Advanced Linear Algebra Quadratic Forms Exam Questions PDF

📚 Quick Study Guide

• 🔢 A quadratic form is a homogeneous polynomial of degree two in $n$ variables. It can be written as $Q(x) = x^T Ax$, where $A$ is a symmetric matrix.
• ➕ The matrix representation of a quadratic form is crucial. Given a quadratic form, you should be able to find the corresponding symmetric matrix and vice versa.
• 🔄 Diagonalization: A quadratic form can be simplified by diagonalizing the matrix $A$. This involves finding an orthogonal matrix $P$ such that $P^T A P = D$, where $D$ is a diagonal matrix.
• 🌟 Principal axes theorem: This theorem states that for any symmetric matrix $A$, there exists an orthogonal matrix $P$ such that $P^T A P$ is a diagonal matrix. The columns of $P$ are the eigenvectors of $A$, and the diagonal entries of $D$ are the eigenvalues of $A$.
• 📊 Congruence: Two matrices $A$ and $B$ are congruent if there exists a non-singular matrix $P$ such that $B = P^T A P$. Congruent matrices represent the same quadratic form under a change of variables.
• 📈 Sylvester's Law of Inertia: This law states that the number of positive, negative, and zero eigenvalues of a symmetric matrix are invariant under congruence transformations. These numbers are called the positive index, negative index, and nullity of the quadratic form, respectively.
• 🔍 Definiteness: A quadratic form is positive definite if $Q(x) > 0$ for all $x \neq 0$. It is positive semi-definite if $Q(x) \geq 0$ for all $x$. Similar definitions exist for negative definite and negative semi-definite forms. If $Q(x)$ takes both positive and negative values, it is indefinite.

Practice Quiz

1. What is the matrix representation of the quadratic form $Q(x, y) = 2x^2 + 4xy + 3y^2$?
   1. $A = \begin{bmatrix} 2 & 4 \\ 4 & 3 \end{bmatrix}$
   2. $A = \begin{bmatrix} 2 & 2 \\ 2 & 3 \end{bmatrix}$
   3. $A = \begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix}$
   4. $A = \begin{bmatrix} 4 & 2 \\ 2 & 6 \end{bmatrix}$
2. Which of the following is a positive definite quadratic form?
   1. $Q(x, y) = x^2 - y^2$
   2. $Q(x, y) = x^2 + 2xy + y^2$
   3. $Q(x, y) = x^2 + y^2$
   4. $Q(x, y) = -x^2 - y^2$
3. If a quadratic form has eigenvalues 2, -1, and 0, what can you say about its definiteness?
   1. Positive definite
   2. Negative definite
   3. Indefinite
   4. Positive semi-definite
4. Let $A = \begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}$. Find an orthogonal matrix $P$ that diagonalizes $A$.
   1. $P = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$
   2. $P = \begin{bmatrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{bmatrix}$
   3. $P = \begin{bmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{bmatrix}$
   4. $P = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$
5. What does Sylvester's Law of Inertia tell us about congruent matrices?
   1. They have the same determinant.
   2. They have the same rank.
   3. They have the same number of positive, negative, and zero eigenvalues.
   4. They are equal.
6. Which of the following statements is true regarding the relationship between a matrix $A$ and its transpose $A^T$ in the context of quadratic forms?
   1. $A$ must be equal to $A^T$.
   2. $A$ must be the inverse of $A^T$.
   3. $A$ must be orthogonal to $A^T$.
   4. $A$ can be any square matrix.
7. The quadratic form $Q(x) = x^T A x$ is said to be negative semi-definite if:
   1. $Q(x) < 0$ for all $x \neq 0$
   2. $Q(x) \leq 0$ for all $x \neq 0$
   3. $Q(x) > 0$ for all $x \neq 0$
   4. $Q(x) \geq 0$ for all $x \neq 0$