Solved Examples: Change of Variables to Simplify Quadratic Forms

📚 Quick Study Guide

• 🔍 A quadratic form is a homogeneous polynomial of degree two in $n$ variables. It can be written as $Q(x) = x^T A x$, where $A$ is a symmetric matrix.
• 💡 The goal is to find a change of variables $x = Py$ such that the quadratic form becomes simpler, ideally diagonal.
• 📝 To do this, find an orthogonal matrix $P$ that diagonalizes $A$, i.e., $P^T A P = D$, where $D$ is a diagonal matrix. The columns of $P$ are the orthonormal eigenvectors of $A$.
• 🔢 The new quadratic form is then $Q(y) = y^T D y = \lambda_1 y_1^2 + \lambda_2 y_2^2 + ... + \lambda_n y_n^2$, where $\lambda_i$ are the eigenvalues of $A$.
• ➕ Completing the square can also be used in simpler cases to eliminate cross-product terms.

🧪 Practice Quiz

1. Question 1: Which of the following transformations would simplify the quadratic form $Q(x, y) = 2x^2 + 4xy + 5y^2$?
   1. $x = u + v, y = u - v$
   2. $x = u - v, y = u + v$
   3. $x = 2u + v, y = u - 2v$
   4. $x = u, y = v$
2. Question 2: If $A = \begin{bmatrix} 5 & 3 \\ 3 & 5 \end{bmatrix}$, what are the eigenvalues used for diagonalizing the corresponding quadratic form?
   1. $\lambda_1 = 2, \lambda_2 = 8$
   2. $\lambda_1 = 3, \lambda_2 = 5$
   3. $\lambda_1 = 1, \lambda_2 = 9$
   4. $\lambda_1 = 0, \lambda_2 = 10$
3. Question 3: What is the general form of a quadratic form in two variables $x$ and $y$?
   1. $ax^2 + bxy + cy^2$
   2. $ax + by + c$
   3. $ax^3 + bxy + cy^3$
   4. $ax^2 + by^2 + c$
4. Question 4: Which matrix represents the quadratic form $Q(x, y) = 3x^2 - 8xy + y^2$?
   1. $\begin{bmatrix} 3 & -4 \\ -4 & 1 \end{bmatrix}$
   2. $\begin{bmatrix} 3 & -8 \\ -8 & 1 \end{bmatrix}$
   3. $\begin{bmatrix} 3 & 0 \\ 0 & 1 \end{bmatrix}$
   4. $\begin{bmatrix} 1 & -4 \\ -4 & 3 \end{bmatrix}$
5. Question 5: If $x = Py$ is a change of variable used to simplify a quadratic form $Q(x)$, what is the new quadratic form $Q(y)$ in terms of $P$ and the original matrix $A$?
   1. $y^T P^T A P y$
   2. $y^T A P y$
   3. $y^T P A P^T y$
   4. $y^T A y$
6. Question 6: Which of the following is NOT a step in simplifying quadratic forms using change of variables?
   1. Finding eigenvalues of the matrix
   2. Finding eigenvectors of the matrix
   3. Calculating the determinant of the matrix
   4. Forming an orthogonal matrix from the eigenvectors
7. Question 7: What type of matrix must $A$ be for the quadratic form $Q(x) = x^T A x$?
   1. Symmetric
   2. Orthogonal
   3. Invertible
   4. Diagonal