Canonical form for matrices and transformations.

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Canonical form for matrices and transformations. 7.3. Jordan form Canonical form for matrices and transformations.

pi=xk_i, -> k=k1  k2….kr1. N nilpotent on Vn. Cyclic decomposition V=Z(a1;N)…  Z(ar;N). p1,….,pr N-annihilators, p i+1|pi , i=1,…,r-1. minpoly N= xk, kn: Nr=0 for some r. Thus xr is in Ann(N). minpoly N divides xr and hence minpolyN=xk for some k. minpolyN divides charpoly N of deg n. pi=xk_i, -> k=k1  k2….kr1.

We now have a rational form for N: Companion matrix of xk_i: kixki-matrix Thus the rational form of N is 1s at one below the diagonal that skips one space after k1-1,k2-1,…, so on.

N is a direct sum of elementary nilpotent matrices A nilpotent nxn-matrix up to similarity <-> positive integers r, k1,…,kr, such that k1+…+kr=n, ki  ki+1 Proof is omitted. r=nullity N. In fact null N=<Nk_i-1ai:i=1,…,r>. ai cyclic vectors. Proof: a in V -> a=f1(N)a1+….+fr(N)ar, fi in F[x], deg fi ki.

Na=0 implies N(fi(N)ai)=0 for each i. xfi(N) ai =0. (N-invariant direct sum property) xfi(N) ai =0. xfi is in N-annihilator of ai xfi is divisible by xk_i. Thus, fi=cixk_i-1, ci in F and a=c1(xk_1-1)(N)a1+….+cr(xk_r-1)(N)ar. Therefore {Nk_1-1a1, …, Nk_r-1ar } is a basis of null N, and dim = r.

Jordan form construction: Let T be a linear operator where charpoly f=(x-c1)d_1….(x-ck)d_k. For example when F=C. minpoly f= (x-c1)r_1….(x-ck)r_k. Let Wi = null (T-ciI)r_i. V=W1…Wr. Let Ti=T|Wi:Wi->Wi. Ni=Ti-ciI is nilpotent. Choose a basis Bi of Wi s.t. Ni is in rational form. Then Ti= Ni+ciI.

[Ti]B_i = There are some gaps of 1s here. If there are no gaps, then it is called the elementary Jordan matrix with char value ci.

A = Jordan form Ai = elementary Jordan matrices The size of elementary Jordan matrices decreases.

Uniquenss of Jordan form: Suppose T is represented by a Jordan matrix. V=W1…Wk. Ai is dixdi-matrix. Ai on Wi. Then charpolyT=(x-c1)d_1….(x-ck)d_k. c1,…,ck, d1,…,dk are determined unique up to order. Wi=null(T-ciI)d_i clearly. Ai is uniquely determined by the rational form for (T-ciI). (Recall rational form is uniquely determined.)

Properties of Jordan matrix: (1) c1,..,ck distinct char.values. ci is repeated di=dimWi=multiplicity in char poly of A. (2) Ai direct sum of ni elementrary Jordan matrices. nidi. (3) Ji1 of Ai is rixri-matrix. ri is multiplicity of the minimal polynomial of T. (So we can read the minimal polynomial from the Jordan form)

Proof of (3): We show that (T-c1I)r_1….(T-ckI)r_k=0. T|Wi=Ai and (Ai-ciI)r_i=0 on Wi. (x-c1)r_1….(x-ck)r_k is the least degree. This can be seen from the largest elementary Jordan matrix J1i. (J1i - ci I)q 0 for q< ri. This completes the proof.