December 31, 2016

This Demonstration gives a portrait of the neighborhood of the spectrum of a matrix ... , including an approximation of the numerical range of matrices ... acting in ... , with low dimension ... . We restrict our attention to upper triangular matrices, since neither the numerical range nor the spectrum is altered by a unitary similarity transformation. For every matrix ... there exists a unitary similarity that transforms ... to upper triangular form (in Mathematica this can be achieved by ... ). Note that the diagonal elements of an upper triangular matrix are exactly the eigenvalues, and an upper triangular matrix that is normal must be diagonal. Choose a dimension! Then some of the entries of a default matrix ... are depicted as locators: all diagonal entries (with a red inner point), some strict upper diagonal entries (with a blue inner point), and no others. Moreover, for your convenience, you can see the convex hulls of the entries of ... (in blue) and of the diagonal entries of ... (in red), which, at the same time, are the eigenvalues of ... . In addition, the whole matrix ... is shown below the graphic and updated as you drag any locator. A yellow point with a red boundary marks the mean of the diagonal entries of ... , which is also the mean of the eigenvalues of ... . In the literature this is usually expressed as ... . We call this point the hub of ... , because it is the natural pivot point for all matrices similar to ... . Indeed, one could claim that the whole question of similarity hinges on this point! For contrast, a light blue point with a dark blue boundary marks the mean of all the entries of ... . Most importantly, a family of yellow rectangles of different orientations is drawn with a green frame around each. Their intersection constitutes an outer approximation of the field of values. The number of those rectangles can be influenced by pushing the slider for the minimal rotational angle ... . To be able to go back to the original matrices, click "reset ... ". To be able to compare results for different dimensions more easily, you can fix the plot range. Now, you can experiment by dragging some of the locators. You will probably be surprised! You can neither create nor delete locators; you can only drag the ones given. Almost every item in the graphics is annotated, so mouseover them to see explanations.