Mathematical Methods I — Linear Algebra & Operators — formula sheet
Every boxed result from the 17 derivations of this unit on one page, for revision. Each title links to the full derivation, with its assumptions and where the formula breaks down. Constants: physical constants table.
Invariance of Dimension of a Vector Space
Every basis of a finitely generated vector space has the same number of elements. That common number is the dimension. It is an intrinsic property of \(V\): it survives any change of basis, any linear coordinate transformation, any choice of frame.
Gram–Schmidt Orthonormalization
Process the vectors in order; at each stage strip off everything the new vector already shares with the directions you have fixed, then scale what remains to unit length. What survives is orthogonal to all predecessors by construction, and nonzero precisely because the inputs were independent. The nested spans (the "flag") are preserved exactly — a triangular relationship between the \(v\)'s and the \(e\)'s that is the \(R\) of the \(QR\) decomposition.
Cauchy–Schwarz and the Triangle Inequality
The overlap of two vectors can never exceed the product of their lengths; the closer \(|\langle u,v\rangle|\) is to \(\|u\|\|v\|\), the more nearly parallel they are, reaching equality exactly when one is a scalar multiple of the other. The triangle inequality then says a straight path is never longer than a detour: the length of a sum is at most the sum of the lengths.
Existence and Uniqueness of the Adjoint
Moving \(A\) across the inner product costs exactly one dagger, and the operator you land on is completely determined by \(A\) and the inner product — no choices, no basis. In an orthonormal basis the dagger is realized as conjugate-transpose, \((A^{\dagger})_{ij}=\overline{A_{ji}}\), but the operator itself predates any basis.
Riesz Representation in Finite Dimensions
Every linear "read-out" you can perform on a finite-dimensional inner-product space is secretly the same as projecting onto one fixed vector and reading its length-weighted overlap. The dual space \( V^* \) is not merely abstractly the same size as \( V \) (both have dimension \( n \)) — it is canonically realised inside \( V \) once the inner product is chosen, the identification independent of which orthonormal basis built \( \mathbf{r}_\varphi \).
Properties of Hermitian Operators
The eigenvectors form an orthonormal basis. Any state can be expanded as |ψ⟩ = Σ cn|an⟩ with cn = ⟨an|ψ⟩ uniquely determined, and Σ|cn|² = 1. The probability interpretation depends entirely on this.
Determinant as the Unique Alternating Multilinear Form
There is one and only one scalar function of a square matrix that scales linearly with each column, flips sign when two columns swap (hence vanishes on dependent columns), and equals \(1\) on the identity. Because "apply \(B\), then take that unique form" is itself such a form, it must be a constant multiple of \(\det\), and the constant is \(\det B\) — giving the multiplicative law for free.
Eigenvalues as Roots of the Characteristic Polynomial
The eigenvalues of \(A\) are exactly the roots of one degree-\(n\) polynomial built from \(A\), and that polynomial is identical in every basis. The spectrum is therefore an intrinsic fingerprint of the operator: similar matrices share the same characteristic polynomial, hence the same eigenvalues with the same algebraic multiplicities. The invariant coefficients \(\operatorname{tr}A=\sum_i\lambda_i\) and \(\det A=\prod_i\lambda_i\) are frame-independent scalars.
Diagonalizability and the Eigenbasis Criterion
An operator can be reduced to independent scalings on \(n\) axes exactly when it supplies enough eigenvectors to build a basis. Two things can go wrong and only two: the eigenvalues may live outside the field (\(p\) does not split), or a repeated eigenvalue may be "defective," its eigenspace \(E_{\lambda_i}\) smaller than the root multiplicity \(a_i\). If neither happens, the eigenspaces tile the whole space as a direct sum. Repetition of an eigenvalue is permitted — the identity is diagonal with a single eigenvalue of multiplicity \(n\) — it is only a shortfall \(g_i<a_i\) that obstructs.
Spectral Theorem for Self-Adjoint Operators
A self-adjoint operator is nothing more than a set of real numbers \( \lambda_k \) attached to mutually orthogonal directions \( |e_k\rangle \); it stretches each direction by its eigenvalue and couples nothing. The unitary \( \hat{U} \) is the change of basis into those principal directions, and the spectral sum writes \( \hat{A} \) as a real-weighted sum of orthogonal projectors \( \hat{P}_k=|e_k\rangle\langle e_k| \) obeying \( \hat{P}_j\hat{P}_k=\delta_{jk}\hat{P}_k \) and \( \sum_k\hat{P}_k=\hat{I} \).
Unitary Operators as Inner-Product Isometries
Preserving all inner products is exactly the algebraic condition \(U^\dagger U=I\) — geometry (angles and lengths are unchanged) and algebra (the adjoint is the inverse) are the same statement. Because the length of an eigenvector cannot change, an eigenvalue can only rotate a vector by a phase, never rescale it, so it must lie on the unit circle.
Simultaneous Diagonalization of Commuting Operators
Commuting self-adjoint operators are two lists of real numbers attached to one shared set of orthogonal directions: the direction \(\lvert e_k\rangle\) is stretched by \(a_k\) under \(\hat{A}\) and by \(b_k\) under \(\hat{B}\). Each common eigenvector carries a joint eigenvalue label \((a_k,b_k)\) — the mathematical origin of a state's simultaneous quantum numbers. When one operator's spectrum is degenerate, the other operator's eigenvalue distinguishes the vectors inside that degenerate block.
Normal Modes from the Generalized Eigenvalue Problem
The system has \(n\) special patterns of motion — normal modes. In mode \(k\) every coordinate oscillates at the single frequency \(\omega_k\) with fixed relative amplitudes given by \(\mathbf{a}_k\). These patterns are "orthogonal" not in the ordinary sense but in the metric set by the masses: \(\mathbf{a}_j^{\mathsf T}\mathbf{M}\mathbf{a}_k=0\) for \(j\ne k\). Writing \(\mathbf{q}(t)=\sum_k Q_k(t)\,\mathbf{a}_k\) turns the coupled system into \(n\) independent oscillators \(\ddot Q_k+\omega_k^2 Q_k=0\).
Variational Characterization of Eigenvalues
The lowest eigenvalue is the global minimum of the Rayleigh quotient, and the \(n\)-th eigenvalue is the smallest value the quotient can be forced to reach on some \(n\)-dimensional subspace. Every eigenvector is a stationary point of \(R\); every stationary value is an eigenvalue. Consequently any normalised trial state gives \(R(\psi)\ge \lambda_1\): a one-sided, always-valid upper bound on the ground state.
Singular Value Decomposition
Any linear map is a rotation/reflection of the domain (\(V^{\dagger}\)), a non-negative rescaling along orthogonal axes (\(\Sigma\)), and a rotation/reflection into the codomain (\(U\)). The right singular vectors \(v_i\) are the principal input axes, the left singular vectors \(u_i\) their images, and \(\sigma_i\) the stretch factors; the number of non-zero \(\sigma_i\) is the rank.
Matrix Exponential and One-Parameter Flows
The matrix exponential is the propagator: it advances any initial state by a time \(t\) along the linear flow \(\dot x=Ax\), and it is the only operator that does so. Composing two advances adds their times (the group law), reversing time inverts the operator, and when the generator is skew-adjoint — the physicist's self-adjoint \(H\) with \(A=-iH/\hbar\) — the propagator is unitary, so lengths, angles, and probabilities are conserved along the flow.
Orthogonal Projections and Spectral Resolution
A self-adjoint idempotent is exactly an orthogonal projection: it splits the space into a subspace it keeps untouched and the perpendicular subspace it annihilates. The spectral theorem then writes any self-adjoint observable as a "menu" of its eigenvalues, each tagged with the projector onto the states that yield it, and those projectors tile the whole space with no overlap and no gaps. Functions of \(A\) act eigenvalue-by-eigenvalue: \(f(A)=\sum_k f(\lambda_k)P_k\).