Quantitative Analysis glossary

Defined only for matrices of the same order: add or subtract corresponding elements. Multiplying a matrix by a scalar (an ordinary number) multiplies every element by it.

Taught in Functions and matrix algebra

One whose highest power is x cubed. Its graph has an S shape with up to two turning points and one point of inflexion, and up to three real roots. Total cost curves in economics are cubic: costs rise steeply at first, flatten as efficiencies are gained, then rise steeply again as capacity is strained, which produces the U-shaped average and marginal cost curves.

Taught in Functions and matrix algebra

Recording quantities by product and region and multiplying by a price vector to get revenues by region; multiplying a bill-of-materials matrix by a production plan to get raw material requirements; and, in linear programming, holding the coefficients of a linear programme for the simplex method.

Taught in Functions and matrix algebra

The product AB exists only when the number of columns in A equals the number of rows in B; the element in row i and column j of the product is the sum of the products of row i of A with column j of B, element by element. The product of an m x n matrix and an n x p matrix is m x p. Matrix multiplication is not commutative: AB and BA are in general different, and one may exist when the other does not.

Taught in Functions and matrix algebra

A system of linear equations is written as A x equals b, where A holds the coefficients, x the unknowns and b the constants; then x equals A inverse times b. This is how a computer solves the normal equations of regression and the constraints of a linear programme.

Taught in Functions and matrix algebra

The least-squares estimates of a regression with several explanatory variables are given by a matrix formula, the inverse of X transpose X, times X transpose y, where X is the matrix of observations on the explanatory variables and y the vector of observations on the dependent variable; the multiple regression in the chapter on regression is computed this way.

Taught in Functions and matrix algebra

The transpose of A, written A with a superscript T, is obtained by writing the rows as columns, so an m x n matrix becomes n x m; the transpose of a product is the product of the transposes in reverse order.

Taught in Functions and matrix algebra

The types are:

• A row matrix has one row and a column matrix one column (both are vectors).
• A square matrix has as many rows as columns.
• A diagonal matrix is square with zeros everywhere except the leading diagonal.
• The identity matrix I is diagonal with ones on the diagonal, and multiplying by it leaves a matrix unchanged.
• A zero (null) matrix has every element zero.
• A symmetric matrix equals its own transpose.
• A singular matrix is a square matrix whose determinant is zero and which therefore has no inverse.

Taught in Functions and matrix algebra