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.