6.6%20Direct%20sum%20decompositions

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6.6 Direct sum decompositions

• The relation to projections

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• W1,,Wk in V subspaces. W1,,Wk are linearly
  independent if a1ak0 implies ai0 for each i.
  
• Lemma. TFAE
• W1,,Wk are independent.
• Bi basis for Wi -gt BB1,,Bk is a basis for W.
• Proof omit
• We write

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• Example
• Example V Mnxn(F)
• W1 all symmetric matrices AtA.
• W2 all antisymmetric matrices At-A.
• Then VW1?W2.
• AA1A2, A1(AAt)/2, A2(A-At)/2.
• Projections EV-gtV, E2E. E is called a
  projection.

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• Example E (x,y,z)-gt(x,y,0).
• Properties Rrange of E. Nnull E.
• b in R lt-gt Ebb.
• (-gt) bEa, EbE(Ea)Eab.
• (lt-) bEb. Done
• I-E is a projection also
• (I-E)(I-E)I-2EE2 1-2EE1-E.
• Im(I-E) null E
• Let a(I-E)b. ab-Eb. EaEb-E2bEb-Eb0. a in
  null E.
• a in null E. Ea0. (I-E)aa-Eaa. a in Im(I-E).

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• a in V. aEa(a-Ea)Ea(I-E)a.
• VIm E ? Im(I-E).
• Ea(I-E)b. EaE2aEb-E2bEb-Eb0. (I-E)b0 also.
• VR ?N.
• Im ER. Im(I-E)N
• Projection is diagonalizable
• Let a1,,ar be a basis of R.
• ar1,, an a basis for N.
• B a1,,an a basis for V which diagonalizes E.

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• Cases where there are a number of projections
  (commuting)
• Theorem 9. If V W1? ?Wk, then there exists k
  linear operators E1,,Ek on V s.t.
• (i) Each Ei is a projection. Ei2Ei.
• (ii) EiEj0 if i?j. (commuting)
• (iii) IE1Ek.
• (iv) Range Ei Wi.
• Conversely, if E1,..,Ek are k linear operators
  satisfying (i)-(iii), then for Wirange Ei, V
  W1? ?Wk.

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• Proof
• (-gt) V W1? ?Wk.
• aa1ak, ai in Wi. Uniquely written.
• Define Eja aj.
• Then Ej2Ej.
• N(Ej) W1??Wj-1?Wj1? ?Wk.
• a E1aEka. IE1Ek.
• Ei Ej0 if i?j. (Wj is in N(Ei)).
• (lt-) Let E1,,Ek linear operators.
• a E1aEka by (iii).
• V W1 Wk.
• The expression is unique.

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• a a1ak. ai in Wi Im Ei. ai Eibi.
• Eja Ej(a1ak) Ej aj Ej Ejbj Ejbj aj .
• By (b) of the Lemma, W1,,Wk are independent.
• V W1? ?Wk
• Note A finite sum of any collection of distinct
  Ei is a projection.
• Check

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6.7. Invariant direct sums

• V W1? ?Wk, Wi T-invariant.
• BB1,,Bk
• Block form

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• Theorem 10 TV-gtV. V W1? ?Wk respective
  E1,,Ek.
• Each Wi is T-invariant lt-gt TEiEiT, i1,,k.
• Proof(lt-) Let a in Wj. aEja.
• TaT(Eja) EjT(a). Ta in Wj.
• Wj is T-invariant.
• (-gt) aE1aEka.
• TaT E1aT Eka.
• Since T(Eja) in Wj , T(Eja)Ejbj for some bj.

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• EjTaEjTE1aEjTEka EjT Eja TEja.Thus,
  EjTTEj
• Theorem 11. T in L(V,V). T is diagonalizable.
  c1,,ck distinct char. Value of T. Then there
  exists projections E1,,Ek s.t.
• (i) Tc1E1ckEk
• (ii) I E1Ek (iii) EiEj0 i?j. (iv) Ei2Ei.
• (v) Range Eichar.v.s. of T ass. ci.
• Conversely, given distinct c1,,ck, E1,,Ek with
  (i)-(iii). Then T is diagonalizable with char.
  values c1,,ck and (iv)(v) also hold.

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• Proof (-gt) T diagonalizable. c1,,ck dist. Char.
  Values.
• V W1? ?Wk , Wi associated with ci.
• aE1aEka.
• TaT E1aT Ekac1 E1ackEka.
• T c1 E1ckEk.
• (lt-) We need to prove (iv)(v) and T is
  diagonalizable.
• (iv) EiI Ei(E1Ek)Ei2.
• T c1 E1ckEk by(i).
• T Ei ciEi by (iii).
• Since Ei is not zero, ci is a char.value.
• T-cI (c1-c)E1(ck-c)Ek .

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• If c?ci for all I, then T-cI has the null-space
  0.
• T has char.values c1,,ck.
• T is diagonalizable since char.v.s. span V.
• (v) null(T-ciI)Im Ei.
• (?) (T-ciI)Eia ((c1-ci)E1(c i-1-ci)E i-1(c
  i1-c i)E i1(ck-ci)Ek ) Eia 0 by (iii).
• (?) If Tacia, then (c1-ci)E1a(c i-1-ci)E
  i-1a(c i1-c i)E i1a(ck-ci)Eka 0.
• (cj-ci)Eja0 for all j ?i.
• Eja0 for all j ?i.
• a in Im Ei.

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• Remark diagonalizable operator T is uniquely
  determined by c1,,ck and E1,,Ek.
• T c1 E1ckEk.
• g(T) g(c1) E1g(ck)Ek
• Proof omitted.
• Let
• Thus, Ej is a polynomial of T and hence commutes
  with T.