Solved examples: Finding extreme values of quadratic forms on the unit sphere

📚 Quick Study Guide

• 📐 A quadratic form is a homogeneous polynomial of degree two in $n$ variables. It can be expressed as $Q(x) = x^T Ax$, where $A$ is a symmetric matrix.
• 🔎 To find the extreme values of $Q(x)$ subject to the constraint $||x|| = 1$ (i.e., $x$ lies on the unit sphere), we need to find the eigenvalues of the matrix $A$.
• 💡 The extreme values of $Q(x)$ are the largest and smallest eigenvalues of $A$.
• 📝 The eigenvectors corresponding to the largest and smallest eigenvalues give the points on the unit sphere where $Q(x)$ attains its maximum and minimum values, respectively.
• 🧪 If $A$ is a symmetric matrix, its eigenvalues are real and its eigenvectors corresponding to distinct eigenvalues are orthogonal.
• 🧭 The method of Lagrange multipliers can also be used to solve this problem. We set up the Lagrangian function $L(x, \lambda) = x^T A x - \lambda(x^T x - 1)$ and solve for the critical points.

Practice Quiz

1. Which of the following statements is true about the extreme values of a quadratic form $Q(x) = x^T A x$ on the unit sphere?

   1. The extreme values are the eigenvalues of $A$.
   2. The extreme values are the singular values of $A$.
   3. The extreme values are always 0 and 1.
   4. The extreme values are the diagonal entries of $A$.

2. Let $A = \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}$. What are the extreme values of $Q(x) = x^T A x$ on the unit circle?

   1. 1 and 3
   2. 2 and 2
   3. 0 and 4
   4. -1 and 3

3. If the eigenvalues of a symmetric matrix $A$ are 5, 2, and -1, what is the minimum value of the quadratic form $Q(x) = x^T A x$ on the unit sphere?

   1. 5
   2. 2
   3. -1
   4. 0

4. Which method can be used to find the extreme values of a quadratic form subject to the unit sphere constraint?

   1. Gaussian elimination
   2. Gram-Schmidt process
   3. Lagrange multipliers
   4. Newton's method

5. What is the relationship between the eigenvectors corresponding to distinct eigenvalues of a symmetric matrix?

   1. Parallel
   2. Orthogonal
   3. Linearly dependent
   4. Equal

6. Consider the quadratic form $Q(x, y) = 3x^2 + 2xy + 3y^2$. What matrix $A$ represents this quadratic form?

   1. $\begin{bmatrix} 3 & 1 \\ 1 & 3 \end{bmatrix}$
   2. $\begin{bmatrix} 3 & 2 \\ 2 & 3 \end{bmatrix}$
   3. $\begin{bmatrix} 3 & 0 \\ 0 & 3 \end{bmatrix}$
   4. $\begin{bmatrix} 1 & 3 \\ 3 & 1 \end{bmatrix}$

7. If the largest eigenvalue of $A$ is 7, what is the maximum value of $Q(x) = x^T A x$ on the unit sphere?

   1. 0
   2. 1
   3. 7
   4. -7