Test Questions on Orthogonal Transformations of Quadratic Forms

📚 Quick Study Guide

• 📐 Quadratic Form: A homogeneous polynomial of degree two in $n$ variables, expressed as $Q(x) = x^T A x$, where $A$ is a symmetric matrix.
• 🔄 Orthogonal Transformation: A linear transformation $x = Py$, where $P$ is an orthogonal matrix ($P^T P = I$). This transformation preserves lengths and angles.
• 🔑 Diagonalization: The goal is to find an orthogonal matrix $P$ such that $P^T A P = D$, where $D$ is a diagonal matrix. The diagonal entries of $D$ are the eigenvalues of $A$.
• ✨ Eigenvalues and Eigenvectors: The eigenvalues $\lambda_i$ are the roots of the characteristic equation $\det(A - \lambda I) = 0$. The eigenvectors $v_i$ satisfy $A v_i = \lambda_i v_i$.
• ➕ Procedure:
  1. Find the eigenvalues of $A$.
  2. Find the corresponding eigenvectors.
  3. Orthogonalize the eigenvectors (if necessary) using the Gram-Schmidt process.
  4. Normalize the orthogonal eigenvectors to obtain an orthonormal basis.
  5. Form the matrix $P$ with the orthonormal eigenvectors as columns.
• 📝 Transformed Quadratic Form: The quadratic form in the new variables $y$ is $Q(y) = y^T D y = \lambda_1 y_1^2 + \lambda_2 y_2^2 + \dots + \lambda_n y_n^2$.

Practice Quiz

1. What is the primary goal of applying an orthogonal transformation to a quadratic form?
   1. To increase the number of variables.
   2. To eliminate the cross-product terms and diagonalize the quadratic form.
   3. To make the quadratic form non-homogeneous.
   4. To change the sign of the eigenvalues.
2. Which of the following properties must a matrix $P$ satisfy to be considered an orthogonal matrix?
   1. $P^2 = I$
   2. $P^T = P$
   3. $P^T P = I$
   4. $P^{-1} = -P$
3. If $A$ is the matrix associated with a quadratic form, what do the diagonal entries of the diagonal matrix $D = P^T A P$ represent after an orthogonal transformation?
   1. The eigenvectors of $A$.
   2. The eigenvalues of $A$.
   3. The trace of $A$.
   4. The determinant of $A$.
4. What is the characteristic equation used to find the eigenvalues of a matrix $A$?
   1. $\det(A + \lambda I) = 0$
   2. $\det(A - \lambda I) = 0$
   3. $\text{trace}(A - \lambda I) = 0$
   4. $\text{trace}(A + \lambda I) = 0$
5. What is the result of applying the transformation $x = Py$ to the quadratic form $Q(x) = x^T A x$?
   1. $Q(y) = y^T A y$
   2. $Q(y) = y^T P A P^T y$
   3. $Q(y) = y^T P^T A P y$
   4. $Q(y) = y^T P A P y$
6. Why is it sometimes necessary to apply the Gram-Schmidt process during the orthogonal transformation of a quadratic form?
   1. To ensure the eigenvectors are linearly dependent.
   2. To ensure the eigenvectors are orthogonal to each other.
   3. To change the eigenvalues of the matrix.
   4. The Gram-Schmidt process is never necessary.
7. Suppose a quadratic form $Q(x, y) = 2x^2 + 2y^2 + 2xy$ is transformed into $Q(u, v) = 3u^2 + v^2$ using an orthogonal transformation. What are the eigenvalues of the original matrix associated with $Q(x, y)$?
   1. 1 and 2
   2. 2 and 2
   3. 3 and 1
   4. 2 and 3