Glossary · Mathematical Methods I

Alphabetical reference for the PHY621 wiki.

Adjoint (\(A^\dagger\)) — Conjugate transpose; defined relative to an inner product by \(\langle A^\dagger u, v\rangle = \langle u, Av\rangle\). → Hermitian matrices

Basis — An independent spanning set. Every vector then has a unique component expansion. → Vector spaces

Characteristic equation — \(\det(A-\lambda I) = 0\); its roots are the eigenvalues. → Characteristic equation

Completeness relation — \(\sum_n|n\rangle\langle n| = \hat1\). Insert it anywhere to introduce a basis; every expansion in the course is one application. → Dirac notation

Convolution — \((f*g)(x) = \int f(x')g(x-x')\,dx'\). Becomes multiplication under transform. → Convolution

Defective matrix — One whose geometric multiplicity is less than its algebraic multiplicity; not diagonalisable. Normal matrices never are. → Eigenvalues & eigenvectors

Degeneracy — Repeated eigenvalues. Orthogonality within the eigenspace must be constructed (Gram–Schmidt), not assumed. → Hermitian matrices

Diagonalization — \(A = SDS^{-1}\) with \(D\) the eigenvalues. Makes functions of operators trivial. → Diagonalization

Dirac delta — A distribution defined by \(\int\delta(x-x')f(x')\,dx' = f(x)\); the continuous Kronecker delta. Not a function. → Dirac delta

Eigenvector — A direction the operator only scales: \(A\mathbf{v} = \lambda\mathbf{v}\). → Eigenvalues & eigenvectors

Fourier series — Expansion of a periodic function in sines and cosines; a basis expansion in an infinite-dimensional inner-product space. → Fourier series

Fourier transform — The continuous limit; equivalently, a change of basis from position to momentum. → Fourier transform

Generalised eigenproblem — \(B\mathbf{a} = \lambda A\mathbf{a}\) with \(A\) positive definite. Its eigenvectors are \(A\)-orthogonal, not orthogonal in the plain dot product. → Normal modes

Gibbs phenomenon — The persistent ~9% overshoot of a truncated Fourier series near a jump. → Fourier series

Green's function — The response to a unit impulse: \(\mathcal{L}G = \delta\). The inverse of a differential operator, written as an integral kernel. → Green's functions

Hermitian — \(A = A^\dagger\). Guarantees real eigenvalues and an orthogonal eigenbasis. → Hermitian matrices

Inner product — The structure supplying length, angle and orthogonality. A choice, not a given — the physically correct one is problem-dependent. → Inner product spaces

Laplace transform — \(\tilde f(s) = \int_0^\infty f(t)e^{-st}dt\); converts initial-value ODEs into algebra. → Laplace transform

Normal matrix — \(AA^\dagger = A^\dagger A\). The exact condition for a complete orthonormal eigenbasis; Hermitian, anti-Hermitian and unitary are all special cases. → Hermitian matrices

Normal mode — An oscillation in which every part moves at one frequency; an eigenvector of the generalised eigenproblem. → Normal modes

Null space (kernel) — \(\{\mathbf{x}: A\mathbf{x} = 0\}\). Physically: zero modes, gauge freedom, non-uniqueness of solutions. → Rank–nullity

Orthonormal — \(\langle e_i, e_j\rangle = \delta_{ij}\); makes component extraction a projection. → Inner product spaces

Parseval's theorem — Norm is preserved under transform: total energy is the same in either basis. → Fourier transform

Rank — Dimension of the range; equals the number of independent rows and of independent columns. → Rank–nullity

Rank–nullity theorem — \(\dim(\ker) + \mathrm{rank} = n\). Settles the solvability of \(A\mathbf{x} = \mathbf{b}\) completely. → Rank–nullity

Schwarz inequality — \(|\langle a,b\rangle| \le |a||b|\). Three lines to prove; yields the triangle inequality and the uncertainty principle. → Inner product spaces

Separation of variables — Reducing a PDE to ODEs by assuming a product solution; works when the geometry matches the coordinate system. → Partial differential equations

Special functions — Legendre, Bessel, Hermite and friends: eigenfunctions of the operators that arise in the standard geometries, orthogonal under their own weight functions. → Special functions

Spectral decomposition — \(A = \sum_n\lambda_n|n\rangle\langle n|\). Turns \(f(A)\) into \(\sum f(\lambda_n)|n\rangle\langle n|\). → Hermitian matrices

Span — The set of all linear combinations of a given set. → Vector spaces

Transfer function — The Fourier/Laplace transform of a Green's function; the response of a linear system per unit input at each frequency. → Convolution

Vector space — A set closed under addition and scalar multiplication, with the usual axioms. Arrows, polynomials, ODE solutions and quantum states all qualify. → Vector spaces

Zero mode — A null vector of an operator; physically, a motion costing no energy, usually protected by a symmetry. → Rank–nullity