On the numerical range of a matrix

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 "¨Uber 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

On the numerical range of a matrix [Translation of R. Kippenhahn (1951). Über den Wertevorrat einer Matrix. Mathematische Nachrichten 6, 193-228]

This is an English translation of the article "ber 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 "ber 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

On the numerical range of a matrix [Translation of R. Kippenhahn (1951). Über den Wertevorrat einer Matrix. Mathematische Nachrichten 6, 193-228]

This is an English translation of the article "ber 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 "ber 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

Numerical approximation of the field of values of the inverse of a large matrix

We consider the approximation of the field of values of the inverse of a large sparse matrix, without explicitly computing the inverse or using its action (i.e., accurately solving a linear system with this matrix). We review results by Manteuffel and Starke and give an alternative that may yield better approximations in practice. We give connections with the harmonic Rayleigh-Ritz approach. Several properties and applications of the studied concepts as well as numerical examples are provided. Key words: Field of values, numerical range, matrix inverse, large sparse matrix, Ritz values, harmonic Rayleigh-Ritz, harmonic Ritz values, GMRES convergence, Arnoldi, numerical radius, numerical abscissa, inner numerical radius, inclusion region

Eigenvalue inclusion regions from inverses of shifted matrices

We consider eigenvalue inclusion regions based on the field of values, pseudospectra, Gershgorin region, and Brauer region of the inverse of a shifted matrix. A family of these inclusion regions is derived by varying the shift. We study several properties, one of which is that the intersection of a family is exactly the spectrum. The numerical approximation of the inclusion sets for large matrices is also examined. Key words: Harmonic Rayleigh–Ritz, inclusion regions, exclusion regions, inclusion curves, exclusion curves, field of values, numerical range, large spars