Matrix Numerical Range
Explore complex quadratic-form values beyond a matrix’s eigenvalues.
Free theory articleExplore complex quadratic-form values beyond a matrix’s eigenvalues.
Concept & interpretation
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.
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.