In this exercise, you will prove that a symmetric matrix is positive definite if and only
if all its eigenvalues are positive.
(a) Show that if A is symmetric positive definite, then all its eigenvalues must be
positive.
Hint: Let λ be an eigenvalue of A. Let x be a corresponding eigenvector with
norm 1.
Hint: xT x = ∥x∥2

To prove that a symmetric matrix \( A \) is positive definite if and only if all its eigenvalues are positive, we first tackle part (a) by showing that if \( A \) is symmetric and positive definite, then all its eigenvalues must be positive.

### Part (a) Proof:

#### Given:
- \( A \) is a symmetric matrix (i.e., \( A = A^T \)).
- \( A \) is positive definite, meaning that for any non-zero vector \( x \in \mathbb{R}^n \), we have \( x^T A x > 0 \).

#### To show:
All eigenvalues \( \lambda \) of \( A \) are positive.

#### Step 1: Eigenvalue definition

Let \( \lambda \) be an eigenvalue of \( A \), and let \( x \) be the corresponding eigenvector such that:
\[
A x = \lambda x
\]
We can also assume, without loss of generality, that the eigenvector \( x \) has unit norm:
\[
\|x\| = 1 \quad \text{or} \quad x^T x = 1
\]

#### Step 2: Use the positive definite property

Since \( A \) is positive definite, for any non-zero vector \( x \), we have:
\[
x^T A x > 0
\]
Now, substitute \( A x = \lambda x \) into this expression:
\[
x^T A x = x^T (\lambda x) = \lambda (x^T x)
\]
Since we have assumed that \( x^T x = 1 \), this simplifies to:
\[
x^T A x = \lambda
\]

#### Step 3: Conclusion

From the positive definite property of \( A \), we know that \( x^T A x > 0 \) for all non-zero \( x \). Therefore:
\[
\lambda > 0
\]
Thus, any eigenvalue \( \lambda \) of the symmetric positive definite matrix \( A \) must be positive.

This proves that if \( A \) is symmetric and positive definite, then all its eigenvalues are positive.

---

### Do you want to continue with part (b) or discuss this part in more detail?

---

#### 5 Related Questions:
1. What is the definition of a positive semi-definite matrix, and how does it differ from positive definite?
2. How are the eigenvalues of a matrix related to its determinant?
3. Can a non-symmetric matrix be positive definite? Why or why not?
4. What are some practical applications of positive definite matrices in real-world problems?
5. How can you numerically determine if a matrix is positive definite?

#### Tip:
Symmetric positive definite matrices are often used in optimization problems, where they ensure that quadratic forms have unique minima.

In this exercise, you will prove that a symmetric matrix is positive definite if and only if all its eigenvalues are positive.
(a) Show that if A is symmetric positive definite, then all its eigenvalues must be positive.
Hint: Let λ be an eigenvalue of A. Let x be a corresponding eigenvector with norm 1.
Hint: xT x = ∥x∥2

Prove that a symmetric matrix is positive definite if and only if all its eigenvalues are positive. This detailed solution covers eigenvalue and eigenvector properties, as well as the positive definite matrix theorem. Ideal for undergraduate students in linear algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation