consapp_matrix.bib

@article{35c3b3c9-7c44-327c-a4e2-a2ca3c5baca3,
 ISSN = {02610523},
 URL = {http://www.jstor.org/stable/108649},
 author = {{Cayley , A.}},
 journal = {Philosophical Transactions of the Royal Society of London},
 pages = {17--37},
 publisher = {The Royal Society},
 title = {A Memoir on the Theory of Matrices},
 urldate = {2026-04-08},
 volume = {148},
 year = {1858}
}
@article{CRILLY1986241,
title = {The rise of Cayley's invariant theory (1841–1862)},
journal = {Historia Mathematica},
volume = {13},
number = {3},
pages = {241-254},
year = {1986},
issn = {0315-0860},
doi = {10.1016/0315-0860(86)90091-1},
author = {{Crilly  T.}},
keywords = {hyperdeterminants, George Boole, partial differential equations, quantics, J. J. Sylvester, multilinear forms},
abstract = {In his pioneering papers of 1845 and 1846, Arthur Cayley (1821–1895) introduced several approaches to invariant theory, the most prominent being the method of hyperdeterminant derivation. This article discusses these papers in the light of Cayley's unpublished correspondence with George Boole, who exercised considerable influence on Cayley at this formative stage of invariant theory. In the 1850s Cayley put forward a new synthesis for invariant theory framed in terms of partial differential equations. In this period he published his memoirs on quantics, the first seven of which appeared in quick succession. This article examines the background of these memoirs and makes use of unpublished correspondence with Cayley's lifelong friend, J. J. Sylvester.}
}
@article{alan1948,
  title = {Rounding-off errors in matrix processes},
  year = {1948},
  issue = {1},
  journal = {The Quarterly Journal of Mechanics and Applied Mathematics},
  pages = {287-308},
  volume = {1},
  author = {{Turing, A. M.}},
url={https://turing.academicwebsite.com/publications/20-rounding-off-errors-in-matrix-processes}
}
@book{saff2025,
title={Matrix Fundamentals},
subtitle={From Equation Solving to Signal Processing},
author={{Saff, E. B., & Snider, A. D.}},
doi={10.1007/978-3-031-97222-5},
publisher={Springer},
address="Cham",
edition={2},
year={2025}
}
@book{gentle2024,
title={Matrix Algebra},
subtitle={Theory, Computations and Applications in Statistics},
author={{Gentle, J. E.}},
series={Springer Texts in Statistics},
doi={10.1007/978-3-031-42144-0},
publisher={Springer},
address="Cham",
year={2024},
edition={3}
}
@Inbook{Shores2018,
author={{Shores, T. S.}},
title="MATRIX ALGEBRA",
bookTitle="Applied Linear Algebra and Matrix Analysis",
year="2018",
publisher="Springer International Publishing",
address="Cham",
pages="65--180",
abstract="In Chapter 1 we used matrices and vectors as simple storage devices. In this chapter matrices and vectors take on a life of their own. We develop the arithmetic of matrices and vectors. Much of what we do is motivated by a desire to extend the ideas of ordinary arithmetic to matrices.",
isbn="978-3-319-74748-4",
doi="10.1007/978-3-319-74748-4_2",
url="https://doi.org/10.1007/978-3-319-74748-4_2"
}
@Inbook{Karpfinger2022a,
author={{Karpfinger, C.}},
title="L R-Zerlegung einer Matrix",
bookTitle="H{\"o}here Mathematik in Rezepten: Begriffe, S{\"a}tze und zahlreiche Beispiele in kurzen Lerneinheiten",
year="2022",
publisher="Springer",
address="Berlin, Heidelberg",
pages="107--117",
abstract="Wir betrachten das Problem, zu einer invertierbaren Matrix {\$}{\$}A {\backslash}in {\backslash}mathbb {\{}R{\}}^{\{}n{\backslash}times n{\}}{\$}{\$}A∈Rn{\texttimes}nund einem Vektor {\$}{\$}b {\backslash}in {\backslash}mathbb {\{}R{\}}^n{\$}{\$}b∈Rneinen Vektor {\$}{\$}x {\backslash}in {\backslash}mathbb {\{}R{\}}^n{\$}{\$}x∈Rnmit {\$}{\$}A {\backslash}, x = b{\$}{\$}Ax=bzu bestimmen; kurz: Wir l{\"o}sen das lineare Gleichungssystem {\$}{\$}A x = b{\$}{\$}Ax=b. Formal erh{\"a}lt man die L{\"o}sung durch {\$}{\$}x = A^{\{}-1{\}} b{\$}{\$}x=A-1b. Aber die Berechnung von {\$}{\$}A^{\{}-1{\}}{\$}{\$}A-1ist bei einer gro{\ss}en Matrix A aufwendig. Die Cramer'sche Regel (siehe Rezept in Abschn. 12.3) ist aus numerischer Sicht zur Berechnung der L{\"o}sung x ungeeignet. Tats{\"a}chlich liefert das Gau{\ss}'sche Eliminationsverfahren, das wir auch in Kap. 9 zur h{\"a}ndischen L{\"o}sung eines LGS empfohlen haben, eine Zerlegung der Koeffizientenmatrix A, mit deren Hilfe es m{\"o}glich ist, ein Gleichungssystem der Form {\$}{\$}A {\backslash}, x = b{\$}{\$}Ax=bmit invertierbarem A zu l{\"o}sen. Diese sogenannte {\$}{\$}L{\backslash}, R{\$}{\$}LR-Zerlegung ist zudem numerisch gutartig. Gleichungssysteme mit bis zu etwa 10000 Zeilen und Unbekannten lassen sich auf diese Weise vorteilhaft l{\"o}sen. F{\"u}r gr{\"o}{\ss}ere Gleichungssysteme sind iterative L{\"o}sungsverfahren zu bevorzugen (siehe Kap. 71).",
isbn="978-3-662-63305-2",
doi="10.1007/978-3-662-63305-2_11",
url="https://doi.org/10.1007/978-3-662-63305-2_11"
}
@Inbook{Karpfinger2022b,
author={{Karpfinger, C.}},
title="Calculating with Matrices",
bookTitle="Calculus and Linear Algebra in Recipes: Terms, phrases and numerous examples in short learning units",
year="2022",
publisher="Springer",
address="Berlin, Heidelberg",
pages="87--100",
abstract="We have already used matrices to solve systems of linear equations: Matrices have been a helpful tool here to represent linear systems of equations economically and clearly. Matrices also serve as a tool in other, manifold ways. This is one reason to consider matrices in their own right, and to clearly illustrate and practice all kinds of manipulations that are possible with them: We will add, multiply, multiply, exponentiate, transpose, and invert matrices. But everything in order.",
isbn="978-3-662-65458-3",
doi="10.1007/978-3-662-65458-3_10",
url="https://doi.org/10.1007/978-3-662-65458-3_10"
}
@article{GRCAR2011163,
title = {How ordinary elimination became Gaussian elimination},
journal = {Historia Mathematica},
volume = {38},
number = {2},
pages = {163-218},
year = {2011},
issn = {0315-0860},
doi = {10.1016/j.hm.2010.06.003},
url = {https://www.sciencedirect.com/science/article/pii/S0315086010000376},
author = {{Grcar, J. F.}},
keywords = {Algebra before 1800, Gaussian elimination, Human computers, Least squares method, Mathematics education},
abstract = {Newton, in notes that he would rather not have seen published, described a process for solving simultaneous equations that later authors applied specifically to linear equations. This method — which Euler did not recommend, which Legendre called “ordinary,” and which Gauss called “common” — is now named after Gauss: “Gaussian” elimination. Gauss’s name became associated with elimination through the adoption, by professional computers, of a specialized notation that Gauss devised for his own least-squares calculations. The notation allowed elimination to be viewed as a sequence of arithmetic operations that were repeatedly optimized for hand computing and eventually were described by matrices.
Zusammenfassung
In Aufzeichnungen, die Newton lieber nicht der Veröffentlichung preisgegeben hätte, beschreibt er den Prozess für die Lösung von simultanen Gleichungen, den spätere Autoren speziell für lineare Gleichungen anwandten. Diese Methode — welche Euler nicht empfahl, welche Legendre “ordinaire” nannte, und welche Gauß “gewöhnlich” nannte — wird nun nach Gauß benannt: Gaußsches Eliminationsverfahren. Die Verbindung des Gaußschen Namens mit Elimination wurde dadurch hervorgebracht, dass professionelle Rechner eine Notation übernahmen, die Gauß speziell für seine eigenen Berechnungen der kleinsten Quadrate ersonnen hatte, welche zuließ, das Elimination als eine Sequenz von arithmetischen Rechenoperationen betrachtet wurde, die wiederholt für Handrechnungen optimisiert wurden und schließlich durch Matrizen beschrieben wurden.}
}