Further Reading — The Characteristic Polynomial and How to Find Eigenvalues

A guide to where the standard texts and free resources treat this chapter's material, with notes on what each does best. Pair these with the chapter; none replaces working the examples by hand.

Core textbooks

  • Gilbert Strang, Introduction to Linear Algebra (6th ed.), Chapter 6, §6.1–6.2. The closest match to this chapter's spirit. Strang introduces $\det(A - \lambda I) = 0$ exactly as we do — through singularity — and his worked $2 \times 2$ and $3 \times 3$ examples, trace/determinant checks, and Markov-matrix treatment parallel ours. Read §6.1 for the characteristic polynomial and §6.2 for where it heads next (diagonalization). His companion video lectures (below) cover this almost verbatim.

  • Sheldon Axler, Linear Algebra Done Right (4th ed.), Chapter 5. The principled counterpoint. Famously, Axler avoids the determinant entirely and defines eigenvalues through the non-injectivity of $T - \lambda I$, proving existence of eigenvalues over $\mathbb{C}$ without the characteristic polynomial. Reading Axler after this chapter is illuminating: it shows that the polynomial, though the standard computational tool, is not logically necessary for the theory. His treatment of why operators on complex vector spaces always have eigenvalues (Theorem 5.19 region) is the cleanest in print. Best for math majors who want to see the determinant-free path.

  • Stephen Boyd & Lieven Vandenberghe, Introduction to Applied Linear Algebra (VMLS), Chapter 17 and the eigenvalue material around it. The applications-first view. Boyd–Vandenberghe care about dynamics $\mathbf{x}_{k+1} = A\mathbf{x}_k$, Markov chains, and population models — exactly our case studies — and treat eigenvalues as the tool for understanding long-run behavior. Free PDF at stanford.edu/~boyd/vmls. Lightest on the by-hand polynomial, heaviest on why you care.

  • David C. Lay, Steven Lay & Judi McDonald, Linear Algebra and Its Applications (6th ed.), Chapter 5, §5.1–5.2. The gentlest, most example-dense treatment, with extensive drill on forming the characteristic polynomial, multiplicity, and the eigenspace as a null space. Its Leslie-matrix and Markov-chain sections are excellent supplements to our two case studies. Best if you want many more worked problems at the ⭐⭐ level.

On multiplicity, defectiveness, and the numerical caution

  • Strang, Introduction to Linear Algebra, §6.2 and §8.3, for the link between geometric multiplicity, diagonalizability, and what goes wrong for defective matrices — the bridge to our Chapter 25.

  • Roger Horn & Charles Johnson, Matrix Analysis (2nd ed.), §1.1–1.4. The rigorous reference for the algebraic-vs-geometric multiplicity inequality $1 \le m_g \le m_a$ (our §24.7 theorem) and for the Cayley–Hamilton theorem (our Math-Major Sidebar). Graduate-level; consult for proofs in full generality.

  • Lloyd N. Trefethen & David Bau, Numerical Linear Algebra, Lectures 24–28. The authoritative, readable account of why you never compute eigenvalues via the characteristic polynomial, and what you do instead — the QR algorithm in particular (our §24.10 and Chapter 38). Lecture 25 on the QR algorithm is a model of clear exposition. Essential before Chapter 38.

  • James Wilkinson, The Algebraic Eigenvalue Problem (1965). The classic that established the numerical pitfalls; the source of the "Wilkinson polynomial" example showing how root-finding amplifies error (referenced in §24.10). Historical but still cited; for the seriously curious.

Free online resources

  • MIT 18.06, Gilbert Strang — Lecture 21 ("Eigenvalues and Eigenvectors"). Free video at ocw.mit.edu. Strang derives $\det(A - \lambda I) = 0$, works examples, and states the trace/determinant facts on the board — the lecture this chapter is most in dialogue with.

  • 3Blue1Brown, Essence of Linear Algebra, "Eigenvectors and eigenvalues" (Chapter 14). The best visual companion to our Geometric Intuition callouts; the animations of vectors staying on their span under a transformation make the invariant-direction picture unforgettable. Watch it alongside §24.1 and §24.9.

  • Khan Academy, Linear Algebra, "Eigen-everything" unit. Step-by-step worked characteristic-polynomial and eigenvector computations at a deliberate pace, with practice problems — ideal reinforcement for the ⭐⭐ tier of our exercises.

  • Wikipedia, "Characteristic polynomial," "Eigenvalue algorithm," and "Power iteration." Reliable for the general formulas (including the $3 \times 3$ characteristic polynomial in terms of trace, the sum of principal $2 \times 2$ minors, and the determinant) and for an overview of the iterative algorithms named in §24.10.

A note on practice

Eigenvalue computation rewards repetition more than reading. After this chapter, the single most useful exercise is to pick random small integer matrices, find their eigenvalues and eigenvectors by hand, and check each against np.linalg.eig — paying attention to the trace/determinant identities, to whether any matrix is defective, and to when the roots come out complex. The fluency you build here is the foundation for diagonalization (Chapter 25), the spectral theorem (Chapter 27), and everything in Part VI.