(a) Let K and L be symmetric n ×n matrices. Prove that 𝐱^T K 𝐱=𝐱^T L 𝐱 for all 𝐱 ∈ℝ^n if and onl

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(a) Let K and L be symmetric n ×n matrices. Prove that 𝐱^T K...

(a) Let $K$ and $L$ be symmetric $n \times n$ matrices. Prove that $\mathbf{x}^T K \mathbf{x}=\mathbf{x}^T L \mathbf{x}$ for all $\mathbf{x} \in \mathbb{R}^n$ if and only if $K=L$. (b) Find an example of two non-symmetric matrices $K \neq L$ such that $\mathbf{x}^T K \mathbf{x}=\mathbf{x}^T L \mathbf{x}$ for all $\mathbf{x} \in \mathbb{R}^n$.

Key Concepts

Non-Symmetric Matrices and Their Quadratic Forms

In contrast to symmetric matrices, non-symmetric matrices can have distinct entries while yielding the same quadratic form because quadratic forms depend only on the symmetric part of a matrix. The symmetric part of any matrix A is given by (A + A^T)/2, and thus two non-symmetric matrices can have identical quadratic forms if their symmetric parts are equal even though the matrices themselves are different.

Polarization Identity

The polarization identity links quadratic forms to their corresponding bilinear forms. It shows that if one knows the quadratic form, one can recover the bilinear form and subsequently the matrix entries in the symmetric case. This identity is a crucial tool in establishing the one-to-one correspondence between symmetric matrices and quadratic forms.

Quadratic Forms

A quadratic form is a homogeneous polynomial of degree two, often expressed as x^T A x for some matrix A. When A is symmetric, the quadratic form uniquely represents the underlying bilinear form. This relationship is fundamental in many areas including optimization and differential equations.

Symmetric Matrices

A symmetric matrix is one that is equal to its own transpose. This property implies that the entries mirror along the main diagonal. When used in quadratic forms, symmetric matrices ensure that the form is well-defined and uniquely associated with a bilinear form, which is critical in proving uniqueness results.

Uniqueness of Matrix Representation via Quadratic Forms

For symmetric matrices, the equality of quadratic forms x^T K x = x^T L x for all vectors x implies that the matrices must be equal. This is because the quadratic form, via the polarization identity, uniquely determines the entries of the symmetric matrix. Thus, if two symmetric matrices yield the same quadratic form over all x, they are necessarily identical.

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