Chapter 25 — Further Reading

Annotated pointers for going deeper on diagonalization, organized by how they complement this chapter. The three standard textbooks are abbreviated Strang (Introduction to Linear Algebra), Axler (Linear Algebra Done Right), and Boyd–Vandenberghe (Introduction to Applied Linear Algebra, free online).

Core textbook treatments

  • Strang, Introduction to Linear Algebra, §6.2 "Diagonalizing a Matrix." The closest companion to this chapter. Strang derives $A = PDP^{-1}$ (he writes it $A = S\Lambda S^{-1}$), emphasizes the power formula $A^k = S\Lambda^k S^{-1}$, and works the Fibonacci and Markov examples in his characteristically concrete, application-driven style. His §6.1 on eigenvalues and §6.3 on systems of differential equations bracket this material; read §6.2 first, then §6.3 for the continuous (matrix-exponential) sequel we preview in §25.7. Strang's MIT OpenCourseWare lectures (18.06, Lectures 21–23) cover exactly this arc on video, free.

  • Axler, Linear Algebra Done Right, Chapter 5 "Eigenvalues, Eigenvectors, and Invariant Subspaces." The rigorous, operator-theoretic view. Axler treats diagonalizability as the existence of a basis of eigenvectors abstractly (without leaning on determinants), and his Theorem on "conditions equivalent to diagonalizability" is the precise §25.3 criterion stated several ways, including the distinct-eigenvalues sufficient condition. Best read after this chapter if you want the cleanest proofs and the direct-sum-of-eigenspaces picture of the Math-Major Sidebar. Axler's deliberate avoidance of determinants makes the why of diagonalizability especially transparent.

  • Boyd & Vandenberghe, Introduction to Applied Linear Algebra (free at stanford.edu/~boyd/vmls). The applications-first complement. Their treatment of eigenvalues centers on dynamical systems $\mathbf{x}_{n+1} = A\mathbf{x}_n$ and Markov chains, matching this chapter's emphasis on powers and long-run behavior. Strong on the engineering reading of the dominant eigenvalue as a growth/decay rate. Free PDF, with companion Julia/Python language exercises.

On the applications in this chapter

  • Linear recurrences and the characteristic equation (Case Study 2). Any discrete mathematics text — e.g. Rosen, Discrete Mathematics and Its Applications, §8.2 — presents the "characteristic-equation method" for solving recurrences. Read it alongside §25.5 to see that the mechanical rule (write $\lambda^2 = a\lambda + b$, find roots, form $c_1\lambda_1^n + c_2\lambda_2^n$) is the diagonalization of the companion matrix. The repeated-root case (the $n\lambda^n$ term) is the discrete face of the defective/Jordan phenomenon.

  • Markov chains and steady states (Case Study 1). Grinstead & Snell, Introduction to Probability (free from the AMS / Dartmouth), Chapter 11, gives the probabilistic theory of finite Markov chains — stationary distributions, the role of the $\lambda = 1$ eigenvalue, and the Perron–Frobenius theorem that guarantees a unique positive steady state for an irreducible chain. This is the rigorous backing for "every column-stochastic matrix has eigenvalue $1$" and connects directly to PageRank (Chapter 29).

  • Population models (Exercise 25). Hal Caswell, Matrix Population Models is the standard reference for Leslie/Lefkovitch matrices in ecology, where the dominant eigenvalue is the central quantity (the asymptotic growth rate) and its eigenvector is the stable stage distribution. A vivid real-world instance of §25.6.1.

Forward-looking and historical

  • The Spectral Theorem (Chapter 27). Strang §6.4 "Symmetric Matrices" and Axler Chapter 7 "Operators on Inner Product Spaces" are the next step: the special, orthogonal diagonalization of symmetric/Hermitian matrices that this chapter's key-takeaways tease. Worth previewing now to see why symmetric matrices are the well-behaved case.

  • Functions of matrices and the matrix exponential (§25.7, Chapter 37). Nicholas Higham, Functions of Matrices: Theory and Computation is the definitive (advanced) treatment of $f(A) = Pf(D)P^{-1}$ and its non-diagonalizable generalizations. For a gentler first look, Strang §6.3 derives $e^{At}$ for solving $\mathbf{x}' = A\mathbf{x}$.

  • History. For the historical arc from quadratic forms and principal axes to the modern eigen-language — Cauchy, Jacobi, Weierstrass, Jordan, Hilbert — see Hawkins, "Cauchy and the spectral theory of matrices," Historia Mathematica (1975), and Chapter 7 of Kline, Mathematical Thought from Ancient to Modern Times. (Several biographical specifics in this chapter's Historical Note are flagged [verify]; these sources are where to confirm them.)

Free interactive resources

  • 3Blue1Brown, Essence of Linear Algebra, "Eigenvectors and eigenvalues" and "A quick trick for computing eigenvalues." The geometric intuition for eigenvectors-as-invariant-directions and diagonalization-as-change-of-coordinates, rendered as animation — the perfect visual companion to §25.2.3.
  • Immersive Math, Linear Algebra (immersivemath.com). Free, fully interactive textbook with manipulable figures for eigenvalues and change of basis.
  • The numpy.linalg.eig and scipy.linalg.eig documentation. Read the notes on eigenvector normalization, ordering, and the complex-output behavior — the exact points raised in this chapter's Computational Note, straight from the source.