Linear Algebra

Section IV. Jordan Form 387IVJordan FormThis section uses material from three optional subsections: Direct Sum,Determinants Exist, and Laplace’s Expansion Formula.The chapter on linear maps shows that every h: V → W can be representedby a partial identity matrix with respect to some bases B ⊂ V and D ⊂ Wthat is, that the partial identity form is a canonical form for matrix equivalence.This chapter considers the special case that the map is a linear transformationt: V → V. The general result still applies so we can get a partial identity withrespect to B, D, but with the codomain equal to the domain we naturally askwhat is possible when the two bases are also equal so that we have Rep B,B (t) —we will find a canonical form for matrix similarity.We began by noting that while a partial identity matrix is the canonical formfor the B, D case, in the B, B case there are some matrix similarity classes withoutone. We therefore extended the forms of interest to the natural generalization,diagonal matrices, and showed that the map or matrix can be diagonalized if itseigenvalues are distinct. But we also gave an example of a matrix that cannotbe diagonalized (because it is nilpotent), and thus diagonal form won’t do asthe canonical form for all matrices.The prior section developed that example. We showed that a linear map isnilpotent if and only if there is a basis on which it acts via disjoint strings. Thatgave us a canonical form that applied to nilpotent matrices.This section wraps up the chapter by showing that the two cases we’vestudied are exhaustive in that for any linear transformation there is a basis suchthat the matrix representation Rep B,B (t) is the sum of a diagonal matrix and anilpotent matrix. This is Jordan canonical form.IV.1Polynomials of Maps and MatricesRecall that the set of square matrices is a vector space under entry-by-entryaddition and scalar multiplication, and that this space M n×n has dimension n 2 .Thus, for any n×n matrix T the n 2 + 1-member set {I, T, T 2 , . . . , T n2 } is linearlydependent and so there are scalars c 0 , . . . , c n 2, not all zero, such thatc n 2T n2 + · · · + c 1 T + c 0 Iis the zero matrix. That is, every transformation has a kind of generalizednilpotency: the powers of a square matrix cannot climb forever without a“repeat.”1.1 Example Rotation of plane vectors π/6 radians counterclockwise is represented