Bilinear form

Let be $L$ ${\ displaystyle L}$ $L$ there is a vector space above the field $K$ ${\ displaystyle K}$ $K$ (most often considered fields $K=\mathbb {R}$ ${\ displaystyle K = \ mathbb {R}}$ $K = {\ mathbb R}$ or $K=\mathbb {C}$ ${\ displaystyle K = \ mathbb {C}}$ $K = {\ mathbb C}$ ).

A bilinear form is a function. $F\colon L\times L\to K$ ${\ displaystyle F \ colon L \ times L \ to K}$ $F \ colon L \ times L \ to K$ linear for each of the arguments :

F(x+z,\,y)=F(x,\,y)+F(z,\,y)

{\ displaystyle F (x + z, \, y) = F (x, \, y) + F (z, \, y)}

{\ displaystyle F (x + z, \, y) = F (x, \, y) + F (z, \, y)}

,

F(x,\,y+z)=F(x,\,y)+F(x,\,z)

{\ displaystyle F (x, \, y + z) = F (x, \, y) + F (x, \, z)}

{\ displaystyle F (x, \, y + z) = F (x, \, y) + F (x, \, z)}

,

F(\lambda x,\,y)=\lambda F(x,\,y)

{\ displaystyle F (\ lambda x, \, y) = \ lambda F (x, \, y)}

{\ displaystyle F (\ lambda x, \, y) = \ lambda F (x, \, y)}

,

F(x,\,\lambda y)=\lambda F(x,\,y)

{\ displaystyle F (x, \, \ lambda y) = \ lambda F (x, \, y)}

{\ displaystyle F (x, \, \ lambda y) = \ lambda F (x, \, y)}

,

here $x,y,z\in L$ ${\ displaystyle x, y, z \ in L}$ $x, y, z \ in L$ and $\lambda \in K.$ ${\ displaystyle \ lambda \ in K.}$ $\ lambda \ in K.$

The bilinear form is a special case of the notion of a tensor (a tensor of rank (0.2)).

Content

Alternative Definition

In the case of finite-dimensional spaces (for example, $\mathbb {R} ^{n}$ ${\ displaystyle \ mathbb {R} ^ {n}}$ ) a different definition is often used.

Let be $L$ ${\ displaystyle L}$ there are many view vectors $x=(x_{1},x_{2},\dots ,x_{n}),$ ${\ displaystyle x = (x_ {1}, x_ {2}, \ dots, x_ {n}),}$ Where $x_{i}\in K,i={\overline {1,n}}$ ${\ displaystyle x_ {i} \ in K, i = {\ overline {1, n}}}$ .

Bilinear forms are called functions. $F\colon L\times L\to K$ ${\ displaystyle F \ colon L \ times L \ to K}$ kind of

F(x,y)=\sum _{i,\,j=1}^{n}a_{ij}x_{i}y_{j},

{\ displaystyle F (x, y) = \ sum _ {i, \, j = 1} ^ {n} a_ {ij} x_ {i} y_ {j},}

Where $x=(x_{1},x_{2},\dots ,x_{n}),$ ${\ displaystyle x = (x_ {1}, x_ {2}, \ dots, x_ {n}),}$ $y=(y_{1},y_{2},\dots ,y_{n}),$ ${\ displaystyle y = (y_ {1}, y_ {2}, \ dots, y_ {n}),}$ but $a_{ij}$ ${\ displaystyle a_ {ij}}$ - some constants from the field $K .$ ${\ displaystyle K.}$

In other words, the bilinear form is a function of two groups of $n$ ${\ displaystyle n}$ variables, which is a homogeneous polynomial of the first degree with respect to variables from each group.

Related definitions

Bilinear form $F$ ${\ displaystyle F}$ called symmetric if $F(x,\,y)=F(y,\,x)$ ${\ displaystyle F (x, \, y) = F (y, \, x)}$ for any vectors $x,y\in L$ ${\ displaystyle x, y \ in L}$ .
Bilinear form $F$ ${\ displaystyle F}$ called antisymmetric (antisymmetric) if $F(x,\,y)=-F(y,\,x)$ ${\ displaystyle F (x, \, y) = - F (y, \, x)}$ for any vectors $x,y\in L$ ${\ displaystyle x, y \ in L}$ .
Vector $x\in L$ ${\ displaystyle x \ in L}$ is called orthogonal (more precisely, left orthogonal ) subspace $M\subset L$ ${\ displaystyle M \ subset L}$ regarding $F$ ${\ displaystyle F}$ , if a $F(x,\,y)=0$ ${\ displaystyle F (x, \, y) = 0}$ for all $y\in M$ ${\ displaystyle y \ in M}$ . The set of vectors $x\in L$ ${\ displaystyle x \ in L}$ orthogonal to a subspace $M\subset L$ ${\ displaystyle M \ subset L}$ relative to this bilinear form $F$ ${\ displaystyle F}$ is called the orthogonal complement of the subspace $M\subset L$ ${\ displaystyle M \ subset L}$ regarding $F$ ${\ displaystyle F}$ and is denoted by $M^{\perp }$ ${\ displaystyle M ^ {\ perp}}$ .
Radical bilinear form $F$ ${\ displaystyle F}$ called the orthogonal complement of the space itself $L$ ${\ displaystyle L}$ regarding $F$ ${\ displaystyle F}$ that is the totality $L^{\perp }$ ${\ displaystyle L ^ {\ perp}}$ vectors $x\in L$ ${\ displaystyle x \ in L}$ for which $F(x,\,y)=0$ ${\ displaystyle F (x, \, y) = 0}$ for all $y\in L$ ${\ displaystyle y \ in L}$ .

Properties

The set of all bilinear forms $W(L,L)$ ${\ displaystyle W (L, L)}$ defined on an arbitrary fixed space is a linear space.
Any bilinear form can be represented as a sum of symmetric and skew-symmetric forms.
With selected basis $e_{1},\ldots ,e_{n}$ ${\ displaystyle e_ {1}, \ ldots, e_ {n}}$ at $L$ ${\ displaystyle L}$ any bilinear form $F$ ${\ displaystyle F}$ uniquely determined by the matrix

{\begin{pmatrix}F(e_{1},\,e_{1})&F(e_{1},\,e_{2})&\ldots &F(e_{1},\,e_{n})\\F(e_{2},\,e_{1})&F(e_{2},\,e_{2})&\ldots &F(e_{2},\,e_{n})\\\vdots &\vdots &\ddots &\vdots \\F(e_{n},\,e_{1})&F(e_{n},\,e_{2})&\ldots &F(e_{n},\,e_{n})\end{pmatrix}},

{\ displaystyle {\ begin {pmatrix} F (e_ {1}, \, e_ {1}) & F (e_ {1}, \, e_ {2}) & \ ldots & F (e_ {1}, \, e_ {n}) \\ F (e_ {2}, \, e_ {1}) & F (e_ {2}, \, e_ {2}) & \ ldots & F (e_ {2}, \, e_ {n} ) \\\ vdots & \ vdots & \ ddots & \ vdots \\ F (e_ {n}, \, e_ {1}) & F (e_ {n}, \, e_ {2}) & \ ldots & F (e_ {n}, \, e_ {n}) \ end {pmatrix}},}

so for any vectors $x=x^{1}e_{1}+x^{2}e_{2}+\cdots +x^{n}e_{n}$ ${\ displaystyle x = x ^ {1} e_ {1} + x ^ {2} e_ {2} + \ cdots + x ^ {n} e_ {n}}$ and $y=y^{1}e_{1}+y^{2}e_{2}+\cdots +y^{n}e_{n}$ ${\ displaystyle y = y ^ {1} e_ {1} + y ^ {2} e_ {2} + \ cdots + y ^ {n} e_ {n}}$

F(x,\,y)={\begin{pmatrix}x^{1}&x^{2}&\ldots &x^{n}\end{pmatrix}}{\begin{pmatrix}F(e_{1},\,e_{1})&F(e_{1},\,e_{2})&\ldots &F(e_{1},\,e_{n})\\F(e_{2},\,e_{1})&F(e_{2},\,e_{2})&\ldots &F(e_{2},\,e_{n})\\\vdots &\vdots &\ddots &\vdots \\F(e_{n},\,e_{1})&F(e_{n},\,e_{2})&\ldots &F(e_{n},\,e_{n})\end{pmatrix}}{\begin{pmatrix}y^{1}\\y^{2}\\\vdots \\y^{n}\end{pmatrix}},

{\ displaystyle F (x, \, y) = {\ begin {pmatrix} x ^ {1} & x ^ {2} & \ ldots & x ^ {n} \ end {pmatrix}} {\ begin {pmatrix} F ( e_ {1}, \, e_ {1}) & F (e_ {1}, \, e_ {2}) & \ ldots & F (e_ {1}, \, e_ {n}) \\ F (e_ {2 }, \, e_ {1}) & F (e_ {2}, \, e_ {2}) & \ ldots & F (e_ {2}, \, e_ {n}) \\\ vdots & \ vdots & \ ddots & \ vdots \\ F (e_ {n}, \, e_ {1}) & F (e_ {n}, \, e_ {2}) & \ ldots & F (e_ {n}, \, e_ {n}) \ end {pmatrix}} {\ begin {pmatrix} y ^ {1} \\ y ^ {2} \\\ vdots \\ y ^ {n} \ end {pmatrix}},}

i.e

F(x,\,y)=\sum _{i,j=1}^{n}f_{ij}\,x^{i}y^{j},\ \quad f_{ij}=F(e_{i},\,e_{j}).

{\ displaystyle F (x, \, y) = \ sum _ {i, j = 1} ^ {n} f_ {ij} \, x ^ {i} y ^ {j}, \ \ quad f_ {ij} = F (e_ {i}, \, e_ {j}).}

It also means that the bilinear form is completely determined by its values on the basis vectors.
Space dimension $W(L,L)$ ${\ displaystyle W (L, L)}$ there is $\dim W(L,L)=(\dim L)^{2}$ ${\ displaystyle \ dim W (L, L) = (\ dim L) ^ {2}}$ .
Although the bilinear matrix $F$ ${\ displaystyle F}$ depends on the choice of basis, the rank of the matrix of the bilinear form in any basis is the same, it is called the rank of the bilinear form $F$ ${\ displaystyle F}$ . A bilinear form is called nondegenerate if its rank is equal to $\dim L$ ${\ displaystyle \ dim L}$ .
For any subspace $M\subset L$ ${\ displaystyle M \ subset L}$ orthogonal complement $M^{\perp }$ ${\ displaystyle M ^ {\ perp}}$ is a subspace $M^{\perp }\subset L$ ${\ displaystyle M ^ {\ perp} \ subset L}$ .
$\dim L^{\perp }=\dim L-r$ ${\ displaystyle \ dim L ^ {\ perp} = \ dim Lr}$ where $r$ ${\ displaystyle r}$ - rank of bilinear form $F$ ${\ displaystyle F}$ .

Convert the bilinear matrix by changing the base

The matrix representing the bilinear form in the new basis is connected with the matrix representing it in the old basis through the matrix, the inverse of the transition matrix to the new basis (Jacobi matrix), through which the coordinates of the vectors are transformed.

In other words, if the coordinates of the vector in the old basis $X^{i}$ ${\ displaystyle X ^ {i}}$ expressed through coordinates in new $x^{i}$ ${\ displaystyle x ^ {i}}$ through the matrix $\beta$ ${\ displaystyle \ beta}$ $X^{i}=\sum \beta _{j}^{i}x^{j}$ ${\ displaystyle X ^ {i} = \ sum \ beta _ {j} ^ {i} x ^ {j}}$ or matrix recording $X=\beta x$ ${\ displaystyle X = \ beta x}$ then bilinear form $F$ ${\ displaystyle F}$ on any vectors $x$ ${\ displaystyle x}$ and $y$ ${\ displaystyle y}$ will be written as

F(x,\,y)=\sum _{i,j}F_{ij}X^{i}Y^{j}=\sum _{i,j,k,m}F_{ij}\beta _{k}^{i}\beta _{m}^{j}x^{k}y^{m}

{\ displaystyle F (x, \, y) = \ sum _ {i, j} F_ {ij} X ^ {i} Y ^ {j} = \ sum _ {i, j, k, m} F_ {ij } \ beta _ {k} ^ {i} \ beta _ {m} ^ {j} x ^ {k} y ^ {m}}

,

that is, the components of the matrix representing the bilinear form in the new basis will be:

f_{km}=\sum _{i,j}F_{ij}\beta _{k}^{i}\beta _{m}^{j}

{\ displaystyle f_ {km} = \ sum _ {i, j} F_ {ij} \ beta _ {k} ^ {i} \ beta _ {m} ^ {j}}

,

or, in matrix notation:

f=\beta ^{T}F\beta

{\ displaystyle f = \ beta ^ {T} F \ beta}

,

\beta =\alpha ^{-1}

{\ displaystyle \ beta = \ alpha ^ {- 1}}

where

\alpha

{\ displaystyle \ alpha}

- direct coordinate transformation matrix

x=\alpha X

{\ displaystyle x = \ alpha X}

.

Relationship with tensor products and the Hom functor

From the universal property of the tensor product, it follows that the bilinear forms on V are in one-to-one correspondence with the set ${\text{Hom}}(V\otimes V,k)$ ${\ displaystyle {\ text {Hom}} (V \ otimes V, k)}$ where k is the main field.

Since the tensor product functor and the Hom functor are conjugate , ${\text{Hom}}(V\otimes V,k)\cong {\text{Hom}}(V,{\text{Hom}}(V,k))$ ${\ displaystyle {\ text {Hom}} (V \ otimes V, k) \ cong {\ text {Hom}} (V, {\ text {Hom}} (V, k))}$ , that is, the bilinear form corresponds to the linear mapping of $V$ ${\ displaystyle V}$ in dual space $V^{*}$ ${\ displaystyle V ^ {*}}$ . This correspondence can be carried out in two ways (since there are two functors of the tensor product — with a fixed left argument and with a fixed right); they are often designated as

$B_{1}({\mathsf {v}})=B({\mathsf {v}},\cdot )$ ${\ displaystyle B_ {1} ({\ mathsf {v}}) = B ({\ mathsf {v}}, \ cdot)}$

$B_{2}({\mathsf {v}})=B(\cdot ,{\mathsf {v}})$ ${\ displaystyle B_ {2} ({\ mathsf {v}}) = B (\ cdot, {\ mathsf {v}})}$ .

Literature

Maltsev A. I. Basics of linear algebra. - M .: Science, 1975.
Gelfand I.M. Lectures on linear algebra. - M .: Science, 1971.
Faddeev DK Lectures on algebra. M .: Science, 1984.
Kostrikin A.I. Introduction to Algebra, Moscow: Nauka, 1977.
Beklemishev D.V. Analytic geometry and linear algebra. - M .: Higher. Shk., 1998. - 320 p.
Gelfand IM , Linear Algebra . Lecture course.
Shafarevich I.R. , Remizov A.O. Linear Algebra and Geometry, - Fizmatlit, Moscow, 2009.