Matrices and Linear Algebra on the Earliest Uses Pages

This is an INDEX to the Matrix and Linear Algebra entries on Jeff Miller’s Earliest Uses Pages

Earliest Known Uses of Some of the Words of Mathematics

Earliest Uses of Various Mathematical Symbols

These pages cover all branches of mathematics. The Words page is organised alphabetically with separate (large) files for each letter (printing out C would take 50 pages, but it’s exceptionally big!) and the Symbols page is organised by subject. The Symbols page has a section Symbols for Matrices and Vectors.

The following is a list of entries on the Words pages. It is only a rough guide to those pages because it is hard to draw a line between matrix terms and general mathematical terms. Matrix indicates there is material on the Symbols for Matrices and Vectors page and Calculus that there is material on the Symbols of Calculus .page.

There are also indexes of terms used in Vector analysis and the one for Probability and Statistics, one of the many fields in which matrices are used. For a general perspective on word-formation in mathematics see my Mathematical Words: Origins and Sources.

Entries

A – B A Adjoint matrix Affine Associative Augmented matrix B Basis Bra and ket vectors	C C Canonical correlation Canonical form Cauchy-Schwarz inequality Cayley-Hamilton theorem Characteristic value etc. Cholesky decomposition etc. Commutative Companion matrix Condition number etc. Cramer’s rule Cross product Matrix Curl Calculus	D – E D Del Calculus Design matrix Matrix Determinant Matrix Distributive Divergence Calculus Dot product Matrix E Eigenvalue Elementary operations Eliminant Elimination Equivalent matrix
F – H F Field G Gauss-Jordan Gaussian elimination Matrix Generalized inverse Matrix Gradient Calculus Gram-Schmidt Group H Hadamard product Hat matrix Helmert transformation Hermitian matrix Hessian Hilbert space Householder transformation Hyper determinant	I – K I Idempotent Identity matrix Matrix Inner product Matrix Inverse Matrix J Jacobian Jordan canonical form K Kernel Kronecker product Matrix	L - N L Laplace expansion Latent value etc. Law of inertia Leverage Linear algebra Linear combination Linear dependence Linear equation Matrix Geometry Linear transformation M Markov chain Matrix Matrixx Matrix mechanics Method of least squares Matrix Minor Moore-Penrose inverse Matrix Multicollinearity N Nilpotent Norm Matrix Normal matrix Nullity Null space
O – P O Orthogonal matrix Orthogonal vectors (see perpendicularity Geometry Outer product P Pauli matrices Permanent Perron-Frobenius theorem Pfaffian Pivot Positive definite Projection Proper value etc.	Q – R Q QR algorithm Quadratic form Quaternion R Rank Rayleigh quotient Regression Matrix Relaxation	S S Scalar Matrix Scalar product Matrix Schmidt othogonalization Schur complement Schur product Secular equation Signature Similar matrix Simultaneous equations Matrix Singular matrix Singular value decomposition Skew symmetric matrix Space Spectrum Square matrix
T T Tensor Trace Transpose Matrix Triangle inequality	U – V U Unitary matrix V Vandermonde determinant Vector Matrix Vector space	W – Z W Wronskian X Y Z Zero matrix Matrix

History

Although matrices as abstract objects were introduced in the 19^th century, historians, such as Katz (ch. 15.5), often begin their account of matrices with the Chinese scholars of the Han period (200-100 BC) who solved linear equations by means of Gaussian elimination. Much of today’s matrix theory was developed in the 18^th and 19^th centuries as determinant theory; the history of that subject is followed contribution by contribution by Muir. More matrix theory was ‘concealed’ in Hamilton’s quaternion analysis from which developed the vector analysis of Gibbs; this history is given by Crowe. Another relevant parallel story was Die Ausdehnungslehre of Grassmann. The principal architects of matrix theory in the 19^th century were Cayley and Frobenius. Monographs on matrix theory in English began to appear in the 1930s and those by MacDuffee and Wedderburn have many historical references. Later the formulations used in functional analysis, particularly in Hilbert space theory, influenced presentations of the theory of finite-dimensional vector spaces. Kline (ch. 33) begins his account of matrices by emphasising their convenience rather than their mathematical significance and, indeed beginning in the 1920s with the matrix mechanics formulation of quantum mechanics, matrices were adopted in many subjects. Matrices were particularly prominent in the post-war development of numerical analysis; the books by Householder and Farebrother contain much valuable historical information.

References

• Morris Kline Mathematical Thought from Ancient to Modern Times. Oxford University Press, 1972.
• Victor J. Katz, A History of Mathematics: An Introduction. Harper Collins, 1993, or Addison Wesley, 1998.
• Thomas Muir The Theory of Determinants in the Historical Order of Development. Four volumes, 1906-24 (reprinted by Dover Books 1960). A supplement published in 1930 covers 1900-1920.
• M. J. Crowe A History of Vector Analysis .1967. Reprinted by Dover Books 1987.
• C. C. MacDuffee, The Theory of Matrices, Springer 1933.
• J. H. M. Wedderburn Lectures on Matrices AMS 1934.
• A. S. Householder The Theory of Matrices in Numerical Analysis (1964). Reprinted by Dover Books 1975/2006.
• R. W. Farebrother Linear Least Squares Computations Marcel Dekker 1988.

See also

• MacTutor history: Matrices and determinants
• MacTutor history: Abstract linear spaces
• Marie A. Vituli: A Brief History of Linear Algebra and Matrix Theory
• Alessandro Perotti History of Linear Algebra
• SIAM conference on the history of numerical linear algebra Oct. 2009
• Gene Herman: Glossary of matrix algebra terms
• TUU: Glossar zur Linearen Algebra
• Rainer Baule: Mathematisches Wörterbuch
• Roger Kraft: Linear Algebra on the Internet.

John Aldrich, University of Southampton, Southampton, UK. (home). Most recent changes June 2010.