Glossary - H

All terms beginning with the letter 'H' are shown below:

homogeneous coordinates: In R3, the representation of .x; y; ´/ as .X; Y; Z; H / for any H ¤ 0, where x D X=H , y D Y =H , and ´ D Z=H . In R2 , H is usually taken as 1, and the homogeneous coordinates of .x; y/ are written as .x; y; 1/. homogeneous equation: An equation of the form Ax D 0, possibly written as a vector equation or as a system of linear equations. ! " v homogeneous form of (a vector) v in Rn : The point vQ D 1 in RnC1 . Householder reflection: A transformation x 7! Qx, where Q D I " 2uuT and u is a unit vector .uTu D 1/. hyperplane (in Rn ): A flat in Rn of dimension n " 1. Also: a translate of a subspace of dimension n " 1.
Adjugate (or classical adjoint): The matrix adj A formed from a square matrix A by replacing the .i; j /-entry of A by the .i; j /-cofactor, for all i and j , and then transposing the resulting matrix.
Affine combination:

A linear combination of vectors (points in Rn ) in which the sum of the weights involved is 1.

Affine dependence relation:

An equation of the form c1 v1 C ! ! ! C cp vp D 0, where the weights c1 ; : : : ; cp are not all zero, and c1 C ! ! ! C cp D 0.

Affine hull (or affine span) of a set S:

The set of all affine combinations of points in S , denoted by aff S.

Affinely dependent set:

A set fv1 ; : : : ; vp g in Rn such that there are real numbers c1 ; : : : ; cp , not all zero, such that c1 C ! ! ! C cp D 0 and c1 v1 C ! ! ! C cp vp D 0.

Affinely independent set:

A set fv1 ; : : : ; vp g in Rn that is not affinely dependent.

Affine set (or affine subset):

A set S of points such that if p and q are in S , then .1 " t/p C t q 2 S for each real number t.

Affine transformation:

A mapping T W Rn ! Rm of the form T .x/ D Ax C b, with A an m # n matrix and b in Rm.

Algebraic multiplicity:

The multiplicity of an eigenvalue as a root of the characteristic equation.

Angle (between nonzero vectors u and v in R2 or R3/:

The angle # between the two directed line segments from the origin to the points u and v. Related to the scalar product by u ! v D kuk kvk cos #

Associative law of multiplication:

A.BC/ D .AB/C , for all A, B, C.

attractor (of a dynamical system in R2):

The origin when all trajectories tend toward 0.

Augmented matrix:

A matrix made up of a coefficient matrix for a linear system and one or more columns to the right. Each extra column contains the constants from the right side of a system with the given coefficient matrix.

Auxiliary equation:

A polynomial equation in a variable r, created from the coefficients of a homogeneous difference equation.