Symmetric bilinear form

<h2 id="formal-definition">Formal definition</h2>
Let V be a vector space of dimension n over a field K. A <a href="/facts/Function_(mathematics)/CdReEKCh">map</a> 
 
 
 
 B
 :
 V
 ×
 V
 →
 K
 
 
 {\displaystyle B:V\times V\rightarrow K}
 
 is a symmetric bilinear form on the space if:

<ul><li>
 
 
 
 B
 (
 u
 ,
 v
 )
 =
 B
 (
 v
 ,
 u
 )
  
 
 ∀
 u
 ,
 v
 ∈
 V
 
 
 {\displaystyle B(u,v)=B(v,u)\ \quad \forall u,v\in V}
 
</li>
<li>
 
 
 
 B
 (
 u
 +
 v
 ,
 w
 )
 =
 B
 (
 u
 ,
 w
 )
 +
 B
 (
 v
 ,
 w
 )
  
 
 ∀
 u
 ,
 v
 ,
 w
 ∈
 V
 
 
 {\displaystyle B(u+v,w)=B(u,w)+B(v,w)\ \quad \forall u,v,w\in V}
 
</li>
<li>
 
 
 
 B
 (
 λ
 v
 ,
 w
 )
 =
 λ
 B
 (
 v
 ,
 w
 )
  
 
 ∀
 λ
 ∈
 K
 ,
 ∀
 v
 ,
 w
 ∈
 V
 
 
 {\displaystyle B(\lambda v,w)=\lambda B(v,w)\ \quad \forall \lambda \in K,\forall v,w\in V}
 
</li></ul>
The last two axioms only establish linearity in the first argument, but the first axiom (symmetry) then immediately implies linearity in the second argument as well.

<h2 id="examples">Examples</h2>
Let V = Rn, the n dimensional real vector space. Then the standard <a href="/facts/Dot_product/tNz8MLjT">dot product</a> is a symmetric bilinear form, B(x, y) = x ⋅ y. The matrix corresponding to this bilinear form (see below) on a <a href="/facts/Standard_basis/sLc9L8xx">standard basis</a> is the identity matrix.
Let V be any vector space (including possibly infinite-dimensional), and assume T is a linear function from V to the field. Then the function defined by B(x, y) = T(x)T(y) is a symmetric bilinear form.
Let V be the vector space of continuous single-variable real functions. For 
 
 
 
 f
 ,
 g
 ∈
 V
 
 
 {\displaystyle f,g\in V}
 
 one can define 
 
 
 
 
 B
 (
 f
 ,
 g
 )
 =
 
 ∫
 
 0
 
 
 1
 
 
 f
 (
 t
 )
 g
 (
 t
 )
 d
 t
 
 
 
 {\displaystyle \textstyle B(f,g)=\int _{0}^{1}f(t)g(t)dt}
 
. By the properties of <a href="/facts/Integral/V7lyV997">definite integrals</a>, this defines a symmetric bilinear form on V. This is an example of a symmetric bilinear form which is not associated to any symmetric matrix (since the vector space is infinite-dimensional).

<h2 id="matrix-representation">Matrix representation</h2>
Let 
 
 
 
 C
 =
 {
 
 e
 
 1
 
 
 ,
 …
 ,
 
 e
 
 n
 
 
 }
 
 
 {\displaystyle C=\{e_{1},\ldots ,e_{n}\}}
 
 be a basis for V. Define the n × n matrix A by 
 
 
 
 
 A
 
 i
 j
 
 
 =
 B
 (
 
 e
 
 i
 
 
 ,
 
 e
 
 j
 
 
 )
 
 
 {\displaystyle A_{ij}=B(e_{i},e_{j})}
 
. The matrix A is a <a href="/facts/Symmetric_matrix/08CGwjNl">symmetric matrix</a> exactly due to symmetry of the bilinear form. If we let the n×1 matrix x represent the vector v with respect to this basis, and similarly let the n×1 matrix y represent the vector w, then 
 
 
 
 B
 (
 v
 ,
 w
 )
 
 
 {\displaystyle B(v,w)}
 
 is given by :

x
          
            
              T
            
          
        
        A
        y
        =
        
          y
          
            
              T
            
          
        
        A
        x
        .
      
    
    {\displaystyle x^{\mathsf {T}}Ay=y^{\mathsf {T}}Ax.}

Suppose C' is another basis for V, with :

[
            
              
                
                  
                    e
                    
                      1
                    
                    ′
                  
                
                
                  ⋯
                
                
                  
                    e
                    
                      n
                    
                    ′
                  
                
              
            
            ]
          
        
        =
        
          
            [
            
              
                
                  
                    e
                    
                      1
                    
                  
                
                
                  ⋯
                
                
                  
                    e
                    
                      n
                    
                  
                
              
            
            ]
          
        
        S
      
    
    {\displaystyle {\begin{bmatrix}e'_{1}&\cdots &e'_{n}\end{bmatrix}}={\begin{bmatrix}e_{1}&\cdots &e_{n}\end{bmatrix}}S}

with S an invertible n×n matrix.
Now the new matrix representation for the symmetric bilinear form is given by

A
          ′
        
        =
        
          S
          
            
              T
            
          
        
        A
        S
        .
      
    
    {\displaystyle A'=S^{\mathsf {T}}AS.}

<h2 id="orthogonality-and-singularity">Orthogonality and singularity</h2>
Two vectors v and w are defined to be orthogonal with respect to the bilinear form B if B(v, w) = 0, which, for a symmetric bilinear form, is equivalent to B(w, v) = 0.
The radical of a bilinear form B is the set of vectors orthogonal with every vector in V. That this is a subspace of V follows from the linearity of B in each of its arguments. When working with a matrix representation A with respect to a certain basis, v, represented by x, is in the radical if and only if

A
        x
        =
        0
        ⟺
        
          x
          
            
              T
            
          
        
        A
        =
        0.
      
    
    {\displaystyle Ax=0\Longleftrightarrow x^{\mathsf {T}}A=0.}

The matrix A is singular if and only if the radical is nontrivial.
If W is a subset of V, then its <a href="/facts/Orthogonal_complement/a13oQBsz">orthogonal complement</a> W⊥ is the set of all vectors in V that are orthogonal to every vector in W; it is a subspace of V. When B is non-degenerate, the radical of B is trivial and the dimension of W⊥ is dim(W⊥) = dim(V) − dim(W).

<h2 id="orthogonal-basis">Orthogonal basis</h2>
A basis 
 
 
 
 C
 =
 {
 
 e
 
 1
 
 
 ,
 …
 ,
 
 e
 
 n
 
 
 }
 
 
 {\displaystyle C=\{e_{1},\ldots ,e_{n}\}}
 
 is orthogonal with respect to B if and only if :

B
        (
        
          e
          
            i
          
        
        ,
        
          e
          
            j
          
        
        )
        =
        0
         
        ∀
        i
        ≠
        j
        .
      
    
    {\displaystyle B(e_{i},e_{j})=0\ \forall i\neq j.}

When the <a href="/facts/Characteristic_(algebra)/D2VzcQaG">characteristic</a> of the field is not two, V always has an orthogonal basis. This can be proven by <a href="/facts/Mathematical_induction/KxAPqNrH">induction</a>.
A basis C is orthogonal if and only if the matrix representation A is a <a href="/facts/Diagonal_matrix/tjIAHrE6">diagonal matrix</a>.

<h3>Signature and Sylvester's law of inertia</h3>
In a more general form, <a href="/facts/Sylvester%27s_law_of_inertia/3dyKLMeI">Sylvester's law of inertia</a> says that, when working over an <a href="/facts/Ordered_field/an0prSgG">ordered field</a>, the numbers of diagonal elements in the diagonalized form of a matrix that are positive, negative and zero respectively are independent of the chosen orthogonal basis. These three numbers form the <a href="/facts/Signature_(quadratic_form)/l2n1jXPe">signature</a> of the bilinear form.

<h3>Real case</h3>
When working in a space over the reals, one can go a bit a further. Let 
 
 
 
 C
 =
 {
 
 e
 
 1
 
 
 ,
 …
 ,
 
 e
 
 n
 
 
 }
 
 
 {\displaystyle C=\{e_{1},\ldots ,e_{n}\}}
 
 be an orthogonal basis.
We define a new basis 
 
 
 
 
 C
 ′
 
 =
 {
 
 e
 
 1
 
 ′
 
 ,
 …
 ,
 
 e
 
 n
 
 ′
 
 }
 
 
 {\displaystyle C'=\{e'_{1},\ldots ,e'_{n}\}}

e
 
 i
 
 ′
 
 =
 
 
 {
 
 
 
 
 e
 
 i
 
 
 
 
 
 if 
 
 B
 (
 
 e
 
 i
 
 
 ,
 
 e
 
 i
 
 
 )
 =
 0
 
 
 
 
 
 
 
 e
 
 i
 
 
 
 B
 (
 
 e
 
 i
 
 
 ,
 
 e
 
 i
 
 
 )
 
 
 
 
 
 
 if 
 
 B
 (
 
 e
 
 i
 
 
 ,
 
 e
 
 i
 
 
 )
 >
 0
 
 
 
 
 
 
 
 e
 
 i
 
 
 
 −
 B
 (
 
 e
 
 i
 
 
 ,
 
 e
 
 i
 
 
 )
 
 
 
 
 
 
 if 
 
 B
 (
 
 e
 
 i
 
 
 ,
 
 e
 
 i
 
 
 )
 <
 0
 
 
 
 
 
 
 
 
 {\displaystyle e'_{i}={\begin{cases}e_{i}&{\text{if }}B(e_{i},e_{i})=0\\{\frac {e_{i}}{\sqrt {B(e_{i},e_{i})}}}&{\text{if }}B(e_{i},e_{i})>0\\{\frac {e_{i}}{\sqrt {-B(e_{i},e_{i})}}}&{\text{if }}B(e_{i},e_{i})<0\end{cases}}}

Now, the new matrix representation A will be a diagonal matrix with only 0, 1 and −1 on the diagonal. Zeroes will appear if and only if the radical is nontrivial.

<h3>Complex case</h3>
When working in a space over the complex numbers, one can go further as well and it is even easier.
Let 
 
 
 
 C
 =
 {
 
 e
 
 1
 
 
 ,
 …
 ,
 
 e
 
 n
 
 
 }
 
 
 {\displaystyle C=\{e_{1},\ldots ,e_{n}\}}
 
 be an orthogonal basis.
We define a new basis 
 
 
 
 
 C
 ′
 
 =
 {
 
 e
 
 1
 
 ′
 
 ,
 …
 ,
 
 e
 
 n
 
 ′
 
 }
 
 
 {\displaystyle C'=\{e'_{1},\ldots ,e'_{n}\}}
 
 :

e
          
            i
          
          ′
        
        =
        
          
            {
            
              
                
                  
                    e
                    
                      i
                    
                  
                
                
                  
                    if 
                  
                  
                  B
                  (
                  
                    e
                    
                      i
                    
                  
                  ,
                  
                    e
                    
                      i
                    
                  
                  )
                  =
                  0
                
              
              
                
                  
                    e
                    
                      i
                    
                  
                  
                    /
                  
                  
                    
                      B
                      (
                      
                        e
                        
                          i
                        
                      
                      ,
                      
                        e
                        
                          i
                        
                      
                      )
                    
                  
                
                
                  
                    if 
                  
                  
                  B
                  (
                  
                    e
                    
                      i
                    
                  
                  ,
                  
                    e
                    
                      i
                    
                  
                  )
                  ≠
                  0
                
              
            
            
          
        
      
    
    {\displaystyle e'_{i}={\begin{cases}e_{i}&{\text{if }}\;B(e_{i},e_{i})=0\\e_{i}/{\sqrt {B(e_{i},e_{i})}}&{\text{if }}\;B(e_{i},e_{i})\neq 0\\\end{cases}}}

Now the new matrix representation A will be a diagonal matrix with only 0 and 1 on the diagonal. Zeroes will appear if and only if the radical is nontrivial.

<h2 id="orthogonal-polarities">Orthogonal polarities</h2>
Let B be a symmetric bilinear form with a trivial radical on the space V over the field K with <a href="/facts/Characteristic_(algebra)/D2VzcQaG">characteristic</a> not 2. One can now define a map from D(V), the set of all subspaces of V, to itself:

α
        :
        D
        (
        V
        )
        →
        D
        (
        V
        )
        :
        W
        ↦
        
          W
          
            ⊥
          
        
        .
      
    
    {\displaystyle \alpha :D(V)\rightarrow D(V):W\mapsto W^{\perp }.}

This map is an orthogonal polarity on the <a href="/facts/Projective_space/7MSpA9cM">projective space</a> PG(W). Conversely, one can prove all orthogonal polarities are induced in this way, and that two symmetric bilinear forms with trivial radical induce the same polarity if and only if they are equal up to scalar multiplication.

<ul><li>Adkins, William A.; Weintraub, Steven H. (1992). Algebra: An Approach via Module Theory. <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">Graduate Texts in Mathematics</a>. Vol. 136. <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-97839-9. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0768.00003">0768.00003</a>.</li>
<li><a href="/facts/John_Milnor/LTf24paN">Milnor, J.</a>; Husemoller, D. (1973). Symmetric Bilinear Forms. <a href="/facts/Ergebnisse_der_Mathematik_und_ihrer_Grenzgebiete/b55RDyX6">Ergebnisse der Mathematik und ihrer Grenzgebiete</a>. Vol. 73. <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-06009-X. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0292.10016">0292.10016</a>.</li>
<li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/SymmetricBilinearForm.html">"Symmetric Bilinear Form"</a>. <a href="/facts/MathWorld/yc1jodbq">MathWorld</a>.</li></ul>

Symmetric bilinear form open-in-new

Symmetric bilinear form