When is a quadratic form positive definite vs. indefinite? Examples

📚 Quick Study Guide

• 🔢 A quadratic form $Q(x)$ is a homogeneous polynomial of degree 2 in $n$ variables, often written as $Q(x) = x^T Ax$, where $A$ is a symmetric matrix.
• ➕ A quadratic form $Q(x)$ is positive definite if $Q(x) > 0$ for all non-zero vectors $x$. Equivalently, all eigenvalues of $A$ are positive.
• ➖ A quadratic form $Q(x)$ is negative definite if $Q(x) < 0$ for all non-zero vectors $x$. Equivalently, all eigenvalues of $A$ are negative.
• ➕/➖ A quadratic form $Q(x)$ is positive semi-definite if $Q(x) \geq 0$ for all vectors $x$. Equivalently, all eigenvalues of $A$ are non-negative.
• ➖/➕ A quadratic form $Q(x)$ is negative semi-definite if $Q(x) \leq 0$ for all vectors $x$. Equivalently, all eigenvalues of $A$ are non-positive.
• ❓ A quadratic form $Q(x)$ is indefinite if it takes on both positive and negative values. Equivalently, $A$ has at least one positive and one negative eigenvalue.
• 🔑 Sylvester's Criterion: A symmetric matrix $A$ is positive definite if and only if all its leading principal minors are positive.

Practice Quiz

1. What condition must be met for a quadratic form $Q(x) = x^T A x$ to be positive definite?
   1. A) $Q(x) < 0$ for all non-zero $x$
   2. B) $Q(x) > 0$ for all non-zero $x$
   3. C) $Q(x) = 0$ for all $x$
   4. D) $Q(x) \geq 0$ for all $x$
2. Which of the following statements is true about the eigenvalues of a matrix associated with a positive definite quadratic form?
   1. A) All eigenvalues are negative.
   2. B) All eigenvalues are zero.
   3. C) All eigenvalues are positive.
   4. D) Eigenvalues can be positive or negative.
3. A quadratic form $Q(x)$ is indefinite if:
   1. A) $Q(x) > 0$ for all $x$
   2. B) $Q(x) < 0$ for all $x$
   3. C) $Q(x)$ takes on both positive and negative values.
   4. D) $Q(x) = 0$ for all $x$
4. What can you say about the eigenvalues of the matrix associated with an indefinite quadratic form?
   1. A) All eigenvalues are positive.
   2. B) All eigenvalues are negative.
   3. C) There is at least one positive and one negative eigenvalue.
   4. D) All eigenvalues are zero.
5. Which criterion can be used to determine if a symmetric matrix is positive definite by checking the signs of its leading principal minors?
   1. A) Rank-Nullity Theorem
   2. B) Sylvester's Criterion
   3. C) Cayley-Hamilton Theorem
   4. D) Spectral Theorem
6. Consider the quadratic form $Q(x, y) = x^2 - y^2$. Is this quadratic form positive definite, negative definite, or indefinite?
   1. A) Positive definite
   2. B) Negative definite
   3. C) Indefinite
   4. D) Positive semi-definite
7. For a matrix $A$ to represent a positive semi-definite quadratic form, its eigenvalues must be:
   1. A) All positive
   2. B) All negative
   3. C) All non-negative
   4. D) All non-positive