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numerical-range:examples:3x3

In the case of a $3\times 3$ matrix there are four classes of numerical ranges [1]:

  • Numerical range with zero line segments (ovular)
  • Numerical range with one line segment
  • Numerical range with two line segments
  • Numerical range with three line segments (triangular)

Example 1

  • The matrix $M=\begin{pmatrix}1&1&1\\0&\ee^{\frac{2\ii}{3}}&1\\0&0&\ee^{\frac{4\ii}{3}}\end{pmatrix}$ has an ovular numerical range

Example of a numerical range whose boundary includes zero line segments.
Example of an ovular numerical range

Example 2

  • The matrix $M=\begin{pmatrix}1&\frac{5-3\ii}{8}&\frac{3}{4}\\0&\frac{5+3\ii}{8}&\frac{3}{4}\\ -\frac{3}{4}&-\frac{3}{4}&-\frac{5}{4}\end{pmatrix}$ has a numerical range whose boundary includes one line segment

Example of a numerical range whose boundary includes one line segment.
Example of a numerical range whose boundary includes one line segment

Example 3

  • The matrix $M=\begin{pmatrix}1&1&0\\0&\ee^{\frac{2\ii}{3}}&0\\0&0&\ee^{\frac{4\ii}{3}}\end{pmatrix}$ has a numerical range whose boundary includes two line segments

Example of a numerical range whose boundary includes two line segments.
Example of a numerical range whose boundary includes two line segments

Example 4

  • The matrix $M=\begin{pmatrix}1&0&0\\0&\ee^{\frac{2\ii}{3}}&0\\0&0&\ee^{\frac{4\ii}{3}}\end{pmatrix}$ has a triangular numerical range. Notice that the matrix is normal.

Example of a numerical range whose boundary includes three line segments.
Example of a triangular numerical range

1. Keeler, D. S., Rodman, L., Spitkovsky, I. M., 1997. The numerical range of 3x3 matrices. Linear Algebra and its Applications, 252, pp.115 - 139.
numerical-range/examples/3x3.txt · Last modified: 2014/02/04 01:54 by lpawela