# Glossary - D

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

decoupled system: A difference equation ykC1 D Ayk , or a differential equation y0 .t/ D Ay.t/, in which A is a diagonal matrix. The discrete evolution of each entry in yk (as a function of k ), or the continuous evolution of each entry in the vector-valued function y.t /, is unaffected by what happens to the other entries as k ! 1 or t ! 1. design matrix: The matrix X in the linear model y D Xˇ C !, where the columns of X are determined in some way by the observed values of some independent variables. determinant (of a square matrix A): The number det A deﬁned inductively by a cofactor expansion along the ﬁrst row of A. Also, ."1/r times the product of the diagonal entries in any echelon form U obtained from A by row replacements and r row interchanges (but no scaling operations). diagonal entries (in a matrix): Entries having equal row and column indices. diagonalizable (matrix): A matrix that can be written in factored form as PDP !1 , where D is a diagonal matrix and P is an invertible matrix. diagonal matrix: A square matrix whose entries not on the main diagonal are all zero. difference equation (or linear recurrence relation): An equation of the form xkC1 D Axk (k D 0; 1; 2; : : :) whose solution is a sequence of vectors, x0 ; x1 ; : : : : dilation: A mapping x 7! r x for some scalar r , with 1 < r . dimension: of a ﬂat S : The dimension of the corresponding parallel subspace. of a set S : The dimension of the smallest ﬂat containing S . of a subspace S : The number of vectors in a basis for S , written as dim S . of a vector space V : The number of vectors in a basis for V , written as dim V . The dimension of the zero space is 0. discrete linear dynamical system: A difference equation of the form xkC1 D Axk that describes the changes in a system (usually a physical system) as time passes. The physical system is measured at discrete times, when k D 0; 1; 2; : : : ; and the state of the system at time k is a vector xk whose entries provide certain facts of interest about the system. distance between u and v: The length of the vector u " v, denoted by dist .u; v/. distance to a subspace: The distance from a given point (vector) v to the nearest point in the subspace. distributive laws: (left) A.B C C / D AB C AC , and (right) .B C C /A D BA C CA, for all A, B , C . domain (of a transformation T ): The set of all vectors x for which T .x/ is deﬁned. dot product: See inner product. dynamical system: See discrete linear dynamical system.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. |