Chapter 16 — Further Reading

Annotated pointers for going deeper on change of basis and similarity, keyed to the three standard texts this book tracks (Strang, Axler, Boyd–Vandenberghe) plus free online resources. Read the chapter first; these expand and re-frame, they do not replace.

Primary textbook references

  • Gilbert Strang, Introduction to Linear Algebra (any recent edition), §7.2 "The Matrix of a Linear Transformation" and §8.2 "Change of Basis." Strang's treatment is the closest in spirit to this chapter: geometric, transformation-first, and explicit that the matrix depends on the chosen basis. His "the matrix changes but the transformation does not" framing is exactly recurring theme #1. Pair this with his §6 chapters on eigenvalues to see the change-of-basis idea cash out as diagonalization.

  • Sheldon Axler, Linear Algebra Done Right (3rd/4th ed.), Chapter 3 (§3.C "Matrices of Linear Maps") and Chapter 5 (§5.A on invariant subspaces, "Eigenvalues and Eigenvectors"). Axler is the rigorous, coordinate-free counterpart to this chapter — he develops linear maps abstractly and treats the matrix as a derived object relative to a basis, which is precisely the object/representation distinction of §16.7. His notation $\mathcal{M}(T)$ for "the matrix of $T$ with respect to chosen bases" makes the basis-dependence visible at every step. The change-of-basis formula appears as the change between two such matrices. Read this if you want the math-major's airtight version of the Math-Major Sidebar.

  • Stephen Boyd & Lieven Vandenberghe, Introduction to Applied Linear Algebra (VMLS), Chapters 5 (linear independence and basis) and 7–10 (matrices, linear functions). The applied, data-centric complement. Boyd–Vandenberghe emphasize coordinates and change of representation in the service of applications (signals, control, data), aligning with this chapter's case studies. Freely available online — see below.

On the eigenvalue payoff (where this chapter leads)

  • Strang, Introduction to Linear Algebra, §6.1–6.2 (eigenvalues, diagonalization). The direct sequel: similarity $B = P^{-1}AP$ becomes diagonalization $A = PDP^{-1}$ when $P$ is built from eigenvectors. This is the Chapter 25 destination previewed throughout Chapter 16.

  • Axler, Chapters 5 and 8 (eigenvalues; operators on complex/real vector spaces, generalized eigenvectors). For the full classification under similarity, culminating in Jordan form (our Chapter 36) — Axler's coordinate-free path makes clear why the canonical forms are what they are.

Online and video resources (free)

  • 3Blue1Brown, Essence of Linear Algebra, the episodes "Change of basis" and "Eigenvectors and eigenvalues." Grant Sanderson's visual treatment of change of basis is the best moving-picture version of this chapter's re-gridding idea; his "translator" metaphor for $P$ and $P^{-1}$ is essentially our "go to old, act, come back" reading of $B = P^{-1}AP$. Watch the change-of-basis episode immediately after this chapter, then the eigenvector episode to see Part V coming.

  • MIT OpenCourseWare 18.06, Linear Algebra (Gilbert Strang), lectures on "The Matrix of a Linear Transformation" and "Change of Basis; Image Compression." Full video lectures, problem sets, and exams, free. Strang's lecture explicitly connects change of basis to wavelet/image compression — a concrete cousin of this chapter's PCA case study.

  • Boyd & Vandenberghe, Introduction to Applied Linear Algebra — free PDF and Julia/Python companion at the VMLS website (stanford.edu/~boyd/vmls). The companion code is a good place to practice building change-of-basis matrices and verifying round trips numerically, in the spirit of this chapter's Build Your Toolkit.

For the applications in the case studies

  • PCA (Case Study 1): Strang, Linear Algebra and Learning from Data, the chapters on the SVD and PCA, treat principal components as the change of basis that diagonalizes the covariance — the precise statement this chapter previews and Chapter 32 proves. For the statistical reading, any multivariate-statistics text's PCA chapter (e.g. the standard treatments) frames it as "rotate to uncorrelated coordinates."

  • Rotated frames in robotics/graphics (Case Study 2): any introductory robotics text's chapter on rigid-body transformations and homogeneous coordinates (the "frames and transforms" material) develops exactly the change-of-basis-plus-translation pipeline; for graphics, the transformation chapters of a standard real-time rendering or computer-graphics text cover model/world/view/clip space as a composition of basis changes. These connect back to this book's own Chapter 12 (computer graphics, homogeneous coordinates).

A note on notation across sources

Watch for differing conventions: some texts define the change-of-basis matrix as the one converting old to new (the inverse of our $P$), and some write the similarity as $PAP^{-1}$ rather than $P^{-1}AP$ depending on whether $P$'s columns are the new or old basis vectors. Always pin down which direction the columns point before trusting a formula — exactly the $P$-versus-$P^{-1}$ discipline of §16.3. The mathematics is identical; only the labeling differs. This book fixes the convention that $P$'s columns are the new basis vectors in old coordinates, so $P$ maps new $\to$ old and $B = P^{-1}AP$ — consistent throughout.