Appendix I. Glossary

This glossary summarizes the main terms used throughout the book. Definitions are stated briefly and emphasize the meaning most relevant to linear algebra.

A

Affine Space

A set obtained by translating a vector subspace. An affine space does not necessarily contain the zero vector.

Algebraic Multiplicity

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

Alternating Form

A multilinear form that changes sign when two arguments are exchanged and becomes zero when two arguments are equal.

Augmented Matrix

A matrix formed by appending the right-hand side vector (b) to the coefficient matrix (A) of a system

$$ Ax=b. $$

B

Backward Error

The size of the perturbation needed to make a computed solution exact for a nearby problem.

Basis

A linearly independent spanning set for a vector space.

Bilinear Form

A function

$$ B:V\times V\to F $$

that is linear in each argument separately.

Block Matrix

A matrix partitioned into submatrices treated as single units.

C

Canonical Form

A standard representative chosen from a class of equivalent matrices or transformations.

Characteristic Polynomial

The polynomial

$$ p_A(\lambda)=\det(\lambda I-A). $$

Its roots are the eigenvalues of (A).

Cholesky Decomposition

A factorization

$$ A=LL^T $$

$$ A=LL^* $$

for positive definite matrices.

Column Space

The span of the columns of a matrix.

Companion Matrix

A matrix associated with a monic polynomial whose characteristic polynomial equals that polynomial.

Complex Conjugate

For

$$ z=a+bi, $$

the conjugate is

$$ \overline{z}=a-bi. $$

Condition Number

A measure of sensitivity of a problem to perturbations in the input.

Coordinate Vector

The vector of coefficients expressing a vector relative to a chosen basis.

D

Determinant

A scalar associated with a square matrix that measures invertibility, signed volume scaling, and orientation change.

Diagonal Matrix

A matrix whose off-diagonal entries are all zero.

Diagonalizable Matrix

A matrix similar to a diagonal matrix.

Dimension

The number of vectors in a basis of a vector space.

Direct Sum

A decomposition of a vector space into subspaces with trivial intersection.

E

Eigenvalue

A scalar (\lambda) such that

$$ Av=\lambda v $$

for some nonzero vector (v).

Eigenvector

A nonzero vector satisfying

$$ Av=\lambda v. $$

Eigenspace

The subspace

$$ \ker(A-\lambda I). $$

Elementary Matrix

A matrix obtained from the identity matrix by one elementary row operation.

Elementary Row Operation

One of the operations:

Operation	Meaning
Row swap	Exchange two rows
Row scaling	Multiply a row by a nonzero scalar
Row replacement	Add a multiple of one row to another

Euclidean Norm

The norm

$$ |x|_2 = \sqrt{x_1^2+\cdots+x_n^2}. $$

F

Field

A set with addition, subtraction, multiplication, and division by nonzero elements satisfying the field axioms.

Forward Error

The difference between a computed solution and the exact solution.

Frobenius Norm

The matrix norm

$$ |A|F = \sqrt{ \sum{i,j}|a_{ij}|^2 }. $$

G

Gaussian Elimination

An algorithm for solving linear systems using elementary row operations.

Geometric Multiplicity

The dimension of the eigenspace associated with an eigenvalue.

Gram Matrix

A matrix of inner products:

$$ G_{ij}=\langle v_i,v_j\rangle. $$

Gram-Schmidt Process

An algorithm that converts a linearly independent set into an orthonormal set.

H

Hermitian Matrix

A complex matrix satisfying

$$ A^*=A. $$

Hessenberg Matrix

A nearly triangular matrix used in eigenvalue algorithms.

Householder Transformation

A reflection used in QR factorization and orthogonalization algorithms.

I

Identity Matrix

The square matrix (I) with ones on the diagonal and zeros elsewhere.

Image

The set of outputs of a function or linear transformation.

Independent Set

A set of vectors whose only linear relation is the trivial relation.

Inner Product

A function

$$ \langle \cdot,\cdot\rangle $$

that generalizes dot products and defines lengths and angles.

Invertible Matrix

A square matrix (A) with a matrix (A^{-1}) satisfying

$$ AA^{-1}=A^{-1}A=I. $$

Isomorphism

A bijective linear transformation.

Iterative Method

An algorithm that approaches a solution through repeated approximation.

J

Jacobian Matrix

The matrix of first partial derivatives of a vector-valued function.

Jordan Block

A matrix of the form

$$ \begin{bmatrix} \lambda & 1 & 0 & \cdots & 0\ 0 & \lambda & 1 & \cdots & 0\ \vdots & \vdots & \vdots & \ddots & \vdots\ 0 & 0 & 0 & \cdots & \lambda \end{bmatrix}. $$

Jordan Canonical Form

A block diagonal matrix built from Jordan blocks and similar to the original matrix.

K

Kernel

The set

$$ \ker(T)={v:T(v)=0}. $$

Krylov Subspace

A subspace generated by vectors

$$ v,Av,A^2v,\ldots. $$

L

Least Squares Problem

An optimization problem minimizing

$$ |Ax-b|^2. $$

Linear Combination

An expression of the form

$$ c_1v_1+\cdots+c_nv_n. $$

Linear Dependence

A relation among vectors where a nontrivial linear combination equals zero.

Linear Independence

The condition that only the trivial linear combination equals zero.

Linear Map

Another term for linear transformation.

Linear System

A collection of linear equations.

Linear Transformation

A function preserving vector addition and scalar multiplication.

LU Decomposition

A factorization

$$ A=LU $$

with (L) lower triangular and (U) upper triangular.

M

Matrix

A rectangular array of scalars.

Matrix Exponential

The matrix function

$$ e^A = I+A+\frac{A^2}{2!}+\cdots. $$

Matrix Norm

A function measuring matrix size.

Minimal Polynomial

The monic polynomial of smallest degree satisfying

$$ m(A)=0. $$

Multilinear Map

A function linear in each argument separately.

N

Nilpotent Matrix

A matrix (A) such that

$$ A^k=0 $$

for some positive integer (k).

Normal Equation

The equation

$$ A^TAx=A^Tb $$

associated with least squares problems.

Normal Matrix

A matrix satisfying

$$ A^A=AA^. $$

Norm

A function measuring vector length or size.

Null Space

Another term for kernel.

Numerical Stability

The property that rounding errors do not grow excessively during computation.

O

Orthogonal Matrix

A real matrix satisfying

$$ Q^TQ=I. $$

Orthogonal Complement

The set of vectors orthogonal to a given set.

Orthogonal Projection

The closest-point projection onto a subspace.

Orthogonality

The condition

$$ \langle u,v\rangle=0. $$

Orthonormal Basis

A basis consisting of mutually orthogonal unit vectors.

P

Partial Pivoting

A row-swapping strategy used in Gaussian elimination for stability.

Permutation Matrix

A matrix obtained by permuting the rows of the identity matrix.

Pivot

A leading nonzero entry used during elimination.

Positive Definite Matrix

A symmetric or Hermitian matrix satisfying

$$ x^TAx>0 $$

$$ x^*Ax>0 $$

for all nonzero (x).

Projection

A linear transformation satisfying

$$ P^2=P. $$

Pseudoinverse

A generalized inverse, often the Moore-Penrose inverse.

Q

QR Decomposition

A factorization

$$ A=QR $$

with (Q) orthogonal or unitary and (R) upper triangular.

Quadratic Form

An expression

$$ x^TAx. $$

R

Rank

The dimension of the image or column space of a matrix.

Reduced Row Echelon Form

A canonical row-equivalent matrix form satisfying specific pivot conditions.

Residual

The vector

$$ r=b-A\widehat{x} $$

for an approximate solution (\widehat{x}).

Row Echelon Form

A triangular-like matrix form obtained during elimination.

Row Space

The span of the rows of a matrix.

S

Scalar

An element of the underlying field.

Schur Decomposition

A factorization

$$ A=QTQ^* $$

with (Q) unitary and (T) upper triangular.

Singular Matrix

A noninvertible square matrix.

Singular Value

The square root of an eigenvalue of

$$ A^*A. $$

Singular Value Decomposition

A factorization

$$ A=U\Sigma V^*. $$

Sparse Matrix

A matrix with many zero entries.

Span

The set of all linear combinations of a collection of vectors.

Spectral Radius

The maximum absolute value of the eigenvalues of a matrix.

Spectral Theorem

A theorem describing diagonalization of symmetric or Hermitian matrices by orthogonal or unitary matrices.

Subspace

A subset closed under vector addition and scalar multiplication.

Symmetric Matrix

A real matrix satisfying

$$ A^T=A. $$

T

Tensor Product

A construction combining vector spaces into a larger multilinear structure.

Trace

The sum of the diagonal entries of a square matrix.

Transformation Matrix

A matrix representing a linear transformation relative to chosen bases.

Transpose

The matrix obtained by interchanging rows and columns.

Triangular Matrix

A matrix with all entries above or below the diagonal equal to zero.

U

Unitary Matrix

A complex matrix satisfying

$$ U^*U=I. $$

Upper Triangular Matrix

A matrix whose entries below the diagonal are zero.

V

Vandermonde Matrix

A matrix of the form

$$ \begin{bmatrix} 1 & x_1 & x_1^2 & \cdots \ 1 & x_2 & x_2^2 & \cdots \ \vdots & \vdots & \vdots & \ddots \end{bmatrix}. $$

Vector

An element of a vector space.

Vector Space

A set with vector addition and scalar multiplication satisfying the vector space axioms.