To find the numerical radius of a $3 \times 3$ matrix, we follow these steps:

Step-by-Step Process

1. Definition: The numerical radius $w(A)$ of a matrix $A$ is defined as: $w(A) = \max_{||x||=1} |x^* A x|$ where $x$ is a vector such that $||x||=1$ (its norm is 1) and $x^*$ is the conjugate transpose of $x$.

2. Matrix Example: Suppose we have a $3 \times 3$ matrix:

   a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix}$$

3. Calculate Eigenvalues: The numerical radius of $A$ is the largest eigenvalue of the matrix $(A^* A)^{1/2}$, where $A^*$ is the conjugate transpose of $A$.

4. Steps to Compute:

   • Compute $A^*$: \overline{a_{11}} & \overline{a_{21}} & \overline{a_{31}} \\ \overline{a_{12}} & \overline{a_{22}} & \overline{a_{32}} \\ \overline{a_{13}} & \overline{a_{23}} & \overline{a_{33}} \end{pmatrix}$$
   • Compute $A^* A$: \overline{a_{11}} & \overline{a_{21}} & \overline{a_{31}} \end{pmatrix} \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix}$$
   • Find the eigenvalues of $A^* A$.
   • The largest eigenvalue of $(A^* A)^{1/2}$ gives the numerical radius $w(A)$.

Example Calculation

Let's use an example matrix $A$:

1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}$$ 1. Compute $$A^*$$: $$A^* = \begin{pmatrix} 1 & 4 & 7 \\ 2 & 5 & 8 \\ 3 & 6 & 9 \end{pmatrix}$$ 2. Compute $$A^* A$$: $$A^* A = \begin{pmatrix} 1 & 4 & 7 \\ 2 & 5 & 8 \\ 3 & 6 & 9 \end{pmatrix} \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} = \begin{pmatrix} 66 & 78 & 90 \\ 78 & 93 & 108 \\ 90 & 108 & 126 \end{pmatrix}$$ 3. Find the eigenvalues of $$A^* A$$. For this example, let's say they are approximately $$\lambda_1, \lambda_2, \lambda_3$$. 4. Compute the square roots of these eigenvalues and find the maximum value among them. This maximum value is the numerical radius $$w(A)$$. To find exact numerical values for the eigenvalues and then compute the numerical radius accurately, we would typically use numerical software or a calculator capable of handling matrix computations. Would you like the detailed calculation or have any specific questions? Here are some additional related questions you might consider: 1. How do you compute the eigenvalues of a matrix? 2. What is the significance of the numerical radius in matrix analysis? 3. Can you explain the difference between the spectral radius and the numerical radius? 4. What are some applications of the numerical radius in mathematics or engineering? 5. How do you find the conjugate transpose of a matrix? 6. What is the relationship between the numerical radius and the matrix norm? 7. Are there any properties of the numerical radius that are particularly useful? 8. Can you provide examples of matrices where the numerical radius is particularly easy to compute? **Tip:** Understanding the concept of matrix eigenvalues is crucial for various applications in linear algebra, including finding the numerical radius.