Question

A bilinear form \(\beta\) on \(\mathbb{R}^{n}\) is a function that assigns to every pair \(\mathbf{x}, \mathbf{y}\) of columns in \(\mathbb{R}^{n}\) a number \(\beta(\mathbf{x}, \mathbf{y})\) in such a way that $$ \begin{array}{l} \beta(r \mathbf{x}+s \mathbf{y}, \mathbf{z})=r \beta(\mathbf{x}, \mathbf{z})+s \beta(\mathbf{y}, \mathbf{z}) \\ \beta(\mathbf{x}, r \mathbf{y}+s \mathbf{z})=r \beta(\mathbf{x}, \mathbf{z})+s \beta(\mathbf{x}, \mathbf{z}) \end{array} $$ for all \(\mathbf{x}, \mathbf{y}, \mathbf{z}\) in \(\mathbb{R}^{n}\) and \(r, s\) in \(\mathbb{R} .\) If \(\beta(\mathbf{x}, \mathbf{y})=\beta(\mathbf{y}, \mathbf{x})\) for all \(\mathbf{x}, \mathbf{y}, \beta\) is called symmetric. a. If \(\beta\) is a bilinear form, show that an \(n \times n\) matrix \(A\) exists such that \(\beta(\mathbf{x}, \mathbf{y})=\mathbf{x}^{T} A \mathbf{y}\) for all \(\mathbf{x}, \mathbf{y}\). b. Show that \(A\) is uniquely determined by \(\beta\). c. Show that \(\beta\) is symmetric if and only if \(A=A^{T}\).

Short answer

The matrix \( A \) exists and is unique, defined by \( \beta(\mathbf{x}, \mathbf{y}) = \mathbf{x}^T A \mathbf{y} \), and \( A \) is symmetric if and only if \( \beta \) is symmetric.

Step-by-step solution

Understanding Bilinear Form

A bilinear form \( \beta \) on \( \mathbb{R}^{n} \) takes two vectors \( \mathbf{x}, \mathbf{y} \) and returns a scalar, maintaining linearity in both arguments as shown by equations: \( \beta(r \mathbf{x}+s \mathbf{y}, \mathbf{z})=r \beta(\mathbf{x}, \mathbf{z})+s \beta(\mathbf{y}, \mathbf{z}) \) and \( \beta(\mathbf{x}, r \mathbf{y}+s \mathbf{z})=r \beta(\mathbf{x}, \mathbf{z})+s \beta(\mathbf{x}, \mathbf{z}) \).

Establish Existence of Matrix A

To show \( \beta(\mathbf{x}, \mathbf{y})=\mathbf{x}^{T} A \mathbf{y} \), define \( A \) by considering \( n \) basis vectors \( \mathbf{e}_i \) in \( \mathbb{R}^n \). Express \( \beta(\mathbf{e}_i, \mathbf{e}_j) = a_{ij} \). Then, for any vectors \( \mathbf{x}, \mathbf{y} \), represented as \( \mathbf{x}=\sum x_i \mathbf{e}_i \) and \( \mathbf{y}=\sum y_j \mathbf{e}_j \), \( \beta(\mathbf{x}, \mathbf{y}) = \sum x_i y_j a_{ij} = \mathbf{x}^T A \mathbf{y} \).

Showing Uniqueness of Matrix A

Consider the requirement that \( \beta(\mathbf{e}_i, \mathbf{e}_j) = a_{ij} \). Each \( a_{ij} \) is defined explicitly through the bilinear form, meaning that any deviation will alter \( \beta \). Thus, the matrix \( A \) determined by these elements uniquely represents the bilinear form.

Symmetry and Transpose Equivalence

For symmetry, \( \beta(\mathbf{x}, \mathbf{y})=\beta(\mathbf{y}, \mathbf{x}) \) implies \( \mathbf{x}^{T} A \mathbf{y} = \mathbf{y}^{T} A \mathbf{x} \). The equivalence of this requires \( \mathbf{x}^{T} A \mathbf{y} = \mathbf{x}^{T} A^{T} \mathbf{y} \) for all \( \mathbf{x}, \mathbf{y} \), which further implies that \( A = A^{T} \).

Key concepts

Matrix Representation

In the fascinating world of linear algebra, bilinear forms have an elegant connection to matrices. When we talk about the matrix representation of a bilinear form \( \beta \), we are basically capturing every interaction between vector pairs in terms of matrix multiplication. This involves finding a special matrix \( A \) such that \( \beta(\mathbf{x}, \mathbf{y}) = \mathbf{x}^T A \mathbf{y} \). How do we find this matrix?

It's done by examining how \( \beta \) acts on a set of basis vectors. For instance, in \( \mathbb{R}^n \), take the standard basis vectors \( \mathbf{e}_i \). The action \( \beta(\mathbf{e}_i, \mathbf{e}_j) \) provides the entries \( a_{ij} \) of the matrix \( A \). With these entries, we can construct \( A \) such that the bilinear form can be expressed as a simple matrix product.

Think of this as translating the abstract concept of \( \beta \) into a concrete form that uses the familiar language of matrices and linear algebra.

Symmetry in Linear Algebra

Symmetry adds a charming layer of elegance to many mathematical structures, and bilinear forms are no exception. When a bilinear form \( \beta \) satisfies \( \beta(\mathbf{x}, \mathbf{y}) = \beta(\mathbf{y}, \mathbf{x}) \) for all vector pairs \( \mathbf{x} \) and \( \mathbf{y} \), it is termed symmetric. This property has intriguing implications in its matrix representation.

The matrix \( A \) corresponding to a symmetric bilinear form also reflects symmetry. In mathematical terms, this means \( A \) is equal to its transpose, i.e., \( A = A^T \). Imagine \( A \) as a mirror image of itself across the diagonal. Each element \( a_{ij} \) will be equal to \( a_{ji} \), ensuring this mirror-like property.

This relationship reveals that symmetry in bilinear forms can be directly observed through the structure of its corresponding matrix. This insight is particularly helpful in fields like physics and computer graphics, where symmetrical properties often play vital roles.

Uniqueness of Bilinear Forms

Uniqueness in mathematics often provides a solid anchor for theoretical concepts, ensuring clarity and consistency. When it comes to bilinear forms, the uniqueness of the matrix \( A \) representing a bilinear form \( \beta \) is like a fingerprint. It's a unique identifier that accurately describes the bilinear interaction between any two vectors.

Why is this matrix unique? It all hinges on how \( \beta \) is defined over a basis. For an \( n \times n \) matrix \( A \), each entry \( a_{ij} \) is explicitly determined by the form \( \beta(\mathbf{e}_i, \mathbf{e}_j) \). If any part of \( A \) differs, \( \beta \) itself would change, violating the original definition. Thus \( A \) must be precisely aligned with \( \beta \), giving us our unique matrix representation.

This uniqueness is essential not just in abstract mathematics but also in applications where precise numerical simulations depend on this property to ensure accurate results.