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

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

Affine

Associative

Augmented matrix

## B

Basis

Bra and ket vectors

## 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

Del  Calculus

Design matrix Matrix

Determinant  Matrix

Distributive

Divergence Calculus

Dot product  Matrix

## E

Eigenvalue

Elementary operations

Eliminant

Elimination

Equivalent matrix

Field

## G

Gauss-Jordan

Gaussian elimination Matrix

Generalized inverse  Matrix

Gram-Schmidt

Group

Hat matrix

Helmert transformation

Hermitian matrix

Hessian

Hilbert space

Householder transformation

Hyper determinant

## I – K

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  Geometry

Linear transformation

Markov chain

Matrix  Matrixx

Matrix mechanics

Method of least squares  Matrix

Minor

Moore-Penrose inverse   Matrix

Multicollinearity

Nilpotent

Norm  Matrix

Normal matrix

Nullity

Null space

# Outer product

## P

Pauli matrices

Permanent

Perron-Frobenius theorem

Pfaffian

Pivot

Positive definite

Projection

Proper value etc.

QR algorithm

Quaternion

# Rank

Rayleigh quotient

Regression Matrix

Relaxation

## S

Scalar

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

Tensor

Trace

Transpose Matrix

Triangle inequality

Unitary matrix

## V

Vandermonde determinant

Vector  Matrix

Vector space

Wronskian

## Z

Zero matrix  Matrix

History

Although matrices as abstract objects were introduced in the 19th 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 18th and 19th 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 19th 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.

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