for each of its columns. A quick calculation will verify that: thus m < dim(v). Then a basis of V with respect to which T has the form: A

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1 Desty of dagoalzable square atrces Studet: Dael Cervoe; Metor: Saravaa Thyagaraa Uversty of Chcago VIGRE REU, Suer 7. For ths etre aer, we wll refer to V as a vector sace over ad L(V) as the set of lear oerators { Α : } A V V. Recall the followg defto: f A s a lear oerator o a vector sace V, ad v V ad λ st A v = λv, the v ad λ are a egevector ad egevalue of A, resectvely. Theore : A atrx s called dagoalzable f t s slar to soe dagoal atrx. If A L(V) has dstct egevalues the A s dagoalzable. Proof: Let w w (assug = ) be the egevectors that corresod to each egevalue. Let W be the atrx that has w w for each of ts colus. A quc calculato wll verfy that: a, a, λ w w w = w w w a, a, λ LHS = Aw Aw A w ad RHS = λw λw λw ad clearly A w = λw. Ad we ow that W s vertble sce the fact that the egevalues of A are dstct les that w w are learly deedet. Thus: a, a, λ = w w w w w w a a λ,, Ths roves the theore. Theore : Suose T L(V) wth odstct egevalues. Let λ λ be the dstct egevalues of T, thus < d(v). The a bass of V wth resect to whch T has the for: A λ where each A s a uer tragular atrx of the for: A λ Proof : let U be the subsace of geeralzed egevectors of T corresodg to λ : { v λ v }, U = V : (T = for soe. It follows fro ths edately that, U = ull(t λ, ad that (T λ s lotet. Note that f A s ay lear oerator o V, the ull(a) s a subsace of V sce t cotas ad clearly satsfes closure uder addto ad scalar ultlcato (these follow fro A beg lear). Before cotug, we eed a crucal lea: U

2 Lea : If N s a lotet lear oerator o a vector sace X, the a bass of X wth resect to whch N has the for: (.e. N has 's o ad below the dagoal). Proof of Lea: Frst ote that N lotet o X st N = [ ] X = ull(n ) { } { } Next, choose a bass b,, b of ull(n) ad exted ths to a bass b,, b of ull(n ), where. We ca do ths because f v ull(n), the N v = [ ], so clearly N(N v) = [ ]. Thus ull(n) ull(n ). Ad sce b,, b are learly deedet vectors that sa ull(n), we ca sa ull(n ) by b,, b ad or ore learly deedet vectors b,, b ull(n ) that do ot deed o b, +, b. 3 We ca ee extedg the bass of ull(n ) to a bass of ull(n ) ad evetually ull(n ). I dog so, we establsh a bass B = { b,, b} of X, sce B s a bass of ull(n ) = X. Now let us cosder N wth resect to ths bass. We ow that by chagg the bass of N, we ca wrte N wth resect to B as the atrx: [Nb Nb Nb Nb ] where each colu s the ( ) vector N b Sce b ull(n), the frst colu wll be etrely. Ths s fact true for each colu through. The ext colu s N b, where Nb ull(n) sce N b = (recall that b ull(n )). + + Nb ull(n) N b s a lear cobato of b b all ozero etres the + colu + + le above the dagoal. Ths s fact true for all colus fro where b b sa ull(n ). Slarly, we ca tae the ext colu, N b, whch s ull(n ) sce b s a bass vector of ull(n ). Thus N b+ deeds o b b ad ay ozero etres the + colu le above the dagoal. We ca cotue ths rocess through colu, thus cofrg that N wth resect to the bass B s of the for:. Ths roves the lea. We ow cotue the roof of the theore. Recall that (T λ s lotet. Thus, by the lea we ust roved, a bass B of U st wth resect to B : λ (T λ U =, ad therefore T U =. λ U

3 { } Moreover, f B s the bass B...B of U where U = U U U the T wth resect to B s the for: U T λ where each T s a uer tragular atrx of the for: T λ Note that ths s the desred for corresodg to our theore. However, we stll eed to show that ths for s ossble for T wth resect to a bass of V. It suffces to show that V = U, the clearly a bass of U s a bass of V. To do ths, cosder the lear oerator S L(V) where S = (T λ. Our cla s that S U =. = To verfy ths, cosder that ull(t λ ull(t λ ull(t λ stateet to the followg lea: + + for ay. We wat to stregthe ths Lea : ull(t λ ull(t λ ull(t λ = ull(t λ = ull(t λ Proof : Suose st ull(t λ = ull( T λ. If x ull(t λ for the (T λ x=. = = + + (T λ (T λ x (T λ x ull(t λ (T λ Thus ull(t λ ull(t λ ull(t λ = ull(t λ. + So ull(t λ = ull(t λ + + ull(t λ ull(t λ ull(t λ = ull(t λ = ull(t λ. + Now we wat to show that ull(t λ = ull(t λ. To rove ths, assue the cotrary,.e.: + ull(t λ ull(t λ ull(t λ ull(t λ. Sce each ull(t λ s a + + subsace of V, ull(t λ ull(t λ d(ull(t λ ) + d(ull(t λ ) + sce the ter left of " " has a lower d tha the oe to the rght. But the d(ull(t λ ) > d V, + whch s a cotradcto sce ull(t λ s a subsace of V. Therefore the followg s true: + + ull(t λ ull(t λ ull(t λ = ull(t λ = ull(t λ. Ths coletes the roof of lea. We aga retur to verfyg that S =. Now cosder S u for soe u U. U u U u= u + u u for u U. Sce atrx ultlcato s dstrbutve, Su= Su + Su S u. Moreover, we ow that,, (T λ ad (T λ are coutable (ths s because ther roduct ether drecto cossts of ters of T of soe order ad ters of TI or IT of soe order. Ad clearly T coutes wth T ad I coutes wth ay atrx). So S u = (T λ (T λ (T λ (T λ (T λ u. Of course, (T λ u = + sce (T λ u. Thus Su= ad we have rove our cla that S U =, whch gves U ull(s).

4 Yet suose u ull(s). The ( (T λ )( u)=. Therefore for soe, (T λ )( u)=. = u U u U ull(s) U ull(s) = U. Now, we have show that,, (T λ ad (T λ are coutabl e. Fro ths t follows that S ad T are coutable. For a vector v V, of course S(T v) Ig(S). Yet sce T ad S coute, T(S v) Ig(S) (.e. Ig(S) s varat over T). Let us assue that Ig(S). Ig(S) varat over T w Ig(S) where w s a egevector for T. Moreover, w Ig(S) x V st S x s a egevector of T. By defto, Sx, thus x ull(s). But S x s a egevector of T, so clearly SS x =. Thus, ull(s) ull(s ). λ = = Ths cotradcts lea, sce ull(s) ull(s ) d(ull( (T ) ) < d(ull( (T λ ) ). Therefore Ig(S)=. If we aly the ra-ullty theore to S:V V, we get: = d(ull(s)) + d(ig(s)). Ig(S)= d(ig(s))=, so = d(ull(s)). We showed earler that U = ull(s), so du=. Ad U beg a subsace of V ad du= U=V. Thus, a bass of U s also a bass of V. Ths roves the theore. Theore 3: A st A L(V), A has dstct egevalues, ad A T. Proof: Theore ( lght of the recet observato) shows that T st T T ad T λ T ca be wrtte: where λ are egevalues λ but ot ecessarly dstct fro oe aother ( does ot ly λ λ ). λ T Now let A be where λ λ ad, = λ = λ λ (the egevalues of each A are dstct). The fact that λ λ A T etrywse A T. But T T osgular atrx P st T = P T P. Now A T P A P T sce atrx ultlcato s - - cotuous (ths s farly easy to verfy: f v v the surely A v v etrywse Av A v. Ad f a sequece of atrces X X, the clearly the colu vectors coverge, thus AX AX. Ad sce A P A P, the egevalues of P AP are equal to thoseof A. Thus P AP s a sequece of atrces wth dstct egevalues ad - P AP T. Ths roves the theore. Observato: If T L(V) ad a bass of V wth resect to whch T s tragular, ths s equvalet to sayg - that T L(V) st T s tragular ad T T (T s slar to T ),.e. osgular atrx P st T = P T P.

5 Corollary: Theores,, ad 3 ly that ay square atrx s a lt ot of a sequece of square atrces wth dstct egevalues. By defto the, square atrces wth dstct egevalues are dese L(V). Ad Theore shows that ay square atrx wth dstct egevalues s dagolzable, thus the dagoalzable atrces are also dese L(V). REFERENCES: Axler, Sheldo. Lear Algebra Doe Rght htt://laetath.org