We began by saying that a matrix is a transformation of space with a life of its own, and asked three questions: how matrices are like numbers and how they differ, what a matrix really does, and what stays the same. Here is what the chapter’s nine questions showed.
1. How are matrices like numbers, and how are they different?
Matrices can be added and multiplied, but Section 7.1 showed which familiar rules fail. Multiplication is not commutative, so \((\mathbf{A} - \mathbf{B})(\mathbf{A} + \mathbf{B})\) need not be \(\mathbf{A}^2 - \mathbf{B}^2\); there are zero divisors, so \(\mathbf{AB} = \mathbf{0}\) does not force \(\mathbf{BA} = \mathbf{0}\); and a quadratic matrix equation can have infinitely many roots. Each failure had a geometric explanation, in terms of what the matrices do to the plane. What survives, crucially, is that determinants multiply, \(\det(\mathbf{AB}) = \det\mathbf{A}\det\mathbf{B}\). Section 7.3 showed that this property essentially characterises the determinant, and Section 7.8 used it to prove Euler’s four-square identity, the key step in Lagrange’s theorem that every whole number is a sum of four squares.
2. What does a matrix really do?
Several sections answered this by finding the right point of view.
Eigenvectors (Section 7.2) are the directions a matrix does not turn. In their coordinates a symmetric matrix is just a stretch, and the image of a circle is an ellipse whose axes and area can be read off.
The determinant (Section 7.3) is the signed area scale factor: shears preserve area, swaps reverse orientation, singular matrices squash the plane flat.
Trace and determinant (Section 7.4) decide, via the Cayley–Hamilton theorem, whether a matrix returns to the identity after a few steps. A matrix with \(\mathbf{M}^8 = \mathbf{I}\) need not be a rotation; it is a rotation seen in skewed coordinates. And because traces of integer matrices are integers, crystals cannot have five-fold symmetry.
Rotations in three dimensions (Section 7.5) are built from the cross product by Rodrigues’ formula, the three-dimensional analogue of \(e^{\mathrm{i}\theta} = \cos\theta + \mathrm{i}\sin\theta\).
Hyperbolic rotations (Section 7.6) move points along a hyperbola and add their hyperbolic angles. They are the Lorentz boosts of special relativity, and their invariant lines are the paths of light.
3. What stays the same?
Invariants were everywhere. Rotations preserve \(x^2 + y^2\), Lorentz boosts preserve \(x^2 - y^2\), and similar matrices share their trace and determinant. In Section 7.7, a matrix evolving by \(\dot{\mathbf{M}} = \mathbf{MN} - \mathbf{NM}\) kept its trace and determinant, and so its eigenvalues, constant, however much its entries changed: the mechanism behind the conservation laws of solitons. And in Section 7.9, the invariant was a number, \(V - E + F = 2\), which holds for every convex polyhedron and pins the regular ones down to just five. That number, the Euler characteristic, was one of the first invariants of topology.
4. A toolkit for linear algebra
| Method | Where you used it |
|---|---|
| Geometric counterexamples | Rules that fail (7.1) |
| Eigenvectors as natural axes | The ellipse (7.2); boosts (7.6) |
| Probing with special matrices | The determinant (7.3) |
| Cayley–Hamilton | Finite order (7.4) |
| Splitting into parallel and perpendicular parts | Rodrigues’ formula (7.5) |
| Choosing the right parameter | Rapidity (7.6) |
| \(\operatorname{tr}(\mathbf{MN}) = \operatorname{tr}(\mathbf{NM})\) | Conserved quantities (7.7) |
| Multiplicativity of determinants | Four squares (7.8) |
| Double counting | Platonic solids (7.9) |
Where next
Eigenvalues and eigenvectors govern the vibrations of systems of springs and masses in Section 11.2.
The eigenvectors of a covariance matrix appear in statistics in Section 13.7.
Linear differential equations, whose solutions are built from matrix exponentials, are the subject of Chapter 8.
3Blue1Brown, Essence of Linear Algebra: a free video series that shows what matrices do. Watch it first.
Gilbert Strang, Introduction to Linear Algebra: the classic first course, with a strong emphasis on understanding and applications.
Sheldon Axler, Linear Algebra Done Right: a more abstract approach, centred on linear maps; free online.
David Richeson, Euler’s Gem: Euler’s formula and the birth of topology.
Kaplansky joked that he and Halmos thought without coordinates but computed with matrices “like fury” behind closed doors. That captures what this chapter has tried to show: matrices are concrete and computable, but behind every calculation there is geometry (a stretch, a rotation, a projection, a boost) and an invariant that does not depend on how you write it down. Learning to move between the two views is one of the most valuable skills a mathematician can have.