| Curl: |
The curl of a vector field
\[
\nabla \times {\bf F} = \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} \right) {\bf i} - \left( \frac{\partial F_z}{\partial x} - \frac{\partial F_x}{\partial z} \right) {\bf j} + \left( \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right) {\bf k}
\]
is a vector field that describes the local rate of rotation or shear.
|
| 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. |
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 ΠB
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.
