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- If is Hermitian, then it is normal, so it is the convex hull of its eigenvalues, which are all real.
Conversely, assume is on the real line. Decompose , where is a Hermitian matrix, and an anti-Hermitian matrix. Since is on the imaginary line, if , then would stray from the real line. Thus , and is Hermitian. (en)
- For , if is normal, then it has a full eigenbasis, so it reduces to .
Since is normal, by the spectral theorem, there exists a unitary matrix such that , where is a diagonal matrix containing the eigenvalues of .
Let . Using the linearity of the inner product, that , and that are orthonormal, we have: (en)
- Expanding this derivative: (en)
- Let . We have .
By Cauchy–Schwarz,
For the other one, let , where are Hermitian.
Since is on the real line, and is on the imaginary line, the extremal points of appear in , shifted, thus both . (en)
- Since the above holds for all , we must have:
For any and , substitute into the equation: Choose and , then simplify, we obtain for all , thus . (en)
- Let satisfy these properties. Let be the original numerical range.
Fix some matrix . We show that the supporting planes of and are identical. This would then imply that since they are both convex and compact.
By property , is nonempty. Let be a point on the boundary of , then we can translate and rotate the complex plane so that the point translates to the origin, and the region falls entirely within . That is, for some , the set lies entirely within , while for any , the set does not lie entirely in .
The two properties of then imply that and that inequality is sharp, meaning that has a zero eigenvalue. This is a complete characterization of the supporting planes of .
The same argument applies to , so they have the same supporting planes. (en)
- By affineness of , we can translate and rotate the complex plane, so that we reduce to the case where has a sharp point at , and that the two supporting planes at that point both make an angle with the imaginary axis, such that since the point is sharp.
Since , there exists a unit vector such that .
By general property , the numerical range lies in the sectors defined by: At , the directional derivative in any direction must vanish to maintain non-negativity. Specifically: (en)
- is the image of a continuous map from the closed unit sphere, so it is compact.
For any of unit norm, project to the span of as . Then is a filled ellipse by the previous result, and so for any , let , we have (en)
- The elements of are of the form , where is projection from to a one-dimensional subspace.
The space of all one-dimensional subspaces of is , which is a 2-sphere. The image of a 2-sphere under a linear projection is a filled ellipse.
In more detail, such are of the form where , satisfying , is a point on the unit 2-sphere.
Therefore, the elements of , regarded as elements of is the composition of two real linear maps and , which maps the 2-sphere to a filled ellipse. (en)
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