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Numerical Methods in Matrix Computations

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Abstract

The eigenvalues and eigenvectors of a matrix play an important role in many settings in physics and engineering.

The eigenvalue problem has a deceptively simple formulation, yet the determination of accurate solutions presents a wide variety of challenging problems.

—J. H. Wilkinson, The Algebraic Eigenvalue Problem, 1965.

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Notes

  1. 1.

    Wilkinson received the Chauvenet Prize of the Mathematical Association of America 1987 for this exposition of the ill-conditioning of polynomial zeros.

  2. 2.

    Marie Ennemond Camille Jordan (1838–1922), French mathematician, professor at École Polytechnique and Collége de France. Jordan made important contributions to finite group theory, linear and multilinear algebra, as well as differential equations. His paper on the canonical form was published in 1870.

  3. 3.

    Vera Nikolaevna Kublanovskaya (1920–2012), Russian mathematician, was one of the founders of modern linear algebra. She came from a small village on the Lake Beloye east of Leningrad and began studies in Leningrad to become a teacher. There she was encouraged to pursue a career in mathematics by D. K. Faddeev. After surviving the siege of Leningrad, she graduated in 1948 and joined the Steklov institute. Here she became responsible for selecting matrix algorithm for BESM, the first electronic computer in the USSR. She is most widely known as one of the inventors of the QR algorithm and her work on canonical forms.

  4. 4.

    Jacopo Francesco Riccati (1676–1754), Italian mathematician. His works on hydraulics and differential equations were used by the city of Venice in regulating the canals. The Riccati differential equation \(y^\prime = c_0(x) + c_1(x)y + c_2(x)y^2\) is named after him. The algebraic Riccati equation, which also is quadratic, is named in analogy to this.

  5. 5.

    Semyon Aranovich Geršgorin (1901–1933), Russian mathematician, who worked at the Leningrad Mechanical Engineering Institute. He published his circle theorem 1931 in [93, 1931].

  6. 6.

    John William Strutt (1842–1919) succeeded his father as third Baron Rayleigh in 1873. Unable to follow a conventional academic career, he performed scientific experiments at his private laboratory for many years. In his major text The Theory of Sound he studied the mechanics of a vibrating string and explained wave propagation. From 1879 to 1884 he held a position in experimental physics at the University of Cambridge. He held many official positions, including President of the London Mathematical Society and President of the Royal Society. In 1904 he and Sir William Ramsey were awarded the Nobel prize for the discovery of the inert gas argon.

  7. 7.

    This result, first published by Toeplitz [228, 1918], is not trivial. Householder [132, 1964], Sect. 3.3.2, gives a proof due to Hans Schneider (unpublished).

  8. 8.

    Helmut Wielandt (1910–2001) was a student of Schmidt and Schur in Berlin. His initial work was in group theory. In 1942 he became attached to the Aerodynamics Research Institute in Göttingen and started to work on vibration theory. He contributed greatly to matrix theory and did pioneering work on computational methods for the matrix eigenvalue problem; see [133, 1996].

  9. 9.

    Hermann Günter Grassmann (1809–1877), German mathematician, was born in Stettin. He studied theology, languages, and philosophy at University of Berlin. As a teacher at the Gymnasium in Stettin he took up mathematical research on his own. In 1844 he published a highly original textbook, in which the symbols representing geometric entities such as points, lines, and planes were manipulated using certain rules. Later his work became used in areas such as differential geometry and relativistic quantum mechanics. Sadly, the leading mathematicians of his time failed to recognize the importance of his work.

  10. 10.

    Heinz Rutishauser (1918–1970) Swiss mathematician, a pioneer in computing, and the originator of many important algorithms in Scientific Computing. In 1948 he joined the newly founded Institute for Applied Mathematics at ETH in Zürich. He spent 1949 at Harvard with Howard Aiken and at Princeton with John von Neumann to learn about electronic computers. Rutishauser was one of the leaders in the international development of the programming language Algol. His qd algorithm [206, 1954] had great impact on methods for eigenvalue calculations.

  11. 11.

    John Francis, born in London 1934, is an English computer scientist. In 1954 he worked for the National Research Development Corporation (NRDC). In 1955–1956 he attended Cambridge University, but did not complete a degree and returned to work for NRDC, now as an assistant to Christopher Strachey. Here he developed the QR algorithm, but by 1962 left the field of numerical analysis. He had no idea of the great impact the QR algorithm has had until contacted by Gene Golub and Frank Uhlig in 2007.

  12. 12.

    Ralph Byers (1955–2007) made a breakthrough in his PhD thesis from Cornell University in 1983 by finding a strongly stable numerical methods of complexity \(O(n^3)\) for Hamiltonian and symplectic eigenvalue problems. He was also instrumental in developing methods based on the matrix sign function for the solution of Riccati equations for large-scale control problems. His work on the multishift QR algorithm was rewarded with the SIAM Linear Algebra prize in 2003 and the SIAM Outstanding paper prize in 2005.

  13. 13.

    This name was chosen because of the connection with Schur’s work on bounded analytic functions.

  14. 14.

    Carl Gustaf Jacob Jacobi (1805–1851), German mathematician, started teaching mathematics at the University of Berlin already in 1825. In 1826 he moved to the University of Königsberg, where Bessel held a chair. In the summer of 1829 he visited Gauss in Göttingen and Legendre and Fourier in Paris and published a paper containing fundamental advances on the theory of elliptic functions. In 1832 he was promoted to full professor. Like Euler, Jacobi was a proficient calculator who drew a great deal of insight from immense computational work.

  15. 15.

    William Rowan Hamilton (1805–1865), Irish mathematician, professor at Trinity College, Dublin. He made important contributions to classical mechanics, optics, and algebra. His greatest contribution is the reformulation of classical Newton mechanics now called Hamiltonian mechanics. He is also known for his discovery of quaternions as an extension of (two-dimensional) complex numbers to four dimensions.

  16. 16.

    Laub [165, 1979] used a similar idea for computing the Hamiltonian Schur form using information from an unstructured Schur form.

  17. 17.

    Henry Eugène Padé (1863–1953), French mathematician and student of Charles Hermite, gave a systematic study of these approximations in his thesis 1892.

  18. 18.

    Henry Briggs (1561–1630), English mathematicians, fellow of St. John’s College, Oxford, was greatly interested in astronomy, which involved much heavy calculations. He learned about logarithms by reading Napier’s text from 1614. Briggs constructed logarithmic tables to 14 decimal places that were published in 1624.

  19. 19.

    John Napier (1550–1617) came from a wealthy Scottish family and devoted much time to running his estate and working on Protestant theology. He studied mathematics as a hobby and undertook long calculations. His work on logarithms was published in Latin 1614. An English translation by E. Wright appeared in 1616.

  20. 20.

    Wassily Leontief (1905–1999) was a Russian–American economist and Nobel laureate 1973. Educated first in St Petersburg, he left USSR in 1925 to earn his PhD in Berlin. In 1931 he went to the United States, where in 1932 he joined Harvard University. Around 1949 he used the computer Harvard Mark II to model 500 sectors of the US economy, one of the first uses of computers for modeling.

  21. 21.

    Oskar Perron (1880–1975), German mathematician held positions at Heidelberg and Munich. His work covered a wide range of topics. He also wrote important textbooks on continued fractions, algebra, and non-Euclidean geometry.

  22. 22.

    Named after Andrei Andreevic Markov (1856–1922). Markov attended lectures by Chebyshev at St Petersburg University, where graduated in 1884. His early work was in number theory, analysis, and continued fractions. He introduced Markov chains in 1908 (see also [175, 1912]), which started a new branch in probability theory.

  23. 23.

    It is known that if this condition holds, \(A\) belongs to a set that forms a multiplicative group under ordinary matrix multiplication.

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  1. Department of Mathematics, Linköping University, Linköping, Sweden

    Åke Björck

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Björck, Å. (2015). Matrix Eigenvalue Problems. In: Numerical Methods in Matrix Computations. Texts in Applied Mathematics, vol 59. Springer, Cham. https://doi.org/10.1007/978-3-319-05089-8_3

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