Matrix Numerical Range

The numerical range, also known as the field of values, of a matrix A is the set of all possible complex numbers z*Az where z is a complex unit vector (z*z = 1). For a 2x2 matrix, the Toeplitz-Hausdorff theorem guarantees that this region is a closed, convex elliptical disk (which can degenerate into a line segment or a point) with the eigenvalues of A as the foci of the ellipse. Exploring the numerical range allows us to visualize properties of A that are not captured by eigenvalues alone.

Worked Example

Let's calculate a specific point in the numerical range for the "Triangular ellipse" matrix preset. The matrix is:

A = [ -1, 2 ] [  0, 1 ]

The values from the preset are a = -1, d = 1, c = 2. Let's use a unit vector z defined by the mix angle α = 30° and phase angle φ = 180°.

Calculate the components of z = (cos(α), exp(iφ)sin(α)):
cos(30°) = √3 / 2 ≈ 0.866
sin(30°) = 1 / 2 = 0.5
exp(i 180°) = -1
So, z = (√3/2, -0.5).
The quadratic form evaluates to:
z*Az = a cos²(α) + d sin²(α) + c cos(α) sin(α) exp(iφ)
Substitute the values:
z*Az = (-1)(√3/2)² + (1)(1/2)² + (2)(√3/2)(1/2)(-1)
Simplify the terms:
(-1)(3/4) = -0.75
(1)(1/4) = 0.25
(2)(√3/2)(1/2)(-1) = -√3 / 2 ≈ -0.8660254
Add them together:
Real part = -0.75 + 0.25 - 0.8660254 = -1.3660254
Imaginary part = 0

The resulting point in the complex plane is exactly -1.3660254 + 0i, demonstrating how complex phase angles affect the evaluated point even when the matrix is real.

Interactive Experiment

You can reproduce this behavior using the Matrix Numerical Range interactive module.

Select Triangular ellipse from the Matrix drop-down menu.
Select 30 from the Mix angle degrees drop-down.
Select 180 from the Phase degrees drop-down.
Observe that the Point real value updates to exactly -1.3660254.
Change the Phase degrees to 90 to see the point move to the top edge of the ellipse boundary, showing a non-zero imaginary component.

Limitations

This visualization is limited to 2x2 real triangular matrices A = [[a, c], [0, d]] to allow exact analytical bounds for the ellipse semimajor and semiminor axes. The general numerical range of larger matrices N > 2 may form more complex shapes and is typically approximated computationally using the field of values of its Hermitian projection rather than purely analytical bounds. Numerical precision effects are present at the edges of the double-precision floating-point format.

References

For a comprehensive mathematical look into the numerical range, see the following reference:

What Is the Numerical Range of a Matrix? - Nicholas J. Higham (2023). Covers properties and examples of the field of values.

Matrix Numerical Range