If A is a matrix representing a linear operator α with respect to some ...
If A is a matrix representing a linear operator α with respect to some ...
If A is a matrix representing a linear operator α with respect to some ...
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<strong>If</strong> A <strong>is</strong> a <strong>matrix</strong> <strong>representing</strong> a <strong>linear</strong> <strong>opera<strong>to</strong>r</strong> <strong>α</strong> <strong>with</strong> <strong>respect</strong> <strong>to</strong> <strong>some</strong> bas<strong>is</strong> then we define<br />
the character<strong>is</strong>tic polynomial of <strong>α</strong> <strong>to</strong> be ∆<strong>α</strong>(t) = det (<strong>α</strong>−tid) = det (A−tI). Th<strong>is</strong> doesn’t<br />
depend on the <strong>matrix</strong> that represents <strong>α</strong> and thus th<strong>is</strong> <strong>is</strong> well defined.<br />
(2) Recal that the algebraic multiplicity, am(λ), of an eigenvalue λ <strong>is</strong> the multiplicity<br />
of λ as a root of ∆A(t) (or ∆<strong>α</strong>(t)). The geometric multiplicity of λ <strong>is</strong> the dimension of the<br />
eignspace EA(λ) (or E<strong>α</strong>(λ)). We know that we always have am(λ) ≥ gm(λ).<br />
Example. In the example above we have<br />
<br />
<br />
∆A(t) = det(A − tI) = −t 1 <br />
<br />
1 −t = t2 − 1 = (t − 1)(t + 1) = mA(t).<br />
We will later see that minimal polynomial and the character<strong>is</strong>tic polynomial are strongly<br />
related and that the latter <strong>is</strong> always a multiple of the minimal polynomial. Here am(1) =<br />
am(−1) = gm(1) = gm(−1) = 1.<br />
Lemma 3.1 Let p be a polynomial such that p(<strong>α</strong>) = 0 then every eigenvalue of <strong>α</strong> <strong>is</strong> a<br />
root of p. In particular every eigenvalue of <strong>α</strong> <strong>is</strong> a root of m<strong>α</strong>.<br />
Proof Let v = 0 be an eigenvec<strong>to</strong>r <strong>with</strong> <strong>respect</strong> <strong>to</strong> λ and suppose p(t) = a0+a1t+. . .+akt k .<br />
Then p(<strong>α</strong>) = 0 gives us<br />
0 = p(<strong>α</strong>) v<br />
= (a0id + a1<strong>α</strong> + · · · + ak<strong>α</strong> k )v<br />
= (a0 + a1λ + · · · + akλ k )v<br />
= p(λ)v.<br />
As v = 0 it follows that p(λ) = 0. ✷<br />
We now turn <strong>to</strong> a remarkable fact. It turns out that any <strong>linear</strong> <strong>opera<strong>to</strong>r</strong> <strong>α</strong> : V → V<br />
sat<strong>is</strong>fies the character<strong>is</strong>tic polynomial ∆<strong>α</strong>(t).<br />
Theorem 3.2 (Cayley-Hamil<strong>to</strong>n). For any n × n <strong>matrix</strong> A we have ∆A(A) = 0. Equivalently,<br />
for any <strong>linear</strong> <strong>α</strong> : V → V we have ∆<strong>α</strong>(<strong>α</strong>) = 0.<br />
Proof Suppose<br />
∆A(t) = det (A − tI) = a0 + a1t + · · · + ant n .<br />
We must show that ∆A(A) = a0I + a1A + · · ·+ anA n = 0 as a <strong>matrix</strong>. We begin <strong>with</strong> the<br />
observation that<br />
adj(A − tI) = B0 + B1t + · · · + Bn−1t n−1 ,<br />
where each Bi <strong>is</strong> an n × n <strong>matrix</strong>. Furthermore, the highest power of t on the right hand<br />
side <strong>is</strong> n−1, because each minor <strong>is</strong> an (n−1)×(n−1) determinant. The adjugate formula<br />
tells us that<br />
(A − tI)adj(A − tI) = det (A − tI)I.<br />
39
That <strong>is</strong><br />
(A − tI)(B0 + B1t + · · · + Bn−1t n−1 ) = (a0 + a1t + · · · + ant n )I.<br />
Comparing the coefficients of t i for i = 0, 1, . . ., n we see that<br />
AB0 = a0I,<br />
AB1 − B0 = a1I,<br />
ABn−1 − Bn−2 = an−1I,<br />
−Bn−1 = anI.<br />
Multiplying these equations from the left by I, A, . . .,A n <strong>respect</strong>ively we get<br />
Adding up these n + 1 equations, we get<br />
which <strong>is</strong> the required formula. ✷.<br />
.<br />
AB0 = a0I,<br />
A 2 B1 − AB0 = a1A,<br />
A n Bn−1 − A n−1 Bn−2 = an−1A n−1 ,<br />
−A n Bn−1 = anA n .<br />
0 = a0I + a1A + · · · + anA n ,<br />
Remark. It follows from the Cayley-Hamil<strong>to</strong>n Theorem that m<strong>α</strong>(t) divides ∆<strong>α</strong>(t). Next<br />
we are going <strong>to</strong> see that these have the same roots.<br />
Proposition 3.3 . The roots of m<strong>α</strong> are prec<strong>is</strong>ely the eigenvalues of <strong>α</strong>.<br />
Proof By the remark above, we have that m<strong>α</strong> divides ∆<strong>α</strong> and thus every root of m<strong>α</strong> <strong>is</strong><br />
a root of ∆<strong>α</strong> and therefore an eigenvalue of <strong>α</strong>. The converse follows from Lemma 3.1. ✷<br />
Remark. It follows from th<strong>is</strong> last proposi<strong>to</strong>n and the Cayley-Hamil<strong>to</strong>n Theorem that,<br />
over C, if λ1, . . .,λk are the d<strong>is</strong>tinct eigenvalues of λ and<br />
then<br />
<strong>with</strong> 1 ≤ si ≤ ri for all 1 ≤ i ≤ k.<br />
∆<strong>α</strong>(t) = (λ1 − t) r1 · · ·(λk − t) rk ,<br />
m<strong>α</strong>(t) = (t − λ1) s1 · · ·(t − λk) sk<br />
II. Invariant subspaces and primary decompositions<br />
A. Invariant subspaces<br />
Definition Let <strong>α</strong> : V → V be a <strong>linear</strong> <strong>opera<strong>to</strong>r</strong>. We say that a subspace W of V <strong>is</strong><br />
<strong>α</strong>-invariant if <strong>α</strong>(W) ⊆ W.<br />
40<br />
.
<strong>If</strong> W <strong>is</strong> <strong>α</strong>-invariant, then the restriction of <strong>α</strong> <strong>to</strong> W <strong>is</strong> the <strong>linear</strong> <strong>opera<strong>to</strong>r</strong><br />
<strong>α</strong>|W : W → W : w ↦→ <strong>α</strong>(w).<br />
Examples. (1) The subspaces {0} and V are always <strong>α</strong>-invariant.<br />
(2) Let λ be an eigenvalue of <strong>α</strong> and v <strong>is</strong> an eigenvec<strong>to</strong>rs <strong>with</strong> <strong>respect</strong> <strong>to</strong> λ then the<br />
one dimensional subspace Kv <strong>is</strong> <strong>α</strong>-invariant. Th<strong>is</strong> <strong>is</strong> because <strong>α</strong>(rv) = r<strong>α</strong>(v) = rλv ∈ Kv.<br />
(3) Let <strong>α</strong> : R 3 → R 3 be the <strong>linear</strong> <strong>opera<strong>to</strong>r</strong> that rotates every vec<strong>to</strong>r 30 degrees around<br />
the z-ax<strong>is</strong> (counter clockw<strong>is</strong>e). Here Re3 and Re1 + Re2 are <strong>α</strong>-invariant.<br />
A <strong>linear</strong> <strong>opera<strong>to</strong>r</strong> <strong>α</strong> : V → V <strong>is</strong> diagonal<strong>is</strong>able if there <strong>is</strong> a bas<strong>is</strong> (v1, . . ., vn) for V<br />
cons<strong>is</strong>ting of eigenvec<strong>to</strong>rs. Suppose that eigenvalue of for vi <strong>is</strong> λi. Then<br />
V = Kv1 ⊕ Kv2 ⊕ · · · ⊕ Kvn<br />
and the <strong>matrix</strong> for <strong>α</strong> <strong>with</strong> <strong>respect</strong> <strong>to</strong> the bas<strong>is</strong> (v1, . . .,vn) <strong>is</strong> the diagonal <strong>matrix</strong><br />
⎛<br />
⎞<br />
More generally suppose that<br />
⎜<br />
⎝<br />
λ1<br />
λ2<br />
. ..<br />
λn<br />
⎟<br />
⎠ .<br />
V = V1 ⊕ V2 ⊕ · · · ⊕ Vk<br />
where V1, . . ., Vk are <strong>α</strong>-invariant subspaces. Let Vi be a bas<strong>is</strong> for Vi and let <strong>α</strong>i = <strong>α</strong>|Vi :<br />
Vi → Vi be the restriction of <strong>α</strong> on Vi. Let Ai be the <strong>matrix</strong> <strong>representing</strong> <strong>α</strong>i <strong>with</strong> <strong>respect</strong> <strong>to</strong><br />
the bas<strong>is</strong> Vi. Then the <strong>matrix</strong> <strong>representing</strong> <strong>α</strong> <strong>with</strong> <strong>respect</strong> <strong>to</strong> the bas<strong>is</strong> V = V1∪V2∪· · ·∪Vk<br />
<strong>is</strong> the <strong>matrix</strong><br />
⎛<br />
⎞<br />
⎜<br />
A = ⎜<br />
⎝<br />
A1<br />
A2<br />
Conversely if we have any <strong>linear</strong> <strong>opera<strong>to</strong>r</strong>s <strong>α</strong>i : Vi → Vi <strong>with</strong> <strong>matrix</strong> Ai <strong>with</strong> <strong>respect</strong> <strong>to</strong><br />
Vi. Then we get a <strong>linear</strong> map (denoted <strong>α</strong>1 ⊕ <strong>α</strong>2 ⊕ · · · ⊕ <strong>α</strong>k) from V <strong>to</strong> V that acts on Vi<br />
like <strong>α</strong>i. Thus for<br />
<strong>α</strong> = <strong>α</strong>1 ⊕ <strong>α</strong>2 ⊕ · · · ⊕ <strong>α</strong>k : V → V<br />
we have <strong>α</strong>(vi) = <strong>α</strong>i(vi) if vi ∈ Vi. So the subspaces V1, . . .,Vk will be <strong>α</strong>-invariant and the<br />
<strong>matrix</strong> for <strong>α</strong> will be the <strong>matrix</strong> A as above that we will often denote by A1 ⊕A2 ⊕· · ·⊕Ak.<br />
Example. Suppose V1 = Fv1 ⊕ Fv2 and V2 = Fv3 ⊕ Fv4. Suppose furthermore that the<br />
<strong>linear</strong> <strong>opera<strong>to</strong>r</strong>s <strong>α</strong>1 : V1 → V1 and <strong>α</strong>2 : V2 → V2 are defined by<br />
. ..<br />
Ak<br />
⎟<br />
⎠ .<br />
<strong>α</strong>1(v1) = 2v1 + v2 <strong>α</strong>1(v2) = v1 − v2<br />
<strong>α</strong>2(v3) = v4 <strong>α</strong>2(v4) = v3.<br />
41
Then <strong>α</strong>1 ⊕ <strong>α</strong>2 : V1 ⊕ V2 → V1 ⊕ V2 has <strong>matrix</strong><br />
⎛ ⎞<br />
2 1 0 0<br />
⎜<br />
A = ⎜ 1 −1 0 0 ⎟<br />
⎝ 0 0 0 1 ⎠<br />
0 0 1 0<br />
=<br />
<br />
A1 0<br />
= A1 ⊕ A2<br />
0 A2<br />
where<br />
A1 =<br />
2 1<br />
1 −1<br />
<br />
0 1<br />
, A2 =<br />
1 0<br />
The aim <strong>is</strong> <strong>to</strong> break V in<strong>to</strong> a direct sum V1 ⊕ V2 ⊕ · · · ⊕ Vk where k <strong>is</strong> as big as possible.<br />
The following lemma <strong>is</strong> going <strong>to</strong> be useful.<br />
Lemma 3.4 Suppose <strong>α</strong>, β : V → V are <strong>linear</strong> <strong>opera<strong>to</strong>r</strong>s such that <strong>α</strong>β = β<strong>α</strong>. Then ker β<br />
and imβ are <strong>α</strong>-invariant.<br />
Proof <strong>If</strong> w ∈ ker β then<br />
β(<strong>α</strong>(w)) = <strong>α</strong>(β(w)) = <strong>α</strong>(0) = 0.<br />
hence <strong>α</strong>(w) ∈ ker β. Th<strong>is</strong> shows that ker β <strong>is</strong> <strong>α</strong>-invariant. To see that imβ <strong>is</strong> <strong>α</strong>-invariant,<br />
notice that if v = β(u) then <strong>α</strong>(v) = <strong>α</strong>(β(u)) = β(<strong>α</strong>(u)) ∈ im β. ✷<br />
B. Primary Decompositions<br />
Suppose that <strong>α</strong> : V → V has a decomposition<br />
<strong>α</strong> = <strong>α</strong>1 ⊕ · · · ⊕ <strong>α</strong>k<br />
<strong>with</strong> <strong>respect</strong> <strong>to</strong> decomposition V = V1 ⊕ · · · ⊕ Vk where V1, . . .,Vk are <strong>α</strong>-invariant. As<br />
before, we pick for each Vi a bas<strong>is</strong> Vi and we let Ai be the <strong>matrix</strong> <strong>representing</strong> <strong>α</strong>i <strong>with</strong><br />
<strong>respect</strong> <strong>to</strong> Vi. Then<br />
⎛<br />
⎞<br />
⎜<br />
A = A1 ⊕ · · · ⊕ Ak = ⎜<br />
⎝<br />
<strong>is</strong> the <strong>matrix</strong> repsenting <strong>α</strong> <strong>with</strong> <strong>respect</strong> <strong>to</strong> V = V1 ∪ · · · ∪ Vk. Notice that if f <strong>is</strong> any<br />
polynomial in K[t] then<br />
⎛<br />
⎞<br />
f(A1)<br />
⎜ f(A2) ⎟<br />
f(A) = ⎜<br />
⎝<br />
. ..<br />
⎟<br />
⎠<br />
f(Ak)<br />
= f(A1) ⊕ · · · ⊕ f(Ak).<br />
Thus f(A) = 0 if and only if mAi |f for all i = 1, . . .,k. Th<strong>is</strong> implies that the mA <strong>is</strong> the<br />
42<br />
A1<br />
A2<br />
. ..<br />
<br />
.<br />
Ak<br />
⎟<br />
⎠
least common multiple of mA1, . . .,mAk . Equivalently m<strong>α</strong> <strong>is</strong> the least common multiple<br />
of m<strong>α</strong>1, . . .,m<strong>α</strong>k . In particular if m<strong>α</strong>1, . . .,m<strong>α</strong>k are pairw<strong>is</strong>e comprime, then<br />
m<strong>α</strong> = m<strong>α</strong>1 · · ·m<strong>α</strong>k .<br />
Thus a decomposition of V in<strong>to</strong> <strong>α</strong>-invariant subspaces leads <strong>to</strong> a fac<strong>to</strong>rization of the<br />
minimal polynomial. Our next aim <strong>is</strong> <strong>to</strong> see that one can reverse th<strong>is</strong> procedure so a<br />
fac<strong>to</strong>rization of the minimal polynomial leads <strong>to</strong> a decomposition of V in<strong>to</strong> <strong>α</strong>-invariant<br />
subspaces.<br />
Lemma 3.5 Let <strong>α</strong> : V → V be a <strong>linear</strong> <strong>opera<strong>to</strong>r</strong> whose minimal polynomial has a fac<strong>to</strong>rization<br />
m<strong>α</strong>(t) = p1(t)p2(t)<br />
where p1 and p2 are monic polynomials that are coprime. Let V1 = imp2(<strong>α</strong>) and V2 =<br />
imp1(<strong>α</strong>). Then<br />
(1) The subspaces V1 and V2 are <strong>α</strong>-invariant.<br />
(2) V = V1 ⊕ V2.<br />
(3) The minimal polynomial of <strong>α</strong>i = <strong>α</strong>|Vi <strong>is</strong> pi(t).<br />
(4) V1 = kerp1(<strong>α</strong>) and V2 = kerp2(<strong>α</strong>).<br />
Proof (1) Notice that p1(<strong>α</strong>) and p2(<strong>α</strong>) commute <strong>with</strong> <strong>α</strong> and thus V1, V2 are <strong>α</strong>-invariant<br />
by Lemma 3.4.<br />
(2) As p1 and p2 are coprime, there are polynomials a1, a2 ∈ K[t] such that 1 = a1(t)p1(t)+<br />
a2(t)p2(t). Hence<br />
id = p2(<strong>α</strong>)a2(<strong>α</strong>) + p1(<strong>α</strong>)a1(<strong>α</strong>)<br />
Thus for any v ∈ V , we have<br />
v = id(v) = [p2(<strong>α</strong>)a2(<strong>α</strong>)](v) + [p1(<strong>α</strong>)a1(<strong>α</strong>)](v) ∈ im p2(<strong>α</strong>) + imp1(<strong>α</strong>) = V1 + V2.<br />
Th<strong>is</strong> shows that V = V1 + V2. To see that the sum <strong>is</strong> direct, suppose v ∈ V1 ∩ V2, say<br />
v = p2(<strong>α</strong>)(v2) = p1(<strong>α</strong>)(v1). Then<br />
v = a1(<strong>α</strong>)(p1(<strong>α</strong>)(v) + a2(<strong>α</strong>)p2(<strong>α</strong>)(v)<br />
= [a1(<strong>α</strong>)p1(<strong>α</strong>)p2(<strong>α</strong>)](v2) + [a2(<strong>α</strong>)p2(<strong>α</strong>)p1(<strong>α</strong>)](v1)<br />
= [a1(<strong>α</strong>)m<strong>α</strong>(<strong>α</strong>)](v2) + [a2(<strong>α</strong>)m<strong>α</strong>(<strong>α</strong>)](v1)<br />
= 0.<br />
Hence V1 ∩ V2 = {0} and V = V1 ⊕ V2.<br />
(3) We have that f(<strong>α</strong>1) = 0 if and only if f(<strong>α</strong>)(v) = 0 for all v ∈ V1. As V1 = imp2(<strong>α</strong>) th<strong>is</strong><br />
happens if and only if [f(<strong>α</strong>)(p2(<strong>α</strong>)](v) = 0 for all v ∈ V . As m<strong>α</strong> <strong>is</strong> the minimal polynomial<br />
for <strong>α</strong>, th<strong>is</strong> happens if and only if m<strong>α</strong> = p1p2 divides fp2. But th<strong>is</strong> happens if and only if<br />
p1|f. Hence p1 <strong>is</strong> the minimal polynomial of <strong>α</strong>1. Similarly p2 <strong>is</strong> the minimal polynomial<br />
of <strong>α</strong>2.<br />
(4) As p1(<strong>α</strong>)p2(<strong>α</strong>)(v) = m<strong>α</strong>(v) = 0 for all v ∈ V , it <strong>is</strong> clear that V1 = imp2(<strong>α</strong>) ⊆ ker p1(<strong>α</strong>).<br />
43
To get equality we just need <strong>to</strong> show that the dimensions are the same. But th<strong>is</strong> follows<br />
from<br />
dim V = dim V1 + dim V2 = dim im p2(<strong>α</strong>) + dim imp1(<strong>α</strong>)<br />
and (using the nullity rank theorem from year 1)<br />
dimV = dim ker p1(<strong>α</strong>) + dim im P1(<strong>α</strong>).<br />
comparing the two equations we see that dim ker p1(<strong>α</strong>) = dimim p2(<strong>α</strong>) = dim V1. Similarly<br />
one shows that V2 = ker p2(<strong>α</strong>). ✷<br />
Now let P be the set of all irreducibles in K[t] that are monic. We have seen earlier<br />
that these form a set of prime representatives for K[t]. Using Lemma 3.5 and induction<br />
on k, we get one of the main results about the structure of <strong>linear</strong> <strong>opera<strong>to</strong>r</strong>s.<br />
Theorem 3.6 (Primary Decomposition) Let <strong>α</strong> : V → V be a <strong>linear</strong> <strong>opera<strong>to</strong>r</strong> whose<br />
minimal polynomial has a fac<strong>to</strong>rization<br />
m<strong>α</strong>(t) = p1(t) n1 · · ·pk(t) nk<br />
where the p1, . . ., pk are d<strong>is</strong>tinct primes in P. Let qi = p ni<br />
i and let Vi = ker qi(<strong>α</strong>).<br />
(1) The subspaces V1, . . .,Vk are <strong>α</strong>-invariant,<br />
(2) V = V1 ⊕ · · · ⊕ Vk,<br />
(3) the minimal polynomial of <strong>α</strong>i = <strong>α</strong>|Vi <strong>is</strong> qi = p ni<br />
i .<br />
It <strong>is</strong> not difficult <strong>to</strong> see (sheet 9) that if <strong>α</strong> <strong>is</strong> a diagonal<strong>is</strong>able <strong>linear</strong> <strong>opera<strong>to</strong>r</strong> <strong>with</strong> (d<strong>is</strong>tinct)<br />
eigenvalues λ1, . . .,λk, then<br />
m<strong>α</strong>(t) = (t − λ1)(t − λ2) · · ·(t − λk).<br />
The next result shows that the converse <strong>is</strong> also true.<br />
Theorem 3.7 The <strong>linear</strong> map <strong>α</strong> : V → V <strong>is</strong> diagonal<strong>is</strong>able iff<br />
for <strong>some</strong> d<strong>is</strong>tinct λ1, . . .,λk ∈ K.<br />
m<strong>α</strong>(t) = (t − λ1)(t − λ2) · · ·(t − λk)<br />
Proof By the remark above we have that the minimal polynomial of a diagonal<strong>is</strong>able<br />
<strong>linear</strong> map <strong>is</strong> a product of d<strong>is</strong>tinct <strong>linear</strong> fac<strong>to</strong>rs. For the converse we make use of the<br />
Primary Decomposition Theorem. According <strong>to</strong> it we have that<br />
V = ker (<strong>α</strong> − λ1id) ⊕ · · · ⊕ ker (<strong>α</strong> − λkid)<br />
= E<strong>α</strong>(λ1) ⊕ · · · ⊕ E<strong>α</strong>(λk).✷<br />
Remark. Suppose the dimension of E<strong>α</strong>(λi) <strong>is</strong> di and Ii <strong>is</strong> the identity <strong>matrix</strong> in Mdi (K).<br />
44
Pick a bas<strong>is</strong> Vi for E<strong>α</strong>(λi) for i = 1, . . ., k. The <strong>matrix</strong> for <strong>α</strong> in Theorem 3.7 <strong>with</strong> <strong>respect</strong><br />
<strong>to</strong> the bas<strong>is</strong> V1 ∪ V2 ∪ · · · ∪ Vk <strong>is</strong> then the diagonal <strong>matrix</strong><br />
⎛<br />
⎞<br />
⎜<br />
⎝<br />
λ1I1<br />
λ2I2<br />
45<br />
. ..<br />
λkIk<br />
⎟<br />
⎠ .