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Bilinear Form

Sep 25, 2024, 6 min read

  • Let be a vector space over . A function is called a bilinear form on if is linear in each variable when the other variable is held fixed. That is and

    The set of all bilinear forms on is denoted

  • (Friedberg e6.7.2) For any bilinear form

    • If for any the functions are defined by
      Then and are linear
    • If then
    • If is defined by , then is a bilinear form.

  • The sum and scalar product of bilinear forms are defined as follows

  • (Friedberg 6.25) For any vector space , is a vector space with the sum and scalar product of bilinear forms as addition and scalar multiplication

  • The Matrix Representation of a bilinear form with respect to basis can be defined as follows. Let be a matrix and be an ordered basis. We associate with to by defining

    We denote this as

  • (Friedberg 6.26) For any -dimensional vector space over field and any basis for . is a vector space isomorphism.

  • (Friedberg 6.26.1)

  • (Friedberg 6.26.2) Let be an -dimensional vector space over the field with basis . If and , then

  • (Friedberg 6.26.3) For any field , positive integer and , there exists a unique matrix where is the standard ordered basis for such that

    In fact

  • Two matrices, are said to be congruent if there exists an invertible matrix such that

  • (Friedberg 6.27) Let be a finite dimensional vector space with bases and and let be the change of coordinate matrix changing -coordinates to -coordinates. Then, for any ,

    In particular, and are congruent.

  • (Friedberg 6.27.1) Let be an -dimensional vector space with basis . Suppose that and .

    If is congruent to , then there exists a basis for such that .

    In fact if for an invertible matrix , then changes -coordinates into -coordinates.

    • Intuition: Apply a change of coordinate matrix to (Friedberg 6.26.2)

  • (Friedberg Lem.6.29) Let be a nontrivial symmetric bilinear form on a vector space over a field not of characteristic then there exists an element such that .

Symmetric Bilinear Forms

  • A bilinear form over a vector space is symmetric if

  • (Friedberg 6.28) Let be a finite-dimensional vector space. For the following are equivalent

    • is symmetric.
    • For any basis for , is a symmetric matrix.
    • There exists a basis for such that is a symmetric matrix.

  • (Friedberg 6.28.1) Let be a finite dimensional vector space. For any , if is diagonalizable, then is symmetric.

  • A bilinear form on a finite dimensional vector space is called diagonalizable if there exists a basis for such that is a diagonal matrix

  • (Friedberg 6.29) Let be a finite dimensional vector space over a field of characteristic two. Then every symmetric bilinear form on is diagonalizable.

  • (Friedberg 6.29.1) Let be a field that is not of characteristic two. If is a symmetric matrix, then is congruent to a diagonal matrix.

  • (Friedberg 6.30) Let be a finite dimensional real inner product space and a symmetric bilinear form on . Then there exists an orthonormal basis for such that is a diagonal matrix

Sylvester’s Laws of Innertia

  • Any two matrix representations of a bilinear form have the same rank. Thus, the rank of a bilinear form is the rank of any of its representation.

  • Sylvester’s Law of Inertia states the following: Let be a symmetric bilinear form on a finite dimensional real vector space . Then the number of positive diagonal entries and the number of negative diagonal entries of any diagonal representation of are both independent of the diagonal representation.

    • Proof: Suppose we have diagonal representations with respect to and with and positive entries respectively. Without loss of generality, assume . Let and . We have

      Define be defined by

      Clearly is linear and and therefore .

      Choose such that . Thus for and for . Represent in terms of the bases and . That is

      Note that for , we have by our stipulation. Hence for all such . Similarly whenever Thus, for some since .

      Now consider and compute it using both representations of . We end up with the claim that both and , a contradiction. Therefore .

  • The index is the number of positive diagonal entries of a diagonal representation of a symmetric bilinear form on a real vector space. The same term is used for a diagonal matrices congruent to a real symmetric matrix

  • The signature of the form is the difference between positive and negative diagonal entries in a diagonal representation of a symmetric bilinear form. The same term is used for a diagonal matrices congruent to a real symmetric matrix

  • Sylvester’s Law of Inertia states that rank, index, and signature are invariants of a bilinear form (and diagonal matrices congruent to a real symmetric matrix).

  • Sylvester’s Law of Inertia for Matrices. Let be a symmetric matrix. Then, the number of positive diagonal entries and the number of negative diagonal entries of any diagonal matrix congruent to is independent of the choice of the diagonal matrix.

  • Sylvester’s Law of Inertia (Cor.2) Two real symmetric matrices are congruent if and only if they have the same index, signature, and rank.

  • In the theory of real symmetric matrices, we define a canonical form matrix

    Missing \end{bmatrix}J_{pr} = \begin{bmatrix} I_p & O & O \\ O & -I_{r-p} & O \\

O &O &O \end{bmatrix}

You can't use 'macro parameter character #' in math mode * **Sylvester's Law of Inertia** (*Cor.2*) A real symmetric matrix $A$ has index $p$ and rank $r$ if and only if $A$ is congruent to $J_{pr}$. # Links * [[Linear Algebra by Friedberg Insel and Spence]]