Linear Algebra

Section IV. Jordan Form 395IV.2Jordan Canonical FormWe are looking for a canonical form for matrix similarity. This subsectioncompletes this program by moving from the canonical form for the classes ofnilpotent matrices to the canonical form for all classes.2.1 Lemma A linear transformation on a nontrivial vector space is nilpotent ifand only if its only eigenvalue is zero.Proof Let the linear transformation be t: V → V. If t is nilpotent then thereis an n such that t n is the zero map, so t satisfies the polynomial p(x) = x n =(x − 0) n . By Lemma 1.10 the minimal polynomial of t divides p, so the minimalpolynomial has only zero for a root. By Cayley-Hamilton, Theorem 1.8, thecharacteristic polynomial has only zero for a root. Thus the only eigenvalue of tis zero.Conversely, if a transformation t on an n-dimensional space has only thesingle eigenvalue of zero then its characteristic polynomial is x n . Lemma 1.9says that a map satisfies its characteristic polynomial so t n is the zero map.Thus t is nilpotent.QEDThe phrase “nontrivial vector space” is there because on a trivial space {⃗0} theonly transformation is the zero map, which has no eigenvalues because there areno associated nonzero eigenvectors.2.2 Corollary The transformation t−λ is nilpotent if and only if t’s only eigenvalueis λ.Proof The transformation t − λ is nilpotent if and only if t − λ’s only eigenvalueis 0. That holds if and only if t’s only eigenvalue is λ, because t(⃗v) = λ⃗v if andonly if (t − λ) (⃗v) = 0 · ⃗v.QEDWe already have the canonical form that we want for the case of nilpotentmatrices, that is, for each matrix whose only eigenvalue is zero. Corollary III.2.15says that each such matrix is similar to one that is all zeroes except for blocksof subdiagonal ones.2.3 Lemma If the matrices T − λI and N are similar then T and N + λI are alsosimilar, via the same change of basis matrices.Proof With N = P(T − λI)P −1 = PTP −1 − P(λI)P −1 we have N = PTP −1 −PP −1 (λI) since the diagonal matrix λI commutes with anything, and so N =PTP −1 − λI. Therefore N + λI = PTP −1 .QED2.4 Example The characteristic polynomial of( )2 −1T =1 4