# Glossary - C

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

Cauchy–Schwarz inequality: change of basis: jhu; vij $kuk!kvk for all u, v. See change-of-coordinates matrix. CONFIRMING PAGES change-of-coordinates matrix (from a basis B to a basis C): A matrix C P B that transforms B-coordinate vectors into Ccoordinate vectors: Œx!C D P Œx!B . If C is the standard basis for Rn , then C C B P is sometimes written as PB . B characteristic equation (of A): det.A " "I / D 0. characteristic polynomial (of A): det.A " "I / or, in some texts, det."I " A/. Cholesky factorization: A factorization A D RTR, where R is an invertible upper triangular matrix whose diagonal entries are all positive. closed ball (in Rn ): A set fx W kx " pk < ıg in Rn , where p is in Rn and ı > 0. closed set (in Rn ): A set that contains all of its boundary points. codomain (of a transformation T W Rn ! Rm /: The set Rm that contains the range of T . In general, if T maps a vector space V into a vector space W , then W is called the codomain of T. coefﬁcient matrix: A matrix whose entries are the coefﬁcients of a system of linear equations. cofactor: A number Cij D ."1/i Cj det Aij , called the .i; j /cofactor of A, where Aij is the submatrix formed by deleting the i th row and the j th column of A. cofactor expansion: A formula for det A using cofactors associated with one row or one column, such as for row 1: det A D a11 C11 C ! ! ! C a1n C1n column–row expansion: The expression of a product AB as a sum of outer products: col1 .A/ row1 .B/ C ! ! ! C coln .A/ rown .B/, where n is the number of columns of A. column space (of an m # n matrix A): The set Col A of all linear combinations of the columns of A. If A D Œa1 ! ! ! an !, then Col A D Span fa1 ; : : : ; an g. Equivalently, Col A D fy W y D Ax for some x in Rn g column sum: The sum of the entries in a column of a matrix. column vector: A matrix with only one column, or a single column of a matrix that has several columns. commuting matrices: Two matrices A and B such that AB D BA. compact set (in Rn ): A set in Rn that is both closed and bounded. companion matrix: A special form of matrix whose characteristic polynomial is ."1/n p."/ when p."/ is a speciﬁed polynomial whose leading term is "n . complex eigenvalue: A nonreal root of the characteristic equation of an n # n matrix. complex eigenvector: A nonzero vector x in C n such that Ax D "x, where A is an n # n matrix and " is a complex eigenvalue. component of y orthogonal to u (for u ¤ 0): The vector y!u y" u. u!u composition of linear transformations: A mapping produced by applying two or more linear transformations in succession. If the transformations are matrix transformations, say left-multiplication by B followed by left-multiplication by A, then the composition is the mapping x 7! A.B x/. condition number (of A): The quotient #1 =#n , where #1 is the largest singular value of A and #n is the smallest singular value. The condition number is C1 when #n is zero. conformable for block multiplication: Two partitioned matrices A and B such that the block product AB is deﬁned: The column partition of A must match the row partition of B. consistent linear system: A linear system with at least one solution. constrained optimization: The problem of maximizing a quantity such as xTAx or kAxk when x is subject to one or more constraints, such as xTx D 1 or xTv D 0. consumption matrix: A matrix in the Leontief input–output model whose columns are the unit consumption vectors for the various sectors of an economy. contraction: A mapping x 7! r x for some scalar r , with 0$ r \$ 1. controllable (pair of matrices): A matrix pair .A; B/ where A is n # n, B has n rows, and rank Œ B AB A2 B !!! An!1 B ! D n Related to a state-space model of a control system and the difference equation xkC1 D Axk C B uk .k D 0; 1; : : :/. convergent (sequence of vectors): A sequence fxk g such that the entries in xk can be made as close as desired to the entries in some ﬁxed vector for all k sufﬁciently large. convex combination (of points v1 ; : : : ; vk in Rn ): A linear combination of vectors (points) in which the weights in the combination are nonnegative and the sum of the weights is 1. convex hull (of a set S ): The set of all convex combinations of points in S , denoted by: conv S . convex set: A set S with the property that for each p and q in S , the line segment pq is contained in S . coordinate mapping (determined by an ordered basis B in a vector space V ): A mapping that associates to each x in V its coordinate vector Œx!B . coordinates of x relative to the basis B D f b1 ; : : : ; bn g : The weights c1 ; : : : ; cn in the equation x D c1 b1 C ! ! ! C cn bn . coordinate vector of x relative to B: The vector Œx!B whose entries are the coordinates of x relative to the basis B. covariance (of variables xi and xj , for i ¤ j ): The entry sij in the covariance matrix S for a matrix of observations, where xi and xj vary over the i th and j th coordinates, respectively, of the observation vectors. covariance matrix (or sample covariance matrix): The p # p matrix S deﬁned by S D .N " 1/!1 BB T , where B is a p # N matrix of observations in mean-deviation form. CONFIRMING PAGES Cramer’s rule: A formula for each entry in the solution x of the equation Ax D b when A is an invertible matrix. cross-product term: A term cxi xj in a quadratic form, with i ¤ j. cube: A three-dimensional solid object bounded by six square faces, with three faces meeting at each vertex.
 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.