.archive.org/web/20170829203442/https://www.cyut.edu.tw/~ckhung/b/la/hermitian.en.php">Archived</a> 2017-08-29 at the <a href="/facts/Wayback_Machine/nmQ3a6JC">Wayback Machine</a>, by Chao-Kuei Hung from Chaoyang University, gives a more geometric explanation.</li>
<li><a href="http://www.mathpages.com/home/kmath306/kmath306.htm">"Hermitian Matrices"</a>. <i>MathPages.com</i>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Archibald, Tom (2010-12-31), Gowers, Timothy; Barrow-Green, June; Leader, Imre (eds.), "VI.47 Charles Hermite", The Princeton Companion to Mathematics, Princeton University Press, p. 773, doi:10.1515/9781400830398.773a, ISBN 978-1-4008-3039-8, retrieved 2023-11-15 <a href="978-1-4008-3039-8" target="_blank">978-1-4008-3039-8</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Ribeiro, Alejandro. "Signal and Information Processing" (PDF). <a href="https://www.seas.upenn.edu/~ese2240/wiki/Lecture%20Notes/sip_PCA.pdf" target="_blank">https://www.seas.upenn.edu/~ese2240/wiki/Lecture%20Notes/sip_PCA.pdf</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>"MULTIVARIATE NORMAL DISTRIBUTIONS" (PDF). <a href="https://dspace.mit.edu/bitstream/handle/1721.1/121170/6-436j-fall-2008/contents/lecture-notes/MIT6_436JF08_lec15.pdf" target="_blank">https://dspace.mit.edu/bitstream/handle/1721.1/121170/6-436j-fall-2008/contents/lecture-notes/MIT6_436JF08_lec15.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Lau, Ivan. "Hermitian Spectral Theory of Mixed Graphs" (PDF). <a href="https://www.sfu.ca/~iplau/Edinburgh_CS_Project.pdf" target="_blank">https://www.sfu.ca/~iplau/Edinburgh_CS_Project.pdf</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Liu, Jianxi; Li, Xueliang (February 2015). "Hermitian-adjacency matrices and Hermitian energies of mixed graphs". Linear Algebra and Its Applications. 466: 182–207. doi:10.1016/j.laa.2014.10.028. <a href="https://doi.org/10.1016%2Fj.laa.2014.10.028" target="_blank">https://doi.org/10.1016%2Fj.laa.2014.10.028</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Frankel, Theodore (2004). The Geometry of Physics: an introduction. Cambridge University Press. p. 652. ISBN 0-521-53927-7. <a href="0-521-53927-7" target="_blank">0-521-53927-7</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Physics 125 Course Notes Archived 2022-03-07 at the Wayback Machine at California Institute of Technology <a href="http://www.hep.caltech.edu/~fcp/physics/quantumMechanics/angularMomentum/angularMomentum.pdf" target="_blank">http://www.hep.caltech.edu/~fcp/physics/quantumMechanics/angularMomentum/angularMomentum.pdf</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>Trefethan, Lloyd N.; Bau, III, David (1997). Numerical linear algebra. Philadelphia, PA, USA: SIAM. p. 34. ISBN 0-89871-361-7. OCLC 1348374386. <a href="0-89871-361-7" target="_blank">0-89871-361-7</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
<li id="fn:9"><p>Horn, Roger A.; Johnson, Charles R. (2013). Matrix Analysis, second edition. Cambridge University Press. ISBN 9780521839402. <a href="9780521839402" target="_blank">9780521839402</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li>
<li id="fn:10"><p>Also known as the Rayleigh–Ritz ratio; named after Walther Ritz and Lord Rayleigh. <a href="/wiki/Walther_Ritz" target="_blank">/wiki/Walther_Ritz</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li>
<li id="fn:11"><p>Horn, Roger A.; Johnson, Charles R. (2013). Matrix Analysis, second edition. Cambridge University Press. ISBN 9780521839402. <a href="9780521839402" target="_blank">9780521839402</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li>
<li id="fn:12"><p>Parlet B. N. The symmetric eigenvalue problem, SIAM, Classics in Applied Mathematics,1998 <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li>
<li id="fn:13"><p>Horn, Roger A.; Johnson, Charles R. (2013). Matrix Analysis, second edition. Cambridge University Press. ISBN 9780521839402. <a href="9780521839402" target="_blank">9780521839402</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li>
</ol>

Hermitian matrix open-in-new

Hermitian matrix