To find \( W(A) \), the numerical range (or field of values) of a \( 3 \times 3 \) matrix \( A \), you need to follow these steps:

### 1. Definition of \( W(A) \)
For a matrix \( A \), the numerical range \( W(A) \) is defined as:
\[
W(A) = \{ \langle Ax, x \rangle : x \in \mathbb{C}^n, \|x\| = 1 \}
\]
where \( \langle Ax, x \rangle = x^* A x \), and \( x \) is a vector with unit norm (\( \|x\| = 1 \)).

### 2. Steps to Compute \( W(A) \)
1. **Eigenvalue Calculation**: Find the eigenvalues of \( A \) since the convex hull of these eigenvalues will be within \( W(A) \).
2. **Compute \( \langle Ax, x \rangle \)**: For various unit vectors \( x \), compute \( \langle Ax, x \rangle \).
3. **Plot the Results**: Plot the set of all such values \( \langle Ax, x \rangle \) in the complex plane to visualize \( W(A) \).

### Example Calculation
Let's say matrix \( A \) is given by:
\[
A = \begin{pmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{pmatrix}
\]
To compute \( W(A) \):
1. Find eigenvalues of \( A \).
2. Consider a general unit vector \( x \) of the form \( x = (e^{i\theta_1}, e^{i\theta_2}, e^{i\theta_3})^T \), where \( \theta_1, \theta_2, \theta_3 \) are real numbers.
3. Calculate \( \langle Ax, x \rangle = x^* A x \) for various values of \( x \).

Would you like to proceed with a specific example, or would you like to input a particular \( 3 \times 3 \) matrix?

Learn how to find the numerical range W(A) of a 3 by 3 matrix A with a detailed step-by-step solution. This problem covers linear algebra concepts including numerical ranges, eigenvalues, and unit vectors in complex space. Suitable for advanced college-level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation