Practical examples of negative semi-definite quadratic forms

📚 Quick Study Guide

• 🔍 A quadratic form $Q(x)$ is negative semi-definite if $Q(x) \leq 0$ for all $x$.
• 🔢 A symmetric matrix $A$ corresponds to a negative semi-definite quadratic form if all its eigenvalues are non-positive ($\lambda_i \leq 0$).
• 📊 Hessian matrices in optimization problems can be used to determine if a critical point is a local maximum. If the Hessian is negative semi-definite, it indicates a local maximum.
• 📝 For a 2x2 matrix $A = \begin{bmatrix} a & b \\ b & c \end{bmatrix}$, $A$ is negative semi-definite if $a \leq 0$, $c \leq 0$, and $ac - b^2 \geq 0$.
• 💡 Common examples include functions representing potential energy where higher values indicate less stability or cost functions where lower values are desirable.

Practice Quiz

1. Which of the following quadratic forms is negative semi-definite?

   1. $Q(x, y) = x^2 + y^2$
   2. $Q(x, y) = -x^2 - y^2$
   3. $Q(x, y) = x^2 - y^2$
   4. $Q(x, y) = x^2 + 2xy + y^2$

2. For what values of $a$ is the matrix $\begin{bmatrix} -2 & 1 \\ 1 & a \end{bmatrix}$ negative semi-definite?

   1. $a \leq -\frac{1}{2}$
   2. $a \geq -\frac{1}{2}$
   3. $a \leq \frac{1}{2}$
   4. $a \geq \frac{1}{2}$

3. Which of the following matrices could represent a negative semi-definite quadratic form?

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

4. Consider the function $f(x, y) = -x^2 - y^2 + 2x + 4y$. At which point is the Hessian matrix negative semi-definite, indicating a potential local maximum?

   1. (0, 0)
   2. (1, 2)
   3. (-1, -2)
   4. (2, 1)

5. A cost function $C(x, y)$ is modeled by a negative semi-definite quadratic form. What does this imply about minimizing the cost?

   1. The cost function has a unique minimum.
   2. The cost function has a unique maximum.
   3. The cost function has infinitely many minima.
   4. The cost function has no minimum.

6. If a matrix $A$ has eigenvalues -2 and 0, what can you conclude about the quadratic form associated with $A$?

   1. It is positive definite.
   2. It is negative definite.
   3. It is positive semi-definite.
   4. It is negative semi-definite.

7. Which condition MUST be true for a 2x2 matrix $\begin{bmatrix} a & b \\ b & c \end{bmatrix}$ to be negative semi-definite?

   1. $a > 0$ and $c > 0$
   2. $a < 0$ and $c < 0$
   3. $a \leq 0$, $c \leq 0$, and $ac - b^2 \geq 0$
   4. $a \geq 0$, $c \geq 0$, and $ac - b^2 \leq 0$