On the numerical range of a matrix

This is an English translation of the article “Über den Wertevorrat einer Matrix” by Rudolf Kippenhahn, Mathematische Nachrichten 6 (1951), 193–228. This article is often cited by mathematicians who work in the area of numerical ranges, thus it is hoped that this translation may be useful. Some notation and wording has been changed to make the article more in line with present articles on the subject written in English. In Part 1 of this article Kippenhahn characterized the numerical range of a matrix as being the convex hull of a certain algebraic curve that is associated to the matrix. More than 55 years later this “boundary generating curve” is still a topic of current research, and “Über den Wertevorrat einer Matrix” is almost always present in the bibliographies of articles on this topic. In Part 2, the author initiated the study of a generalization of the numerical range to matrices with quaternion entries. The translators note that in Theorem 36, it is stated incorrectly that this set of points in 4-dimensional space is convex. A counterexample to this statement was given in 1984.[I] In the notes at the end of this article the translators pinpoint the flaw in the argument. In the opinion of the translators, this error does not detract from the overall value and significance of this article. The translators also have corrected some minor misprints in the original without comment.