Linear Algebra

Recall (doc/postdoc). Not a first course — spectral and structure language as used in papers.

Job: finite-dimensional vector spaces and linear maps as matrices; canonical forms and spectral theory as the entrance to functional analysis, representation theory, and numerical linear algebra.

Core

Linear algebra is the classification of \(T\in L(V)\) up to change of basis (Jordan / rational / spectral forms) and the geometry of bilinear and sesquilinear forms. Dimension, rank, and determinant are the basic invariants; over \(\mathbb{C}\) or algebraically closed fields, almost everything reduces to eigenvalues and generalized eigenspaces.

Layer Content
Spaces / maps \(V\) over \(K\); \(L(V,W)\); bases; change of basis \(P^{-1}AP\)
Matrix algebra mult., invertibility, block form
Determinants \(\det\in\bigwedge^n V^*\); volume / characteristic poly
Spectral eigenvalues, diagonalization, Jordan, Cayley–Hamilton
Inner product orthonormal bases; self-adjoint / normal / unitary (spectral thm)
Multilinear exit tensors, exterior algebra \(\to\) manifolds / reps

Reading map

Page Focus
Vectors \(\mathbb{R}^n\) geometry: dot, cross, proj
Matrices \(L(K^n,K^m)\) coordinates; multiplication
Determinants \(\det\); volume / invertibility criteria
Matrix inversion \(A^{-1}\); Cramer; adjugate
Vector spaces basis, dim, rank–nullity
Eigenvalues \(\chi_A\); diagonalization; Jordan
Functional analysis Infinite-dim operators; spectrum
Representation theory Linear actions of groups / Lie algebras

Dependency order

  1. Vectors / Matrices \(\to\) Vector spaces.
  2. Determinants \(\to\) Eigenvalues (characteristic poly needs det).
  3. Inner-product spectral theory \(\to\) FA / Lie groups.

Interfaces

Area Contact
Abstract algebra Modules over PID \(\Rightarrow\) rational/Jordan form; \(\operatorname{End}_K(V)\)
Differential geometry Tangent maps \(Df\); metrics as positive-definite forms
PDE / DE Linear systems \(\dot x=Ax\); fundamental matrix \(e^{tA}\)
Quantum Observables as self-adjoint operators (finite-dim prototype)
Numerical analysis Conditioning, SVD, iterative eigensolvers

Notation (paper-standard)

  • \(L(V,W)\), \(\operatorname{End}(V)\), \(\mathrm{GL}(V)\), \(\mathrm{SL}\), \(\mathrm{O}\), \(\mathrm{U}\)
  • \(\operatorname{rank}\), \(\operatorname{nullity}\), \(\chi_A(t)=\det(tI-A)\), \(m_A\) minimal poly
  • \(\langle\cdot,\cdot\rangle\), \(A^*\), \(A^\dagger\); SVD \(A=U\Sigma V^*\)
  • \(V\otimes W\), \(\bigwedge^k V\), \(\operatorname{tr}\), \(\det\)

Density (invariants)

\[ \operatorname{rank}A+\operatorname{nullity}A=n,\quad Av=\lambda v,\quad \det(AB)=\det A\det B,\quad A=PDP^{-1}\ \text{(diag.\ case)},\quad \chi_A(A)=0 \]

Structure theorem (one-liner): over a field, \(V\) as \(K[t]\)-module via \(T\) decomposes into cyclic primary summands \(\Leftrightarrow\) rational form; over algebraically closed \(K\), Jordan form.

See it

Jordan vs diagonalizable. \(N=\begin{pmatrix}0&1\\0&0\end{pmatrix}\) has \(\chi_N(t)=t^2\) and \(N\neq0\), so not diagonalizable; generalized eigenspace for \(\lambda=0\) is all of \(K^2\). In dynamical systems, \(\dot x=Nx\) produces linear (secular) growth \(x(t)=(I+tN)x_0\) — the finite-dim prototype of Jordan blocks in linear PDE and Floquet theory.

Check

Over algebraically closed fields, is every matrix diagonalizable?

No — Jordan blocks for repeated eigenvalues with deficient geometric multiplicity (e.g. nonzero nilpotents). Diagonalizable \(\Leftrightarrow\) minimal polynomial square-free.

Why does \(\det(e^A)=e^{\operatorname{tr}A}\)?

\(\chi\)-continuity / triangularization over \(\overline{K}\): eigenvalues of \(e^A\) are \(e^{\lambda_i}\) and \(\det=\prod e^{\lambda_i}=e^{\sum\lambda_i}=e^{\operatorname{tr}A}\). Holds for real \(A\) by complexification.

Pitfalls

  • Algebraic vs geometric multiplicity: \(\dim\ker(A-\lambda I)\le\) algebraic mult.; equality for all \(\lambda\) \(\Leftrightarrow\) diagonalizable.
  • Real vs complex spectral theory: real matrices may need complex eigenvalues or \(2\times2\) rotation blocks (real canonical form).
  • Conditioning of eigenproblems: non-normal matrices can have pseudospectra far from eigenvalues; \(\|e^{tA}\|\) not controlled by \(\operatorname{Re}\sigma(A)\) alone without constants.

Refs

  • Axler, Linear Algebra Done Right (operator view)
  • Horn–Johnson, Matrix Analysis
  • Lax, Linear Algebra and Its Applications
  • Roman, Advanced Linear Algebra

Next

Vectors · Vector spaces · Eigenvalues · Functional analysis · Representation theory · DE foundations