On the Numerical Range of a Matrix Translated from the German by Paul F. Zachlin

This is an English translation of the paper “Über den Wertevorrat einer Matrix” by Rudolf Kippenhahn, Mathematische Nachrichten 6 (1951), 193–228. This paper 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 paper more in line with present papers on the subject written in English. In Part 1 of this paper 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 papers 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 paper the translators pinpoint the flaw in the argument. In the opinion of the translators, this error does not significantly detract from the overall value and significance of this paper. In the translation, footnotes in the original version are indicated by superscript Arabic numerals, while superscript Roman numerals in brackets are used to indicate that the translators have a comment about the original paper. All of these comments appear at the end of this paper, and the translators also have corrected some minor misprints in the original without comment. † Department of Mathematics, Case Western Reserve University, Cleveland, OH 44106-7058, USA [email protected]. ‡ Department of Mathematics and Computing Science, Eindhoven University of Technology, PO Box 513, 5600 MB, The Netherlands. [email protected].