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Question Number 188774 by Shrinava last updated on 06/Mar/23

	Answered by aleks041103 last updated on 09/Mar/23

	$D i a g o n l i z e A, i . e . A = P^{- 1} D P, w h e r e D$ $i s d i a g o n a l .$ $N o w y o u c a n c h o s e a s m a n y a s y o u w a n t$ $t w o m o r e d i a g o n a l m a t r i c e s D_{1} a n d D_{2},$ $s u c h t h a t$ $D^{2023} = D_{1}^{2023} + D_{2}^{2023} \Rightarrow$ $P^{- 1} D^{2023} P = P^{- 1} D_{1}^{2023} P + P^{- 1} D_{2}^{2023} P$ $N o w n o t i c e t h a t :$ $P^{- 1} S^{n} P = P^{- 1} S S S . . . S P =$ $= P^{- 1} {S P P}^{- 1} {S P P}^{- 1} . . . . P^{- 1} S P$ $= {(P^{- 1} S P)}^{n}$ $\Rightarrow {(P^{- 1} D P)}^{2023} = A^{2023} = {(P^{- 1} D_{1} P)}^{2023} + {(P^{- 1} D_{2} P)}^{2023}$ $A n d n o w i f B = P^{- 1} D_{1} P a n d C = P^{- 1} D_{2} P,$ $w e h a v e A^{2023} = B^{2023} + C^{2023 .}$ $\Rightarrow f o r e v e r y d i a g o n a l m a t r i x D_{1}, t h e m a t r i c e s$ $B = P^{- 1} D_{1} P a n d C = P^{- 1} \sqrt[2023]{D^{2023} - D_{1}^{2023}} P$ $s a t i s f y$ $A^{2023} = B^{2023} + C^{2023}$ $w h e r e A = P^{- 1} D P a n d D i s d i a g o n a l .$ $P . S . A i s d k a o n l i z a b l e, s i n c e i t i s a s y m m e t r i c$ $m a t r i x w i t h r e a l e n t r i e s .$
	Commented by Shrinava last updated on 09/Mar/23

	$Dear professor, thank you very much$ $the for detailed explanation . .$ ${Let}^{'} s note how in response . ?$
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