| 要約: | Star-convexity of the eigenvalue region for the set of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></semantics></math></inline-formula> stochastic matrices has already been proven, for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula>, by Dmitriev and Dynkin. The star-convexity property enables full determination of the eigenvalue region by its boundary. This study offers a more straightforward proof that extends to other subclasses of the stochastic matrices. Furthermore, the proof is constructive as it includes the explicit construction of the corresponding realizing matrices. Explicit sufficient conditions for star-convexity of the eigenvalue regions of stochastic subclasses are presented. In particular, star-convexity of the eigenvalue region is proved for the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></semantics></math></inline-formula> doubly stochastic and the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></semantics></math></inline-formula> monotone stochastic matrices.
|
|---|
