The $k$-Numerical Range

An algorithm for computing the k-numerical range for any given matrix.

🔍 What is the $k$-Numerical Range?

The $k$-numerical range, $W_k(A)$, is a generalization of the classical numerical range. Instead of computing a single value $x^*Ax$ for a unit vector $x$, this version averages over $k$ orthonormal vectors:

$$W_k(A) = \left\{ \frac{1}{k} \sum_{i=1}^{k} x_i^* A x_i \ : \ x_i \text{ orthonormal} \right\}$$

This set is always:

• Convex (e.g., may appear as shapes like squares, circles, or triangles),
• Compact (Closed and Bounded),
• Useful in matrix theory and applications involving averages of eigenvalues.

🧠 How the Algorithm Works

To trace the boundary of $W_k(A)$, the algorithm:

1. Rotates the matrix by an angle $\theta$,
2. Takes the real part of the rotated matrix,
3. Finds the top $k$ eigenvectors with the largest eigenvalues,
4. Averages the values $x_j^* A x_j$ for those $k$ vectors,
5. Repeats for many angles to trace the boundary.

This approach generalizes a known method (Carl Cowen’s) for the classical case $k = 1$.

📁 Files

• AllKValues.py: The core Python script for computing and plotting the $k$-numerical range.
• Senior_Project_Poster.pdf: Visual summary of the algorithm and examples, suitable for presentations.

✅ Requirements

This project uses:

• numpy
• matplotlib

Install them with:

pip install numpy matplotlib

🖼️ Sample Output

The script includes several plotted examples of the $k$-numerical range for different matrices, demonstrating how the shape evolves as $k$ changes. $A_0$ is my first version of the $k$-numerical range in Python. This naive method primarily used NumPy’s trace function to compute the approximation, with a structure like:

approx = np.trace(X.conj().T @ A @ X) / k

$$A_0 = \begin{bmatrix} i & 0 & 0 & \cdots & 0 \\\ 0 & -i & 0 & \cdots & 0 \\\ 0 & 0 & 1 & \cdots & 0 \\\ \vdots & \vdots & \vdots & \ddots & \vdots \\\ 0 & 0 & 0 & \cdots & 1 \end{bmatrix} \in M_7(\mathbb{C})$$

This naive method was not well-suited for accurately tracing the boundary of the $k$-numerical range. Instead, it generated points from the interior of $W_k(A)$ by averaging over randomly chosen subspaces.

Moving on to $A_1$ where

$$\omega = \frac{-1+i\sqrt{3}}{2}$$

such that,

$$A_1 = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\\ 0 & \omega & 0 & 0 & 0 \\\ 0 & 0 & \omega^2 & 0 & 0 \\\ 0 & 0 & 0 & 0 & 1 \\\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}$$

$A_2$ is the same matrix but $1$ is now a $\frac{1}{2}$

$$A_2 = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\\ 0 & \omega & 0 & 0 & 0 \\\ 0 & 0 & \omega^2 & 0 & 0 \\\ 0 & 0 & 0 & 0 & \frac{1}{2} \\\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}$$

Lastly we have the same matrix $A_3$ but the $1$ becomes a $2$

$$A_3 = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\\ 0 & \omega & 0 & 0 & 0 \\\ 0 & 0 & \omega^2 & 0 & 0 \\\ 0 & 0 & 0 & 0 & 2 \\\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}$$