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Question Number 201343 by gabi last updated on 05/Dec/23

	Answered by aleks041103 last updated on 05/Dec/23
	$A^3 +A^2 +A=0 λ is eigenvalue ⇒λ^3 +λ^2 +λ=λ(λ^2 +λ+1)=0 ⇒λ=0,w,w^∗ where w=e^(2πi/3) and w^∗ is the complex conjugate of w. let A=PLP^( −1) , where L is the Jordan normal form of A. ⇒P(L^3 +L^2 +L)P^( −1) =0 ⇒L^3 +L^2 +L=0 where L is block−diagonal with Jordan blocks i.e. L=diag(J_(11) ,..,J_(1a) ,J_(21) ,...,J_(2b) ,J_(31) ,...,J_(3c) ) where J_(ks) is a Jordan block with k−th eigenvalue ⇒J_(ks) ^( 3) +J_(ks) ^( 2) +J_(ks) =0 J_(ks) =λ_k E_s +S_s where E_s is the identity over the s×s square matrices and S_s is the s×s square matrix with 0s everywhere, except on the top off−diagonal. ex.: J_(14) = [(λ_1 ,1,0,0),(0,λ_1 ,1,0),(0,0,λ_1 ,1),(0,0,0,λ_1 ) ] J_(14) =λ_1 [(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1) ]+ [(0,1,0,0),(0,0,1,0),(0,0,0,1),(0,0,0,0) ]= =λ_1 E_4 +S_4 It is good to note that S_s ^k =0, if k≥s and {S_s ^k }_(k=0) ^(s−1) ={E_s }∪{S_s ^k }_(k=1) ^(s−1) is lin.indep. Then J_(ks) ^( 3) +J_(ks) ^( 2) +J_(ks) =0 ⇒(λE+S)^3 +(λE+S)^2 +λE+S=0 λ^3 E+3λ^2 S+3λS^2 +S^3 +λ^2 E+2λS+S^2 +λE+S=0 ⇒(λ^3 +λ^2 +λ)E+(3λ^2 +2λ+1)S+(3λ+1)S^2 +S^3 =0 (E,S∈M_(s×s) ) if s>3: {E,S,S^2 ,S^3 }⊆{S_s ^k }_(k=0) ^(s−1) is lin.ind. ⇒ { ((λ^3 +λ^2 +λ=0)),((3λ^2 +2λ+1=0)),((3λ+1=0)),((1=0)) :} which is imposs. if s=3: S^3 =0, {E,S,S^2 } are lin.indep. ⇒ { ((λ^3 +λ^2 +λ=0)),((3λ^2 +2λ+1=0)),((3λ+1=0)) :} which is imposs. if s=2: ⇒S^3 =S^2 =0 and {E,S} are lin.ind. ⇒ { ((λ^3 +λ^2 +λ=0)),((3λ^2 +2λ+1=0)) :} 1)⇒λ=0,w,w^∗ ⇒ λ^2 =−λ−1 or λ=0 replace in 2)⇒ −3(λ+1)+2λ+1=−λ−2⇒λ=−2 but (−2)^3 +(−2)^2 +(−2)≠0 or 3(0)^2 +2(0)+1≠0 ⇒imposs. ⇒s=1 ⇒ the jordan blocks are numbers ⇒A is diagonalisable. ⇒rg(A)=dim(ker(A−wE))+dim(ker(A−w^∗ E)) A is diagon. ⇒ker(A−wE)=span({v_1 ,...,v_k }) where {v_1 ,...,v_k } are lin.ind. and Av_m =wv_m ⇒(Av_m )^∗ =A^∗ v_m ^∗ =Av_m ^∗ =(wv_m )^∗ =w^∗ v_m ^∗ where it is used that A^∗ =A, because A∈M_(n×n) (R). ⇒{v_1 ^∗ ,...,v_k ^∗ } are eigenvectors corresponding to eigenvalue w^∗ ≠w. ⇒span({v_1 ^∗ ,...,v_k ^∗ })≤ker(A−w^∗ E) let u∈ker(A−w^∗ E)/span({v_1 ^∗ ,...,v_k ^∗ }) and u≠0 then, Au=w^∗ u ⇒Au^∗ =wu^∗ ⇒u^∗ ∈span({v_1 ,...,v_k }) ⇒u∈span({v_1 ^∗ ,...,v_k ^∗ }) ⇒u∈[span({v_1 ^∗ ,...,v_k ^∗ })]∩[ker(A−w^∗ E)/span({v_1 ^∗ ,...,v_k ^∗ })]={0} ⇒u=0 ⇒ contradiction ⇒ker(A−w^∗ E)=span({v_1 ^∗ ,...,v_k ^∗ }) ⇒rg(A)=dim(ker(A−wE))+dim(ker(A−w^∗ E))= =dim(span({v_1 ,...,v_k }))+dim(span({v_1 ^∗ ,...,v_k ^∗ }))= =k+k=2k ⇒2∣rg(A)$
	$A^{3} + A^{2} + A = 0$ $λ i s e i g e n v a l u e$ $\Rightarrow λ^{3} + λ^{2} + λ = λ (λ^{2} + λ + 1) = 0$ $\Rightarrow λ = 0, w, w^{}$ $w h e r e w = e^{2 π i / 3} a n d w^{} i s t h e c o m p l e x$ $c o n j u g a t e o f w .$ $l e t A = {P L P}^{- 1}, w h e r e L i s t h e J o r d a n n o r m a l$ $f o r m o f A .$ $\Rightarrow P (L^{3} + L^{2} + L) P^{- 1} = 0$ $\Rightarrow L^{3} + L^{2} + L = 0$ $w h e r e L i s b l o c k - d i a g o n a l w i t h J o r d a n b l o c k s$ $i . e .$ $L = d i a g (J_{11}, . ., J_{1 a}, J_{21}, . . ., J_{2 b}, J_{31}, . . ., J_{3 c})$ $w h e r e J_{k s} i s a J o r d a n b l o c k w i t h k - t h$ $e i g e n v a l u e$ $\Rightarrow J_{k s}^{3} + J_{k s}^{2} + J_{k s} = 0$ $J_{k s} = λ_{k} E_{s} + S_{s}$ $w h e r e E_{s} i s t h e i d e n t i t y o v e r t h e s \times s s q u a r e$ $m a t r i c e s a n d S_{s} i s t h e s \times s s q u a r e m a t r i x$ $w i t h 0 s e v e r y w h e r e, e x c e p t o n t h e t o p$ $o f f - d i a g o n a l .$ $e x . :$ $J_{14} = [\begin{matrix} λ_{1} & 1 & 0 & 0 \\ 0 & λ_{1} & 1 & 0 \\ 0 & 0 & λ_{1} & 1 \\ 0 & 0 & 0 & λ_{1} \end{matrix}]$ $J_{14} = λ_{1} [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] + [\begin{matrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{matrix}] =$ $= λ_{1} E_{4} + S_{4}$ $I t i s g o o d t o n o t e t h a t$ $S_{s}^{k} = 0, i f k ⩾ s$ $a n d {S_{s}^{k}}_{k = 0}^{s - 1} = {E_{s}} \cup {S_{s}^{k}}_{k = 1}^{s - 1} i s l i n . i n d e p .$ $T h e n$ $J_{k s}^{3} + J_{k s}^{2} + J_{k s} = 0$ $\Rightarrow {(λ E + S)}^{3} + {(λ E + S)}^{2} + λ E + S = 0$ $λ^{3} E + 3 λ^{2} S + 3 λ S^{2} + S^{3} + λ^{2} E + 2 λ S + S^{2} + λ E + S = 0$ $\Rightarrow (λ^{3} + λ^{2} + λ) E + (3 λ^{2} + 2 λ + 1) S + (3 λ + 1) S^{2} + S^{3} = 0$ $(E, S \in M_{s \times s})$ $i f s > 3 :$ ${E, S, S^{2}, S^{3}} \subseteq {S_{s}^{k}}_{k = 0}^{s - 1} i s l i n . i n d .$ $\Rightarrow {\begin{cases} λ^{3} + λ^{2} + λ = 0 \\ 3 λ^{2} + 2 λ + 1 = 0 \\ 3 λ + 1 = 0 \\ 1 = 0 \end{cases} w h i c h i s i m p o s s .$ $i f s = 3 :$ $S^{3} = 0, {E, S, S^{2}} a r e l i n . i n d e p .$ $\Rightarrow {\begin{cases} λ^{3} + λ^{2} + λ = 0 \\ 3 λ^{2} + 2 λ + 1 = 0 \\ 3 λ + 1 = 0 \end{cases} w h i c h i s i m p o s s .$ $i f s = 2 :$ $\Rightarrow S^{3} = S^{2} = 0 a n d {E, S} a r e l i n . i n d .$ $\Rightarrow {\begin{cases} λ^{3} + λ^{2} + λ = 0 \\ 3 λ^{2} + 2 λ + 1 = 0 \end{cases}$ $1) \Rightarrow λ = 0, w, w^{} \Rightarrow λ^{2} = - λ - 1 o r λ = 0$ $r e p l a c e i n 2) \Rightarrow - 3 (λ + 1) + 2 λ + 1 = - λ - 2 \Rightarrow λ = - 2$ $b u t {(- 2)}^{3} + {(- 2)}^{2} + (- 2) \neq 0$ $o r 3 {(0)}^{2} + 2 (0) + 1 \neq 0$ $\Rightarrow i m p o s s .$ $\Rightarrow s = 1 \Rightarrow t h e j o r d a n b l o c k s a r e n u m b e r s$ $\Rightarrow A i s d i a g o n a l i s a b l e .$ $\Rightarrow r g (A) = d i m (k e r (A - w E)) + d i m (k e r (A - w^{} E))$ $A i s d i a g o n .$ $\Rightarrow k e r (A - w E) = s p a n ({v_{1}, . . ., v_{k}})$ $w h e r e {v_{1}, . . ., v_{k}} a r e l i n . i n d . a n d$ ${A v}_{m} = {w v}_{m}$ $\Rightarrow {({A v}_{m})}^{} = A^{} v_{m}^{} = {A v}_{m}^{} = {({w v}_{m})}^{} = w^{} v_{m}^{}$ $w h e r e i t i s u s e d t h a t A^{} = A, b e c a u s e$ $A \in M_{n \times n} (R) .$ $\Rightarrow {v_{1}^{}, . . ., v_{k}^{}} a r e e i g e n v e c t o r s c o r r e s p o n d i n g$ $t o e i g e n v a l u e w^{} \neq w .$ $\Rightarrow s p a n ({v_{1}^{}, . . ., v_{k}^{}}) ⩽ k e r (A - w^{} E)$ $l e t u \in k e r (A - w^{} E) / s p a n ({v_{1}^{}, . . ., v_{k}^{}}) a n d$ $u \neq 0$ $t h e n, A u = w^{} u$ $\Rightarrow {A u}^{} = {w u}^{} \Rightarrow u^{} \in s p a n ({v_{1}, . . ., v_{k}})$ $\Rightarrow u \in s p a n ({v_{1}^{}, . . ., v_{k}^{}})$ $\Rightarrow u \in [s p a n ({v_{1}^{}, . . ., v_{k}^{}})] \cap [k e r (A - w^{} E) / s p a n ({v_{1}^{}, . . ., v_{k}^{}})] = {0}$ $\Rightarrow u = 0$ $\Rightarrow c o n t r a d i c t i o n$ $\Rightarrow k e r (A - w^{} E) = s p a n ({v_{1}^{}, . . ., v_{k}^{}})$ $\Rightarrow r g (A) = d i m (k e r (A - w E)) + d i m (k e r (A - w^{} E)) =$ $= d i m (s p a n ({v_{1}, . . ., v_{k}})) + d i m (s p a n ({v_{1}^{}, . . ., v_{k}^{}})) =$ $= k + k = 2 k$ $\Rightarrow 2 ∣ r g (A)$
	Commented by aleks041103 last updated on 05/Dec/23

	$W e {c o u l d}^{'} v e s o l v e d i t i n a n e v e n e a s i e r m a n n e r$ $l e t t h e m i n i m a l p o l y n o m i a l o f A b e m_{A} (x) .$ $(b y d e f m_{A} (x) i s t h e l o w e s t o r d e r p o l y n o m i a l$ $t h a t h a s t h e p r o p e r t y m_{A} (A) = 0)$ $s i n c e f (A) = A^{3} + A^{2} + A = 0, t h e n m_{A} ∣ f .$ $t h e s p e c t r u m o f f (x) = 0 i s x = 0, w, w^{*} .$ $\Rightarrow f h a s a s i m p l e s p e c t r u m .$ $s i n c e m_{A} ∣ f \Rightarrow m_{A} h a s a s i m p l e s p e c t r u m t o o .$ $T h e r e i s a t h e o r e m, w h i c h w e e s s e n t i a l l y$ $p r o v e d f o r t h i s s p e c i f i c c a s e, t h a t s t a t e s$ $t h a t t h e m a t r i x A i s d i a g o n a l i z a b l e$ $i f a n d o n l y i f t h e m i n i m a l p o l y n o m i a l$ $o f A h a s s i m p l e s p e c t r u m .$ $T h u s, s i n c e m_{A} h a s s i m p l e s p e c t r u m$ $t h e n A i s d i a h l g o n a l i s a b l e .$ $F r o m h e r e o n f o r w a r d i t i s t h e s a m e .$
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