# References — Chapter ᚈ(T): The caboose *Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#ref-ch-10, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.* [Guide for AI readers](index.md) · [The chapter](ch10.md) *Each entry: one plain sentence saying exactly what it is cited for, its status, then every source with a note saying what that source actually carries. "Own computation" / "Own reading" entries are the author's own work, not outside sources.* ### 10.1 {#r-10.1} The Nine Chapters used an array-based elimination method for systems of simultaneous linear equations, and Gauss later used the method in orbit determination. *Status: confirmed* - [MacTutor, Nine Chapters on the Mathematical Art](https://mathshistory.st-andrews.ac.uk/HistTopics/Nine_chapters/) — Chapter 8 describes systems of up to six simultaneous linear equations, an augmented array, and reduction to triangular form by elementary column operations. - [MacTutor, Matrices and determinants](https://mathshistory.st-andrews.ac.uk/HistTopics/Matrices_and_determinants/) — says Gaussian elimination appeared in the Nine Chapters and was used by Gauss for the orbit of Pallas; “The first to use the term ‘matrix’ was Sylvester in 1850..” ### 10.2 {#r-10.2} Sylvester introduced the word matrix in 1850; Cayley published the first matrix inverse in 1853 and the first abstract definition of a matrix, with addition, multiplication, scalar multiplication and inverses, in his memoir of 1858. *Status: confirmed* - [MacTutor, Matrices and determinants](https://mathshistory.st-andrews.ac.uk/HistTopics/Matrices_and_determinants/) — “The first to use the term ‘matrix’ was Sylvester in 1850”; “by 1853 Cayley had published a note giving, for the first time, the inverse of a matrix”; and the 1858 memoir “is remarkable for containing the first abstract definition of a matrix,” giving addition, multiplication, scalar multiplication and inverses. - [MacTutor, James Joseph Sylvester](https://mathshistory.st-andrews.ac.uk/Biographies/Sylvester/) — links Sylvester’s first use of the word matrix to his 1850 paper. ### 10.3 {#r-10.3} For a linear map, the determinant is the area/volume scale factor; a zero determinant means the map is singular and has no inverse. *Status: confirmed* - [Wikipedia, Determinant](https://en.wikipedia.org/wiki/Determinant) — describes the determinant as the signed scale factor for volume and states that a matrix is invertible exactly when its determinant is nonzero. - **Own computation** (the author's own work) — For A = [[1, 2], [2, 4]], det A = 1·4 − 2·2 = 0; the second row is twice the first, so the image is the line spanned by (1, 2), and no inverse can exist. ### 10.4 {#r-10.4} For A = [[1, 2], [2, 4]], the displayed determinant, trace, Frobenius norm, eigenvalues, rank, nullity and Cayley-Hamilton substitution have the stated values. *Status: confirmed* - **Own computation** (the author's own work) — Direct calculation: det A = 0; tr A = 5; AᵀA has eigenvalues 25 and 0, so the Frobenius norm is 5; the characteristic polynomial is λ²−5λ, giving eigenvalues 5 and 0; im A = span(1, 2) has dimension 1 and ker A = span(2, −1) has dimension 1; A² = [[5, 10], [10, 20]] = 5A. ### 10.5 {#r-10.5} The Kronecker delta is 1 when its indices agree and 0 otherwise; the identity matrix has these delta entries; the notation is traced to Kronecker’s paper on bilinear forms of 1868. Sources differ on the year: the Encyclopedia of Mathematics dates Kronecker’s first use of the symbol to 1866, while the paper the notation is usually traced to was printed in 1868. Corrections welcome. *Status: confirmed* - [Wikipedia, Kronecker delta](https://en.wikipedia.org/wiki/Kronecker_delta) — defines δᵢⱼ as 1 for i=j and 0 otherwise and gives the identity matrix as the matrix of Kronecker deltas, and carries no history section, so the date below rests on the paper itself. - [L. Kronecker, Ueber bilineare Formen, Journal für die reine und angewandte Mathematik 68 (1868), 273–285](https://eudml.org/doc/148042) — the European Digital Mathematics Library record for the paper the delta notation is standardly traced to, which dates it 1868 (volume 68, pages 273–285). - [Kronecker symbol, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Kronecker_symbol) — “The Kronecker symbol was first used by L. Kronecker (1866).” ### 10.6 {#r-10.6} The trace is the sum of diagonal entries and, for a square matrix, the sum of the eigenvalues counted with multiplicity; the English word translates the German Spur, which survives in the older abbreviation sp A. *Status: confirmed* - [Wikipedia, Trace (linear algebra)](https://en.wikipedia.org/wiki/Trace_(linear_algebra)) — defines trace as the sum of diagonal entries and states that it equals the sum of eigenvalues with algebraic multiplicity. - [Earliest Known Uses of Some of the Words of Mathematics, T](https://mathshistory.st-andrews.ac.uk/Miller/mathword/t/) — gives trace as a translation of the German die Spur, used in H. L. Brose’s 1922 translation of Weyl’s Raum, Zeit, Materie, and records Aitken’s “the spur or trace of A. We shall denote it by sp A.” ### 10.7 {#r-10.7} The Frobenius norm is the square root of the sum of squared entries and also the square root of the sum of squared singular values. *Status: confirmed* - [Wikipedia, Matrix norm](https://en.wikipedia.org/wiki/Matrix_norm) — gives the Frobenius norm as the square root of the sum of squared entries and as the Euclidean norm of the singular-value vector. ### 10.8 {#r-10.8} The eigenvalues of A are the roots of det(λI−A), and for this symmetric matrix the eigenvectors for 5 and 0 are perpendicular; the word eigen is Hilbert’s, from 1904, and the idea is older: Euler studied the rotation of a rigid body and found the principal axes, and Lagrange recognised them as the eigenvectors of the inertia matrix. *Status: confirmed* - [Wikipedia, Eigenvalues and eigenvectors](https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors) — defines eigenvalues through Av=λv and the characteristic polynomial det(A−λI). Re-read raw on 17 September 2026 for the history: “In the 18th century, Leonhard Euler studied the rotational motion of a rigid body, and discovered the importance of the principal axes. Joseph-Louis Lagrange realized that the principal axes are the eigenvectors of the inertia matrix.” It does not state the symmetric-matrix orthogonality property — it says “The eigenvectors need not be orthogonal to one another” — so that half is carried by the entry below; an eigenvalue is “the multiplying factor λ (possibly a negative or complex number),” and there is “a direct correspondence between linear transformations from the vector space into itself and n × n square matrices.” - [Symmetric matrix, Wikipedia](https://en.wikipedia.org/wiki/Symmetric_matrix) — (§ Real symmetric matrices): “For every real symmetric matrix A, there exists a real orthogonal matrix Q such that D = QᵀAQ is a diagonal matrix,” and derives that eigenvectors for distinct eigenvalues satisfy ⟨x,y⟩ = 0. - **Own computation** (the author's own work) — A(1, 2)ᵀ = 5(1, 2)ᵀ and A(2, −1)ᵀ = 0; their dot product is 2−2=0. - [Earliest Known Uses of Some of the Words of Mathematics, E](https://mathshistory.st-andrews.ac.uk/Miller/mathword/e/) — David Hilbert introduced Eigenfunktion and Eigenwert in 1904, in “Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen”; Eigenvektor follows in Courant and Hilbert, 1924. ### 10.9 {#r-10.9} A nilpotent matrix has a power equal to zero; N=[[0, 1], [0, 0]] is nonzero with N²=0, and Jordan’s 1870 canonical-form work decomposes matrices using a nilpotent part. Jordan first stated his decomposition theorem in 1870; every complex matrix is D + N with D diagonalizable, N nilpotent and DN = ND. *Status: confirmed* - [Wikipedia, Nilpotent matrix](https://en.wikipedia.org/wiki/Nilpotent_matrix) — defines nilpotence by Aᵏ=0 for some positive k and gives the 2×2 Jordan nilpotent block as the standard example. - [MacTutor, Matrices and determinants](https://mathshistory.st-andrews.ac.uk/HistTopics/Matrices_and_determinants/) — dates Jordan’s canonical form to 1870: “In 1870 the Jordan canonical form appeared in Treatise on substitutions and algebraic equations by Jordan.” The nilpotent part is on the Jordan normal form line. - [Jordan normal form, Wikipedia](https://en.wikipedia.org/wiki/Jordan_normal_form) — “The Jordan normal form is named after Camille Jordan, who first stated the Jordan decomposition theorem in 1870.”; “Any M can be written as a sum D + N where D is semisimple, N is nilpotent, and DN = ND.” ### 10.10 {#r-10.10} Rank is the dimension of the image, nullity the dimension of the kernel, and rank plus nullity equals the dimension of the domain. Sources differ on the year of the word rank: MacTutor places Frobenius’s rank in his matrix paper of 1878, while Miller dates the word Rang to his paper in J. reine angew. Math. 86 (1879). Corrections welcome. *Status: confirmed* - [MathWorld, Rank-Nullity Theorem](https://mathworld.wolfram.com/Rank-NullityTheorem.html) — states dim(V)=dim(Ker(T))+dim(Im(T)) and identifies these dimensions as nullity and rank. - [MacTutor, Matrices and determinants](https://mathshistory.st-andrews.ac.uk/HistTopics/Matrices_and_determinants/) — says Sylvester defined nullity in 1884 and Frobenius defined rank in his 1878 matrix work. - [Jeff Miller, Earliest Known Uses of Some of the Words of Mathematics (R), MacTutor](https://mathshistory.st-andrews.ac.uk/Miller/mathword/r/) — “RANK (of a determinant or matrix) was coined by F. G. Frobenius, who used the German word Rang in his paper ‘Uber homogene totale Differentialgleichungen,’ J. reine angew. Math. Vol. 86 (1879) p.1.” ### 10.11 {#r-10.11} The Cayley-Hamilton theorem says every square matrix satisfies its characteristic polynomial; Hamilton proved a case for quaternions in 1853, Cayley handled the 2×2 case and stated the general one in 1858, and Frobenius proved the general result in 1878. *Status: confirmed* - [MathWorld, Cayley-Hamilton Theorem](https://mathworld.wolfram.com/Cayley-HamiltonTheorem.html) — states A is annihilated by det(xI−A) and works out the 2×2 identity. - [MacTutor, Matrices and determinants](https://mathshistory.st-andrews.ac.uk/HistTopics/Matrices_and_determinants/) — dates Cayley’s 2×2/3×3 result and Frobenius’s general proof to 1878, and notes Hamilton’s special quaternion case. - [Cayley–Hamilton theorem, Wikipedia](https://en.wikipedia.org/wiki/Cayley%E2%80%93Hamilton_theorem) — “A special case of the theorem was first proved by Hamilton in 1853 in terms of inverses of linear functions of quaternions.”; “Cayley in 1858 stated the result for 3 × 3 and smaller matrices, but only published a proof for the 2 × 2 case.”; “The general case was first proved by Ferdinand Frobenius in 1878.” ### 10.12 {#r-10.12} The identity matrix acts as a multiplicative identity; an inverse exists exactly for a nonsingular matrix; transposition swaps rows and columns, and this A is symmetric. The identity matrix has ones on the diagonal and zeros elsewhere, written δᵢⱼ; a real symmetric matrix’s eigenvectors for distinct eigenvalues are perpendicular. *Status: confirmed* - [Wikipedia, Invertible matrix](https://en.wikipedia.org/wiki/Invertible_matrix) — states a square matrix is invertible exactly when its determinant is nonzero and defines the inverse equation AA⁻¹=I. - **Own computation** (the author's own work) — I·A=A, Aᵀ=A because A is symmetric, and A has det A=0, so it has no inverse. - [Identity matrix, Wikipedia](https://en.wikipedia.org/wiki/Identity_matrix) — “the identity matrix of size n is the n × n square matrix with ones on the main diagonal and zeros elsewhere”; “When A is an m × n matrix, it is a property of matrix multiplication that I_m A = A I_n = A.”; “The identity matrix can also be written using the Kronecker delta notation: (I_n)_ij = δ_ij.” - [Transpose, Wikipedia](https://en.wikipedia.org/wiki/Transpose) — “transposition is an operation that flips a matrix over its diagonal; that is, transposition switches the row and column indices of the matrix A to produce another matrix, called the transpose of A.” - [Symmetric matrix, Wikipedia](https://en.wikipedia.org/wiki/Symmetric_matrix) — “To see orthogonality, suppose x and y are eigenvectors corresponding to distinct eigenvalues λ₁, λ₂.” … “Since λ₁ and λ₂ are distinct, we have ⟨x, y⟩ = 0.” ### 10.13 {#r-10.13} Singular values are square roots of eigenvalues of AᵀA; the SVD factors a matrix as UΣVᵀ, and for this rank-one A the sole nonzero singular value is 5 and equals the Frobenius norm; Beltrami and Jordan reached the factorization independently in 1873 and 1874, and Golub and Kahan gave the first stable algorithm for it in 1965. *Status: confirmed* - [Wikipedia, Singular value decomposition](https://en.wikipedia.org/wiki/Singular_value_decomposition) — defines singular values through AᵀA and gives A=UΣVᵀ, with applications including least squares, principal component analysis and image compression. - **Own computation** (the author's own work) — AᵀA = [[5, 10], [10, 20]] has eigenvalues 25 and 0, so the singular values are 5 and 0; the Frobenius norm is therefore 5. - [Nicholas J. Higham, What Is the Singular Value Decomposition?](https://nhigham.com/2020/10/13/what-is-the-singular-value-decomposition/) — “The SVD was introduced independently by Beltrami in 1873 and Jordan in 1874,” and credits Golub with the first reliable algorithms for computing it. Editorial note: Several names are in circulation for one idea. G. W. Stewart’s history of the decomposition (SIAM Review 35, 1993) treats Beltrami in 1873 and Jordan in 1874 as the independent first derivations, with Sylvester, Schmidt and Weyl arriving at it again before the numerical work of the 1960s. The card names the first two and the 1965 algorithm, and leaves the rediscoveries out. ### 10.14 {#r-10.14} Cramer’s rule expresses each unknown using a determinant ratio, and it cannot give a unique solution when det A=0; Maclaurin was teaching it by ~1729 and Cramer published it in 1750. See 10.23. Corrected 11 September 2026: this entry previously gave Maclaurin 1748, which is the posthumous printing of his Treatise of Algebra, two years after his death. *Status: confirmed* - [MacTutor, Gabriel Cramer](https://mathshistory.st-andrews.ac.uk/DSB/Cramer.pdf) — discusses Cramer’s 1750 rule and Maclaurin’s equivalent idea in the 1748 Treatise of Algebra. - [Wikipedia, Cramer’s rule](https://en.wikipedia.org/wiki/Cramer%27s_rule) — gives xᵢ=det(Aᵢ)/det(A) and states the rule applies when the determinant is nonzero. ### 10.15 {#r-10.15} A vector space supports addition and scalar multiplication, a basis is a linearly independent spanning set, and dimension is the common cardinality of its bases; ℝ² has dimension 2 and polynomial space has countably infinite dimension; Peano gave the first axiomatic definition of a vector space in 1888. A spanning set is a basis exactly when it is minimal. *Status: confirmed* - [Wikipedia, Vector space](https://en.wikipedia.org/wiki/Vector_space) — gives the vector-space axioms, basis definition and dimension, including the standard basis of R². - **Own computation** (the author's own work) — The monomials 1,x,x²,… form a countable basis of the polynomial vector space over a field, while (1, 0),(0, 1) is a basis of R². - [MacTutor, Abstract linear spaces](https://mathshistory.st-andrews.ac.uk/HistTopics/Abstract_linear_spaces/) — “The first to give an axiomatic definition of a real linear space was Peano in a book published in Torino in 1888,” the Calcolo geometrico, which Peano wrote after Grassmann’s 1844 Ausdehnungslehre. - [Basis (linear algebra), Wikipedia](https://en.wikipedia.org/wiki/Basis_(linear_algebra)) — “A generating set S is a basis of V if and only if it is minimal, that is, no proper subset of S is also a generating set of V.” ### 10.16 {#r-10.16} The outer product u⊗v has entries uᵢvⱼ; A=(1, 2)⊗(1, 2), and the three-dimensional cross product vanishes for parallel vectors and has parallelogram-area magnitude; Gibbs introduced the dot and the cross in 1881, and Heaviside split the quaternion product the same way independently. The calculus of higher-rank tensors is Ricci-Curbastro’s, developed from 1887 to 1896 and set out with his pupil Levi-Civita in 1900. Gibbs’s 1881 notes were privately printed; the name dot product appears in Gibbs and Wilson’s Vector Analysis of 1901, and for two nonzero vectors the dot product is zero exactly when they are perpendicular. A matrix has rank one exactly when it is a nonzero outer product of two vectors. *Status: confirmed* - [Wikipedia, Outer product](https://en.wikipedia.org/wiki/Outer_product) — defines the outer product as the matrix with entries uᵢvⱼ and notes that rank-one matrices arise this way. - [Ricci calculus, Wikipedia](https://en.wikipedia.org/wiki/Ricci_calculus) — read as raw HTML: the absolute differential calculus was “developed by Gregorio Ricci-Curbastro in 1887–1896, and subsequently popularized in a paper written with his pupil Tullio Levi-Civita in 1900,” the paper being Ricci and Levi-Civita, Méthodes de calcul différentiel absolu et leurs applications, Mathematische Annalen 54 (March 1900), 125–201, doi 10.1007/BF01454201. - [Wikipedia, Cross product](https://en.wikipedia.org/wiki/Cross_product) — defines the cross product in three dimensions, gives its magnitude as parallelogram area, states it is zero for parallel vectors, and dates the notation a · b and a × b to Gibbs in 1881, with Heaviside independently. - **Own computation** (the author's own work) — (1, 2)⊗(1, 2)=[[1, 2], [2, 4]], and (1, 2, 0)×(2, 4, 0)=(0, 0, 0). - [Josiah Willard Gibbs, Wikipedia](https://en.wikipedia.org/wiki/Josiah_Willard_Gibbs) — “Gibbs’s lecture notes on vector calculus were privately printed in 1881 and 1884 for the use of his students, and were later adapted by Edwin Bidwell Wilson into a textbook, Vector Analysis, published in 1901.” - [Jeff Miller, Earliest Known Uses of Some of the Words of Mathematics (D), MacTutor](https://mathshistory.st-andrews.ac.uk/Miller/mathword/d/) — DOT PRODUCT, found in 1901 in Gibbs and Wilson’s Vector Analysis: “A·B This is read A dot B and therefore may often be called the dot product instead of the direct product.” - [Dot product, Wikipedia](https://en.wikipedia.org/wiki/Dot_product) — “Two non-zero vectors a and b are orthogonal if and only if a ⋅ b = 0” - [Rank (linear algebra), Wikipedia](https://en.wikipedia.org/wiki/Rank_(linear_algebra)) — “a matrix is defined to have rank 1 if and only if it can be written as a nonzero product c ⋅ r of a column vector c and a row vector r” ### 10.17 {#r-10.17} The characteristic polynomial p_A(λ)=det(λI−A) has eigenvalues as roots; for a 2×2 matrix its coefficients recover trace and determinant, and Cauchy used the polynomial in 1829 and named the characteristic equation in 1840. *Status: confirmed* - [Wikipedia, Characteristic polynomial](https://en.wikipedia.org/wiki/Characteristic_polynomial) — defines det(λI−A), identifies its roots as eigenvalues, and gives the 2×2 coefficient relation λ²−tr(A)λ+det(A). - [MacTutor, Matrices and determinants](https://mathshistory.st-andrews.ac.uk/HistTopics/Matrices_and_determinants/) — records the historical development of determinant and matrix theory, including Cauchy’s work on determinants. - [Earliest Known Uses of Some of the Words of Mathematics, E](https://mathshistory.st-andrews.ac.uk/Miller/mathword/e/) — Cauchy’s 1829 paper on symmetric determinants is “Sur l’équation à l’aide de laquelle on détermine les inégalités séculaires des mouvements des planètes,” and he introduced the term l’équation caractéristique in the Exercices d’analyse et de physique mathématique of 1840. ### 10.18 {#r-10.18} A linear map is written f:V→W, composition means doing f then g, and matrix multiplication represents composition; A∘A=A²=5A here. *Status: confirmed* - [Wikipedia, Linear map](https://en.wikipedia.org/wiki/Linear_map) — defines linear maps between vector spaces and their composition. - [MacTutor, Matrices and determinants](https://mathshistory.st-andrews.ac.uk/HistTopics/Matrices_and_determinants/) — “Cayley gave a matrix algebra defining addition, multiplication, scalar multiplication and inverses” (1858); earlier Gauss “describes matrix multiplication (which he thinks of as composition so he has not yet reached the concept of matrix algebra),” for the arrays of quadratic forms, not linear maps. - **Own computation** (the author's own work) — A²=[[5, 10], [10, 20]]=5A, so the composition A∘A is represented by 5A. Editorial note: Cayley’s 1858 memoir is where matrix multiplication is defined so that it represents composition. The barred arrow ↦ is Bourbaki’s, but no reference work consulted dates it: the accounts that give a date put it in the Théorie des ensembles of the early 1960s rather than in Bourbaki’s first decade, so the card gives the arrow neither an author nor a year. ### 10.19 {#r-10.19} Two matrix-valued zeros in the numerator are evaluated in their own matrix algebra and then contribute the integer 0 to the scalar factor. Chapter ᚁ(B) applies the same editorial rule to its six laws: each zero-valued law contributes the plain number 0 to the displayed sum. *Status: judgement* - **Own computation** (the author's own work) — N² and A²−5A are the zero matrices. The chapter’s displayed scalar factor treats each matrix equality as contributing its zero value to the surrounding numerical sum; this is an editorial evaluation rule, not a theorem about adding unlike mathematical types. Editorial note: This is a declared convention of the book rather than a theorem, and the disclaimer page states it: every factor is computed inside its own framework and hands back an ordinary finite real before it joins the product. Adding a zero matrix to a zero integer is not an operation supplied by the frameworks used here, so the chapter says what it is doing where its first matrix-valued zeros appear; the later chapters whose zeros are vectors, forms, operators or numbers of some other algebra rest on the same rule. ### 10.20 {#r-10.20} The determinant appears in Seki’s work of 1683 and in Leibniz’s letter to de l’Hôpital of 1693; Gauss introduced the word determinant in 1801 for a different object, and Cauchy used it in its modern sense in 1812. *Status: confirmed* - [MacTutor, Matrices and determinants](https://mathshistory.st-andrews.ac.uk/HistTopics/Matrices_and_determinants/) — “In 1683 Seki wrote Method of solving the dissimulated problems which contains matrix methods written as tables”; “In 1693 Leibniz wrote to de l’Hôpital” giving the condition for a system to have a solution; and “It was Cauchy in 1812 who used ‘determinant’ in its modern sense,” Gauss having introduced the term in 1801 while discussing quadratic forms. ### 10.21 {#r-10.21} The word matrix is Latin for womb, and Sylvester introduced it into mathematics in 1850 for the oblong array out of which systems of determinants are formed. *Status: confirmed* - [Wiktionary, matrix](https://en.wiktionary.org/wiki/matrix) — the Latin entry glosses matrix as “uterus, womb,” then “dam (non-human female animal kept for breeding)” and “source, origin,” formed from mater; the English noun comes “from Old French matrice (“pregnant animal”), or from Latin mātrīx (“dam, womb”),” and its English sense “The womb” is marked “now rare.” - [Earliest Known Uses of Some of the Words of Mathematics, M](https://mathshistory.st-andrews.ac.uk/Miller/mathword/m/) — “The term MATRIX was introduced into mathematics by James Joseph Sylvester (1814-1897) in 1850. Matrix was a long-established word with the meaning of ‘the place from which something else originates.’ For Sylvester the ‘something else’ was a determinant of some description,” quoting his 1850 Philosophical Magazine paper: an oblong arrangement of m lines and n columns “will not in itself represent a determinant, but is, as it were, a Matrix out of which we may form various systems of determinants.” ### 10.22 {#r-10.22} Chapter 8 of the Nine Chapters, Rectangular Arrays, gives eighteen problems solved by column elimination on an array of coefficients; most historians date the text to about 200 BC, with 206 BC to 50 AD covering its different parts; and Gauss devised his notation for symmetric elimination in 1810, having obtained six linear equations in six unknowns from observations of the asteroid Pallas taken between 1803 and 1809. *Status: confirmed* - [MacTutor, Nine Chapters on the Mathematical Art](https://mathshistory.st-andrews.ac.uk/HistTopics/Nine_chapters/) — “Most historians think that the text originated around 200 BC after the burning of the books by Shih Huang Ti,” while “others give dates between 100 BC and 50 AD,” the page settling on “206 BC to 50 AD” for its different parts; chapter 8 gives “18 problems which reduce to solving systems of simultaneous linear equations,” solved with “the augmented matrix of coefficients,” which “is then reduced to triangular form, using elementary column operations as is done today in the method of Gaussian elimination.” - [Wikipedia, Gaussian elimination](https://en.wikipedia.org/wiki/Gaussian_elimination) — names the source as “Chapter Eight: Rectangular Arrays of The Nine Chapters on the Mathematical Art,” where “its use is illustrated in eighteen problems, with two to five equations,” and states that “Carl Friedrich Gauss in 1810 devised a notation for symmetric elimination that was adopted in the 19th century by professional hand computers to solve the normal equations of least-squares problems.” - [MacTutor, Matrices and determinants](https://mathshistory.st-andrews.ac.uk/HistTopics/Matrices_and_determinants/) — “Using observations of Pallas taken between 1803 and 1809, Gauss obtained a system of six linear equations in six unknowns. Gauss gave a systematic method for solving such equations which is precisely Gaussian elimination on the coefficient matrix”; the same page gives no year for that work, which is why the 1810 date is carried by the entry above. ### 10.23 {#r-10.23} Colin Maclaurin was teaching the rule now called Cramer’s rule to his students by 1729, more than twenty years before Cramer published it in 1750; the Treatise of Algebra that carries it was published in 1748, two years after Maclaurin’s death. *Status: confirmed* - [Bruce A. Hedman, An Earlier Date for “Cramer’s Rule,” Historia Mathematica 26 (1999) 365-368](http://professorhedman.com/Cramers.Rule.pdf) — read in full here. Abstract: “Carl B. Boyer has shown that Colin Maclaurin published ‘Cramer’s Rule’ two years before Gabriel Cramer and conjectured that Maclaurin knew the technique as early as 1729. This note establishes Boyer’s conjecture through a heretofore unpublished manuscript in the Edinburgh University Library.” The manuscript is a handwritten copy of the Treatise of Algebra made by John Russell and dated 1729 (Edinburgh University Library, shelfmark Dc. 3.66); its twelfth chapter carries the same form of the rule as the 1748 edition, and Hedman concludes that it “provides conclusive evidence that Maclaurin was teaching his students ‘Cramer’s Rule’ over 20 years before Cramer published it.” Hedman also records that Patrick Murdoch prepared the Treatise for posthumous publication in 1748. - [MacTutor, Colin Maclaurin](https://mathshistory.st-andrews.ac.uk/Biographies/Maclaurin/) — gives Maclaurin’s dates as February 1698 to 14 June 1746 and states that “Maclaurin’s Treatise on algebra was published in 1748, two years after his death.” Editorial note: Added 11 September 2026. The page previously said “Cramer published it in 1750 and Maclaurin had an equivalent rule in 1748,” which dates Maclaurin two years after his own death and names the later man first. Under D1 the prose now takes the earliest defensible year and the first name: Maclaurin, ~1729. ### 10.24 {#r-10.24} An orthogonal projection is a linear map P with P² = P whose kernel is orthogonal to its range, equivalently a self-adjoint idempotent; it sends each vector to the nearest point of the subspace it projects onto, so the error is perpendicular to that subspace. For this chapter’s A = [[1, 2], [2, 4]], A/5 is the orthogonal projection onto the line y = 2x. *Status: confirmed* - [Projection (linear algebra), Wikipedia](https://en.wikipedia.org/wiki/Projection_(linear_algebra)) — read as raw HTML: defines a projection as a linear map with P² = P, and an orthogonal projection as one whose range and kernel are orthogonal complements, equivalently a projection that is self-adjoint; gives the nearest-point and least-squares readings. - **Own computation** (the author's own work) — For A = [[1, 2], [2, 4]] and u = (1, 2): uuᵀ = A and |u|² = 5, so A/5 = uuᵀ/|u|² is symmetric and (A/5)² = A²/25 = 5A/25 = A/5; its range is the line y = 2x, spanned by (1, 2), and its kernel, spanned by (2, −1), is perpendicular to it. So A/5 is the orthogonal projection onto y = 2x, and A = 5·(A/5) has determinant 0, rank 1 and eigenvalues 5 and 0. ### 10.25 {#r-10.25} The spectral theorem for a real symmetric matrix says it is diagonalizable by an orthogonal matrix: there is an orthonormal basis of eigenvectors and the eigenvalues are real. Augustin-Louis Cauchy proved the spectral theorem for symmetric matrices, and John von Neumann generalized it to operators on a Hilbert space. Spectral theory underlies quantum measurement: it attaches a probability measure to an observable’s values in any state. *Status: confirmed* - [Spectral theorem, Wikipedia](https://en.wikipedia.org/wiki/Spectral_theorem) — read as raw HTML: “Augustin-Louis Cauchy proved the spectral theorem for symmetric matrices, i.e., that every real, symmetric matrix is diagonalizable,” and “The spectral theorem as generalized by John von Neumann is today perhaps the most important result of operator theory.” - [Spectral Theorem, MathWorld](https://mathworld.wolfram.com/SpectralTheorem.html) — the statement for a normal bounded operator on a Hilbert space, via a unique resolution of the identity on the Borel subsets of its spectrum. - [Mathematical formulation of quantum mechanics, Wikipedia](https://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics) — “By spectral theory, we can associate a probability measure to the values of A in any state ψ.” ### 10.26 {#r-10.26} Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Sylvester’s ‘womb of a common parent’ definition of a matrix, in ‘On the relation between the minor determinants of linearly equivalent quadratic functions’, 1851. *Status: confirmed* - [The Collected Mathematical Papers of James Joseph Sylvester, vol. I (Cambridge, 1904), Internet Archive full text](https://archive.org/download/collectedmathem01sylvrich/collectedmathem01sylvrich_djvu.txt) — read raw: “37. ON THE RELATION BETWEEN THE MINOR DETERMINANTS OF LINEARLY EQUIVALENT QUADRATIC FUNCTIONS. [Philosophical Magazine, I. (1851), pp. 295 — 305.].” Editorial note: Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim. ### 10.27 {#r-10.27} States that the Nine Chapters on the Mathematical Art used column-elimination methods on an array of coefficients, and that Gauss used the same method in 1810 for orbit computations. Notes that the Nine Chapters reached its final form in the 1st century AD, and that Gauss’s 1810 work solved six equations for the orbit of the asteroid Pallas. *Status: confirmed* - [Gaussian elimination, Wikipedia](https://en.wikipedia.org/wiki/Gaussian_elimination) — traces the method to the Chinese text Nine Chapters on the Mathematical Art and to Gauss’s 1810 notation for symmetric elimination; “The first reference to the book by this title is dated to 179 AD, but parts of it were written as early as approximately 150 BC.” - [The Nine Chapters on the Mathematical Art, Wikipedia](https://en.wikipedia.org/wiki/The_Nine_Chapters_on_the_Mathematical_Art) — carries the dating: “composed by several generations of scholars from the 10th–2nd century BCE, its latest stage being from the 1st century CE.” - [Nine chapters on the Mathematical Art, MacTutor](https://mathshistory.st-andrews.ac.uk/HistTopics/Nine_chapters/) — carries the column operations: “The matrix is then reduced to triangular form, using elementary column operations as is done today in the method of Gaussian elimination.” - [The Quest for Linear Equation Solvers, John L. Gustafson](https://ed-thelen.org/ABC/QuestLinearSolvers-.html) — carries Gauss and Pallas: “To solve the problem of predicting the orbit of Pallas, he had to reduce astronomical data to a least-squares problem… involving six equations in six unknowns.” ### 10.30 {#r-10.30} Defines least squares as minimizing the sum of squared residuals and states its characterization by the normal equations, crediting Legendre 1805 and Gauss’s earlier claimed use. Notes that the residual is orthogonal to the column space of A, that Gauss claimed use since 1795 while Legendre keeps priority of publication, and that orthogonal methods such as the SVD compute it more stably. *Status: confirmed* - [Least squares, Wikipedia](https://en.wikipedia.org/wiki/Least_squares) — defines the least-squares solution, gives the normal equations, and credits Legendre’s 1805 publication and Gauss’s priority claim. - [Linear least squares, Wikipedia](https://en.wikipedia.org/wiki/Linear_least_squares) — carries the orthogonality: “the matrix X(XᵀX)⁻¹Xᵀ represents orthogonal projection onto the column space of X.” - [Legendre, Adrien-Marie, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Legendre,_Adrien-Marie) — carries the priority verdict: “The claim for priority made by Gauss at the time would seem to have been substantiated in the opinion of Laplace, Plackett (1972), Stigler (1977, 1981, 1986), and other authorities, but Legendre retains the priority of publication” - [Numerical methods for linear least squares, Wikipedia](https://en.wikipedia.org/wiki/Numerical_methods_for_linear_least_squares) — carries the stability comparison: “Orthogonal decomposition methods of solving the least squares problem are slower than the normal equations method but are more numerically stable because they avoid forming the product XᵀX.” ### 10.33 {#r-10.33} Defines a zero divisor as a ring element that multiplies with another nonzero element to give zero, and states that this fails in the integers, reals and complex numbers but occurs in rings like the integers mod 6. Notes that the rationals, quaternions and octonions also lack nonzero zero divisors, that 2 × 3 = 0 modulo 6, that they first appear at the sedenions in the Cayley–Dickson ladder and persist above, and that a nonzero matrix squaring to zero is one. *Status: confirmed* - [Zero divisor, Wikipedia](https://en.wikipedia.org/wiki/Zero_divisor) — defines zero divisors, with the example Z/4Z, where the class of 2 is a zero divisor since 2 × 2 = 4 = 0, and “A nonzero commutative ring whose only zero divisor is 0 is called an integral domain.” The mod 6 example is on the Domain line. - [Octonion, Wikipedia](https://en.wikipedia.org/wiki/Octonion) — carries the division algebras and the sedenions: “ℝ, ℂ, or ℍ, and 𝕆 are the only normed division algebras over the real numbers. … The higher-dimensional algebras defined by the Cayley–Dickson construction (starting with the sedenions) all fail to satisfy this property. They all have zero divisors.” - [Domain (ring theory), Wikipedia](https://en.wikipedia.org/wiki/Domain_(ring_theory)) — carries the mod 6 example: “The ring Z/6Z is not a domain, because the images of 2 and 3 in this ring are nonzero elements with product 0.” - [Sedenion, Wikipedia](https://en.wikipedia.org/wiki/Sedenion) — carries the persistence above the sedenions: “All hypercomplex number systems after sedenions that are based on the Cayley–Dickson construction also contain zero divisors.”