Understanding Eigenvalues, Singular Values, and Extremal Values in Quadratic Forms

Quadratic forms and their associated matrices play a fundamental role in many areas of mathematics and engineering. This article delves into the concept of the maximum, minimum, and minimum absolute values of a quadratic form, using the specific matrix and SVD (Singular Value Decomposition) as illustrative examples. We will explore how these values are derived and how they reflect the properties of the underlying matrix.

Introduction to Quadratic Forms

A quadratic form is a second-degree polynomial in multiple variables, typically represented as ( x^T A x ), where ( x ) is an n-dimensional vector and ( A ) is an n×n symmetric matrix.

Eigenvalues and Quadratic Forms

Consider the matrix ( A ) with the property ( x^T A x x^T A^T x ). Changing the matrix ( A ) to ( S frac{AA^T}{2} ) preserves this property. The matrix ( S ) can be expressed in terms of its eigenvalues. For the given matrix ( S ), the eigenvalues are -2, 0, and 4. This means that the maximum value of the quadratic form ( x^T S x ) is 4, the minimum value is -2, and the minimum absolute value is 0.

Singular Value Decomposition (SVD) and Quadratic Forms

The Singular Value Decomposition (SVD) is a factorization of a real or complex matrix. For the given matrix ( M ), the SVD can be written as ( M U Sigma V^T ), where ( U ) and ( V ) are orthogonal matrices, and ( Sigma ) is a diagonal matrix containing the singular values of ( M ).

SVD Example

The given matrix ( M ) and its SVD components are:

M (begin{pmatrix}1 -3 -5 -1 1 1 1 1 1 -1 1 -3 1 -5 1 1 end{pmatrix}) U (begin{pmatrix}-0.812164 -0.536287 0.0376478 0.226647 0.191726 0.1266 0.159478 0.960094 -0.1266 0.191726 -0.960094 0.159478 -0.536287 0.812164 0.226647 -0.0376478 end{pmatrix}) (Sigma) (begin{pmatrix}6.69354 0 0 0 0 4.69354 0 0 0 0 3.36296 0 0 0 0 1.36296 end{pmatrix}) V (begin{pmatrix}-0.191726 0.1266 -0.159478 0.960094 0.812164 -0.536287 -0.0376478 0.226647 0.536287 0.812164 -0.226647 -0.0376478 0.1266 0.191726 0.960094 0.159478 end{pmatrix})

The matrices ( U ) and ( V ) are orthogonal, and the matrix ( Sigma ) contains the singular values of ( M ).

Maximizing and Minimizing the Quadratic Form

One approach to maximizing the quadratic form is to consider the problem ( w^T Sigma w ) under the constraint that ( w ) is a unit vector. The maximum value of ( w^T Sigma w ) is achieved when ( w ) is aligned with the largest singular value, which is 6.69354 in this case.

However, if ( U ) and ( V ) were the same, the problem would be easier. In this case, we could set ( x U^T w ) and directly find the solution. Unfortunately, the SVD of the matrix ( M ) involves complex eigenvectors, making this approach infeasible.

Implications and Conclusions

The SVD of the matrix ( M ) reveals important information about the matrix’s structure and its quadratic form. Despite the complexity of the eigenvectors, the singular values provide insights into the maximum and minimum values of the quadratic form.

Conclusion: Understanding eigenvalues and singular values is crucial for analyzing quadratic forms and their extremal values. The given example demonstrates how to derive these values and highlights the importance of orthogonal matrices in simplifying such problems. Changing the matrix ( M ) slightly can significantly alter the results, making it essential to verify the correctness of the problem formulation.