PU-104 · Mathematical Methods I — Linear Algebra & Operators
This unit builds the linear-algebraic machinery on which all of quantum mechanics, normal-mode analysis, and modern classical mechanics rest: vector spaces, inner products, linear operators, and above all the spectral theory of self-adjoint operators. Starting from abstract axioms and ending with the spectral theorem and its physical incarnations, it teaches the student to see eigenvalue problems, orthogonal decompositions, and unitary symmetries as one coherent structure.
Lectures
| L01 | What Linear Algebra Is For in Physics — |
| L02 | Vector Spaces, Subspaces, and Linear Independence |
| L03 | Bases and the Invariance of Dimension |
| L04 | Linear Maps, Matrices, and Change of Basis |
| L05 | Inner Products and Norms |
| L06 | Cauchy–Schwarz and Geometry of Hilbert Space |
| L07 | Orthogonality and Gram–Schmidt |
| L08 | Dual Spaces and the Riesz Representation |
| L09 | The Adjoint of an Operator |
| L10 | The Determinant: Volume and Multilinearity |
| L11 | The Eigenvalue Problem and the Characteristic Polynomial |
| L12 | Diagonalization and When It Fails |
| L13 | Hermitian Operators: Real Spectra, Orthogonal Modes |
| L14 | The Spectral Theorem for Self-Adjoint Operators |
| L15 | Projection Operators and the Resolution of the Identity |
| L16 | Unitary Operators and Isometries |
| L17 | Commuting Observables and Simultaneous Diagonalization |
| L18 | Functions of Operators and the Spectral Calculus |
| L19 | The Matrix Exponential and Linear Flows |
| L20 | Generators, Unitary Evolution, and Stone's Picture |
| L21 | The Rayleigh Quotient and Variational Eigenvalues |
| L22 | The Min-Max Principle and Eigenvalue Bounds |
| L23 | Coupled Oscillators and Normal Modes |
| L24 | The Generalized Eigenvalue Problem and M-Orthogonality |
| L25 | Quadratic Forms, Signature, and Principal Axes |
| L26 | The Singular Value Decomposition |
| L27 | Least Squares, Pseudoinverses, and Low-Rank Approximation |
| L28 | Tensor Products and Composite Systems |
| L29 | From Finite to Infinite Dimensions: A First Look |
| L30 | Synthesis: Operators as the Language of Physics |
Derivations homed in this unit
Invariance of Dimension of a Vector Space
Any two bases of a finite-dimensional vector space have the same cardinality, so dimension is well-defined.
Gram–Schmidt Orthonormalization
Any linearly independent set in an inner-product space can be converted into an orthonormal set spanning the same subspace.
Cauchy–Schwarz and the Triangle Inequality
The inner product bounds |⟨u,v⟩| ≤ ‖u‖‖v‖, from which the norm's triangle inequality follows.
Existence and Uniqueness of the Adjoint
Every linear operator on a finite-dimensional inner-product space has a unique adjoint defined by ⟨Au,v⟩=⟨u,A†v⟩.
Riesz Representation in Finite Dimensions
Every linear functional on a finite-dimensional inner-product space is inner product with a unique fixed vector.
Properties of Hermitian Operators
Hermitian operators have real eigenvalues and mutually orthogonal eigenvectors for distinct eigenvalues.
Determinant as the Unique Alternating Multilinear Form
The determinant is the unique normalized alternating multilinear function of columns, giving det(AB)=det(A)det(B).
Eigenvalues as Roots of the Characteristic Polynomial
The eigenvalues of an operator are exactly the roots of det(A−λI), which is basis-independent.
Diagonalizability and the Eigenbasis Criterion
An operator is diagonalizable iff its eigenvectors span the space, equivalently algebraic equals geometric multiplicity.
Spectral Theorem for Self-Adjoint Operators
Every self-adjoint operator admits an orthonormal eigenbasis and is diagonalized by a unitary transformation.
Unitary Operators as Inner-Product Isometries
An operator preserves the inner product iff it is unitary (U†U=I), and its eigenvalues lie on the unit circle.
Simultaneous Diagonalization of Commuting Operators
Two commuting self-adjoint operators share a common orthonormal eigenbasis.
Normal Modes from the Generalized Eigenvalue Problem
Coupled small oscillations reduce to the generalized problem Kx=ω²Mx with M-orthogonal normal modes.
Variational Characterization of Eigenvalues
The extrema of the Rayleigh quotient are the eigenvalues, giving the min-max principle for self-adjoint operators.
Singular Value Decomposition
Any linear map factors as A=UΣV† with orthonormal bases and non-negative singular values.
Matrix Exponential and One-Parameter Flows
exp(tA) is the unique solution operator of ẋ=Ax, with self-adjoint generators producing unitary flows.
Orthogonal Projections and Spectral Resolution
Self-adjoint idempotents are orthogonal projectors, and the spectral theorem resolves the identity into them.