Symmetric bilinear form
Encyclopedia
A symmetric bilinear form is a bilinear form on a vector space
that is symmetric. Symmetric bilinear forms are of great importance in the study of orthogonal polarity and quadrics
.
They are also more briefly referred to as just symmetric forms when "bilinear" is understood. They are closely related to quadratic form
s; for the details of the distinction between the two, see ε-quadratic forms.
is a symmetric bilinear form on the space if :
The last two axioms only imply linearity in the first argument, but the first immediately implies linearity in the second argument then too.
be a basis for V. Define the n×n matrix A by 
. The matrix A is a symmetric matrix exactly due to symmetry of the bilinear form. If the n×1 matrix x represents a vector v with respect to this basis, and analogously, y represents w, then 
is given by :
Suppose C' is another basis for V, with :
with S an invertible n×n matrix.
Now the new matrix representation for the symmetric bilinear form is given by
, which is, due to reflexivity, equivalent with 
The radical of a bilinear form B is the set of vectors orthogonal with every other vector in V. One easily checks that this is a subspace of V. 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
The matrix A is singular if and only if the radical is nontrivial.
If W is a subspace of V, then
, the set of all vectors orthogonal with every vector in W, is also subspace. When the radical of B is trivial, the dimension of 
= dim(V) − dim(W).
is orthogonal with respect to B if and only if :
When the characteristic
of the field is not two, there is always an orthogonal basis. This can be proven by induction
.
A basis C is orthogonal if and only if the matrix representation A is a diagonal matrix
.
says that, when working over an ordered field
K, the number of diagonal elements equal to 0, or that are positive or negative, is independent of the chosen orthogonal basis. These three numbers form the signature of the bilinear form.
be an orthogonal basis.
We define a new basis
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.
Let
be an orthogonal basis.
We define a new basis
:
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.
different from 2. One can now define a map from D(V), the set of all subspaces of V, to itself :
This map is an orthogonal polarity on the projective space
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.
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
that is symmetric. Symmetric bilinear forms are of great importance in the study of orthogonal polarity and quadrics
Quadric (projective geometry)
In projective geometry a quadric is the set of points of a projective space where a certain quadratic form on the homogeneous coordinates becomes zero...
.
They are also more briefly referred to as just symmetric forms when "bilinear" is understood. They are closely related to quadratic form
Quadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....
s; for the details of the distinction between the two, see ε-quadratic forms.
Definition
Let V be a vector space of dimension n over a field K. A mapFunction (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
is a symmetric bilinear form on the space if :
The last two axioms only imply linearity in the first argument, but the first immediately implies linearity in the second argument then too.
Matrix representation
Let be a basis for V. Define the n×n matrix A by . The matrix A is a symmetric matrix exactly due to symmetry of the bilinear form. If the n×1 matrix x represents a vector v with respect to this basis, and analogously, y represents w, then is given by :Suppose C' is another basis for V, with :
with S an invertible n×n matrix.
Now the new matrix representation for the symmetric bilinear form is given by
Orthogonality and singularity
A symmetric bilinear form is always reflexive. Two vectors v and w are defined to be orthogonal with respect to the bilinear form B if, which is, due to reflexivity, equivalent with The radical of a bilinear form B is the set of vectors orthogonal with every other vector in V. One easily checks that this is a subspace of V. 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
The matrix A is singular if and only if the radical is nontrivial.
If W is a subspace of V, then
, the set of all vectors orthogonal with every vector in W, is also subspace. When the radical of B is trivial, the dimension of = dim(V) − dim(W).Orthogonal basis
A basis is orthogonal with respect to B if and only if :When the characteristic
Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...
of the field is not two, there is always an orthogonal basis. This can be proven by induction
Mathematical induction
Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers...
.
A basis C is orthogonal if and only if the matrix representation A is a diagonal matrix
Diagonal matrix
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero. The diagonal entries themselves may or may not be zero...
.
Signature and Sylvester's law of inertia
In its most general form, Sylvester's law of inertiaSylvester's law of inertia
Sylvester's law of inertia is a theorem in matrix algebra about certain properties of the coefficient matrix of a real quadratic form that remain invariant under a change of coordinates...
says that, when working over an ordered field
Ordered field
In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Historically, the axiomatization of an ordered field was abstracted gradually from the real numbers, by mathematicians including David Hilbert, Otto Hölder and...
K, the number of diagonal elements equal to 0, or that are positive or negative, is independent of the chosen orthogonal basis. These three numbers form the signature of the bilinear form.
Real case
When working in a space over the reals, one can go a bit a further. Let be an orthogonal basis.We define a new basis
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.
Complex case
When working in a space over the complex numbers, one can go further as well and it is even easier.Let
be an orthogonal basis.We define a new basis
: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.
Orthogonal polarities
Let B be a symmetric bilinear form with a trivial radical on the space V over the field K with characteristicCharacteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...
different from 2. One can now define a map from D(V), the set of all subspaces of V, to itself :
This map is an orthogonal polarity on the projective space
Projective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
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.