# Appendix I. Glossary # 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 $$ A=LL^T $$ or $$ A=LL^* $$ 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 $$ x^TAx>0 $$ or $$ x^*Ax>0 $$ 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. ## W ### Well-Conditioned Problem A problem whose solution changes little under small input perturbations. ## Z ### Zero Matrix A matrix whose entries are all zero. ### Zero Vector The additive identity element of a vector space.