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let-A-1-1-0-1-1-calculate-A-n-2-find-e-A-e-A-3-determine-e-iA-then-cosA-and-sinA-




Question Number 61895 by maxmathsup by imad last updated on 10/Jun/19
let A = (((1       −1)),((0           1)) )  1) calculate A^n   2) find  e^A   ,e^(−A)   3) determine e^(iA)    then  cosA  and sinA .
$${let}\:{A}\:=\begin{pmatrix}{\mathrm{1}\:\:\:\:\:\:\:−\mathrm{1}}\\{\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}}\end{pmatrix} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{A}^{{n}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:\:{e}^{{A}} \:\:,{e}^{−{A}} \\ $$$$\left.\mathrm{3}\right)\:{determine}\:{e}^{{iA}} \:\:\:{then}\:\:{cosA}\:\:{and}\:{sinA}\:. \\ $$
Commented by maxmathsup by imad last updated on 11/Jun/19
we have A =  (((1          0)),((0           1)) )  + (((0         −1)),((0             0)) ) =I +J  we have J^2   = (((0         −1)),((0             0)) ) . (((0          −1)),((0             0)) )  = (((0        0)),((0        0)) )  A^n  =(I +J)^n  =Σ_(k=0) ^n  C_n ^k   J^k  I^(n−k)  = I^n  + n J I^(n−1)  +0 =I +nJ  = (((1         0)),((0          1)) )  + (((0          −n)),((0               0)) ) =  (((1           −n)),((0                1)) )   for n=1  the result is true  let suppise A^n  = (((1        −n)),((0            1)) )  ⇒ A^(n+1)  = A.A^n  = (((1       −1)),((0           1)) ) . (((1        −n)),((0             1)) )  =  ((( 1              −n−1)),((0                      1)) ) =  (((1        −(n+1))),((0                 1)) )   the result is true at term (n+1)  2)  we have  e^A  =Σ_(n=0) ^∞   (A^n /(n!)) =Σ_(n=0) ^∞  (1/(n!))  (((1         −n)),((0              1)) )  =  (((Σ_(n=0) ^∞ (1/(n!))         −Σ_(n=0) ^∞  (n/(n!)))),((0                            Σ_(n=0) ^∞  (1/(n!)))) )  we have e^x  =Σ_(n=0) ^∞  (x^n /(n!)) ⇒Σ_(n=0) ^∞  (1/(n!)) =e   Σ_(n=0) ^∞  (n/(n!)) =Σ_(n=1) ^∞  (1/((n−1)!)) =Σ_(n=0) ^∞  (1/(n!)) =e ⇒ e^A  =  (((e              −e)),((0                   e)) )  we have also e^(−A)  =Σ_(n=0) ^∞  (((−A)^n )/(n!)) =Σ_(n=0) ^∞  (−1)^n  (A^n /(n!)) =Σ_(n=0) ^∞ (((−1)^n )/(n!)) (((1        −n)),((0              1)) )  =  (((Σ_(n=0) ^∞ (((−1)^n )/(n!))       −Σ_(n=0) ^∞  n(((−1)^n )/(n!)))),((0                                  Σ_(n=0) ^∞   (((−1)^n )/(n!)))) )  Σ_(n=0) ^∞   (((−1)^n )/(n!)) =e^(−1)       and  Σ_(n=0) ^∞  ((n(−1)^n )/(n!)) =Σ_(n=1) ^∞  (((−1)^n )/((n−1)!)) =Σ_(n=0) ^∞  (((−1)^(n+1) )/(n!))  =−e^(−1)  ⇒ e^(−A)  =  ((((1/e)            (1/e))),((0                (1/e))) )
$${we}\:{have}\:{A}\:=\:\begin{pmatrix}{\mathrm{1}\:\:\:\:\:\:\:\:\:\:\mathrm{0}}\\{\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}}\end{pmatrix}\:\:+\begin{pmatrix}{\mathrm{0}\:\:\:\:\:\:\:\:\:−\mathrm{1}}\\{\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{0}}\end{pmatrix}\:={I}\:+{J} \\ $$$${we}\:{have}\:{J}^{\mathrm{2}} \:\:=\begin{pmatrix}{\mathrm{0}\:\:\:\:\:\:\:\:\:−\mathrm{1}}\\{\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{0}}\end{pmatrix}\:.\begin{pmatrix}{\mathrm{0}\:\:\:\:\:\:\:\:\:\:−\mathrm{1}}\\{\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{0}}\end{pmatrix}\:\:=\begin{pmatrix}{\mathrm{0}\:\:\:\:\:\:\:\:\mathrm{0}}\\{\mathrm{0}\:\:\:\:\:\:\:\:\mathrm{0}}\end{pmatrix} \\ $$$${A}^{{n}} \:=\left({I}\:+{J}\right)^{{n}} \:=\sum_{{k}=\mathrm{0}} ^{{n}} \:{C}_{{n}} ^{{k}} \:\:{J}^{{k}} \:{I}\:^{{n}−{k}} \:=\:{I}^{{n}} \:+\:{n}\:{J}\:{I}^{{n}−\mathrm{1}} \:+\mathrm{0}\:={I}\:+{nJ} \\ $$$$=\begin{pmatrix}{\mathrm{1}\:\:\:\:\:\:\:\:\:\mathrm{0}}\\{\mathrm{0}\:\:\:\:\:\:\:\:\:\:\mathrm{1}}\end{pmatrix}\:\:+\begin{pmatrix}{\mathrm{0}\:\:\:\:\:\:\:\:\:\:−{n}}\\{\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{0}}\end{pmatrix}\:=\:\begin{pmatrix}{\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:−{n}}\\{\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}}\end{pmatrix}\:\:\:{for}\:{n}=\mathrm{1}\:\:{the}\:{result}\:{is}\:{true} \\ $$$${let}\:{suppise}\:{A}^{{n}} \:=\begin{pmatrix}{\mathrm{1}\:\:\:\:\:\:\:\:−{n}}\\{\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}}\end{pmatrix}\:\:\Rightarrow\:{A}^{{n}+\mathrm{1}} \:=\:{A}.{A}^{{n}} \:=\begin{pmatrix}{\mathrm{1}\:\:\:\:\:\:\:−\mathrm{1}}\\{\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}}\end{pmatrix}\:.\begin{pmatrix}{\mathrm{1}\:\:\:\:\:\:\:\:−{n}}\\{\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}}\end{pmatrix} \\ $$$$=\:\begin{pmatrix}{\:\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\:\:−{n}−\mathrm{1}}\\{\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}}\end{pmatrix}\:=\:\begin{pmatrix}{\mathrm{1}\:\:\:\:\:\:\:\:−\left({n}+\mathrm{1}\right)}\\{\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}}\end{pmatrix}\:\:\:{the}\:{result}\:{is}\:{true}\:{at}\:{term}\:\left({n}+\mathrm{1}\right) \\ $$$$\left.\mathrm{2}\right)\:\:{we}\:{have}\:\:{e}^{{A}} \:=\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{{A}^{{n}} }{{n}!}\:=\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{\mathrm{1}}{{n}!}\:\begin{pmatrix}{\mathrm{1}\:\:\:\:\:\:\:\:\:−{n}}\\{\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}}\end{pmatrix}\:\:=\:\begin{pmatrix}{\sum_{{n}=\mathrm{0}} ^{\infty} \frac{\mathrm{1}}{{n}!}\:\:\:\:\:\:\:\:\:−\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{{n}}{{n}!}}\\{\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{\mathrm{1}}{{n}!}}\end{pmatrix} \\ $$$${we}\:{have}\:{e}^{{x}} \:=\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{{x}^{{n}} }{{n}!}\:\Rightarrow\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{\mathrm{1}}{{n}!}\:={e}\: \\ $$$$\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{{n}}{{n}!}\:=\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{\left({n}−\mathrm{1}\right)!}\:=\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{\mathrm{1}}{{n}!}\:={e}\:\Rightarrow\:{e}^{{A}} \:=\:\begin{pmatrix}{{e}\:\:\:\:\:\:\:\:\:\:\:\:\:\:−{e}}\\{\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{e}}\end{pmatrix} \\ $$$${we}\:{have}\:{also}\:{e}^{−{A}} \:=\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{\left(−{A}\right)^{{n}} }{{n}!}\:=\sum_{{n}=\mathrm{0}} ^{\infty} \:\left(−\mathrm{1}\right)^{{n}} \:\frac{{A}^{{n}} }{{n}!}\:=\sum_{{n}=\mathrm{0}} ^{\infty} \frac{\left(−\mathrm{1}\right)^{{n}} }{{n}!}\begin{pmatrix}{\mathrm{1}\:\:\:\:\:\:\:\:−{n}}\\{\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}}\end{pmatrix} \\ $$$$=\:\begin{pmatrix}{\sum_{{n}=\mathrm{0}} ^{\infty} \frac{\left(−\mathrm{1}\right)^{{n}} }{{n}!}\:\:\:\:\:\:\:−\sum_{{n}=\mathrm{0}} ^{\infty} \:{n}\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}!}}\\{\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}!}}\end{pmatrix} \\ $$$$\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}!}\:={e}^{−\mathrm{1}} \:\:\:\:\:\:{and}\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{{n}\left(−\mathrm{1}\right)^{{n}} }{{n}!}\:=\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{n}} }{\left({n}−\mathrm{1}\right)!}\:=\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{n}+\mathrm{1}} }{{n}!} \\ $$$$=−{e}^{−\mathrm{1}} \:\Rightarrow\:{e}^{−{A}} \:=\:\begin{pmatrix}{\frac{\mathrm{1}}{{e}}\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{1}}{{e}}}\\{\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{1}}{{e}}}\end{pmatrix} \\ $$
Commented by maxmathsup by imad last updated on 11/Jun/19
3) we have e^(iA)  =Σ_(n=0) ^∞   (((iA)^n )/(n!)) =Σ_(n=0) ^∞  (i^n /(n!))  (((1      −n)),((0           1)) )  =  (((Σ_(n=0) ^∞  (i^n /(n!))     −Σ_(n=0) ^∞  ((ni^n )/(n!)))),((0                        Σ_(n=0) ^∞  (i^n /(n!)))) )  we know that e^z  =Σ_(n=0) ^∞   (z^n /(n!)) ⇒Σ_(n=0) ^∞  (i^n /(n!)) =e^i   Σ_(n=0) ^∞  ((ni^n )/(n!)) =Σ_(n=1) ^∞  (i^n /((n−1)!)) =Σ_(n=0) ^∞  (i^(n+1) /(n!)) =i e^i  ⇒  e^(iA)  =  (((   e^i            −i e^i )),((0                   e^i )) )  cosA =Σ_(n=0) ^∞  (((−1)^n  A^(2n) )/(n!)) =Σ_(n=0) ^∞  (((−1)^n )/(n!))  (((1         −2n)),((0             1)) )  =  (((Σ_(n=0) ^∞  (((−1)^n )/(n!))           −2 Σ_(n=0) ^∞  ((n(−1)^n )/(n!)))),((0                                        Σ_(n=0) ^∞  (((−1)^n )/(n:)))) ) =.....
$$\left.\mathrm{3}\right)\:{we}\:{have}\:{e}^{{iA}} \:=\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{\left({iA}\right)^{{n}} }{{n}!}\:=\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{{i}^{{n}} }{{n}!}\:\begin{pmatrix}{\mathrm{1}\:\:\:\:\:\:−{n}}\\{\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}}\end{pmatrix}\:\:=\:\begin{pmatrix}{\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{{i}^{{n}} }{{n}!}\:\:\:\:\:−\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{{ni}^{{n}} }{{n}!}}\\{\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{{i}^{{n}} }{{n}!}}\end{pmatrix} \\ $$$${we}\:{know}\:{that}\:{e}^{{z}} \:=\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{{z}^{{n}} }{{n}!}\:\Rightarrow\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{{i}^{{n}} }{{n}!}\:={e}^{{i}} \\ $$$$\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{{ni}^{{n}} }{{n}!}\:=\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{{i}^{{n}} }{\left({n}−\mathrm{1}\right)!}\:=\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{{i}^{{n}+\mathrm{1}} }{{n}!}\:={i}\:{e}^{{i}} \:\Rightarrow \\ $$$${e}^{{iA}} \:=\:\begin{pmatrix}{\:\:\:{e}^{{i}} \:\:\:\:\:\:\:\:\:\:\:−{i}\:{e}^{{i}} }\\{\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{e}^{{i}} }\end{pmatrix} \\ $$$${cosA}\:=\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{n}} \:{A}^{\mathrm{2}{n}} }{{n}!}\:=\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}!}\:\begin{pmatrix}{\mathrm{1}\:\:\:\:\:\:\:\:\:−\mathrm{2}{n}}\\{\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}}\end{pmatrix} \\ $$$$=\:\begin{pmatrix}{\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}!}\:\:\:\:\:\:\:\:\:\:\:−\mathrm{2}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{{n}\left(−\mathrm{1}\right)^{{n}} }{{n}!}}\\{\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}:}}\end{pmatrix}\:=….. \\ $$
Commented by maxmathsup by imad last updated on 11/Jun/19
forgive  cosA =Σ_(n=0) ^∞  (((−1)^n )/((2n)!)) A^(2n)  =Σ_(n=0) ^∞  (((−1)^n )/((2n)!))  (((1       −2n)),((0              1)) )  =  (((Σ_(n=0) ^∞   (((−1)^n )/((2n)!))            −2 Σ_(n=0) ^∞  (((−1)^n n)/((2n)!)))),((0                                              Σ_(n=0) ^∞  (((−1)^n )/((2n)!)))) )  but Σ_(n=0) ^∞  (((−1)^n )/((2n)!)) =cos(1)    , Σ_(n=0) ^∞  ((n(−1)^n )/((2n)!)) =(1/2)Σ_(n=1) ^∞  (((−1)^n 2n)/((2n)!))  =(1/2) Σ_(n=1) ^∞  (((−1)^n )/((2n−1)!)) =_(n=p+1)    (1/2) Σ_(p=0) ^∞   (((−1)^(p+1) )/((2p+1)!)) =−(1/2)sin(1) ⇒  cosA =  (((cos(1)            sin(1))),((0                         cos(1))) )
$${forgive}\:\:{cosA}\:=\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{n}} }{\left(\mathrm{2}{n}\right)!}\:{A}^{\mathrm{2}{n}} \:=\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{n}} }{\left(\mathrm{2}{n}\right)!}\:\begin{pmatrix}{\mathrm{1}\:\:\:\:\:\:\:−\mathrm{2}{n}}\\{\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}}\end{pmatrix} \\ $$$$=\:\begin{pmatrix}{\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{\left(\mathrm{2}{n}\right)!}\:\:\:\:\:\:\:\:\:\:\:\:−\mathrm{2}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{n}} {n}}{\left(\mathrm{2}{n}\right)!}}\\{\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{n}} }{\left(\mathrm{2}{n}\right)!}}\end{pmatrix} \\ $$$${but}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{n}} }{\left(\mathrm{2}{n}\right)!}\:={cos}\left(\mathrm{1}\right)\:\:\:\:,\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{{n}\left(−\mathrm{1}\right)^{{n}} }{\left(\mathrm{2}{n}\right)!}\:=\frac{\mathrm{1}}{\mathrm{2}}\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{n}} \mathrm{2}{n}}{\left(\mathrm{2}{n}\right)!} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{n}} }{\left(\mathrm{2}{n}−\mathrm{1}\right)!}\:=_{{n}={p}+\mathrm{1}} \:\:\:\frac{\mathrm{1}}{\mathrm{2}}\:\sum_{{p}=\mathrm{0}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{p}+\mathrm{1}} }{\left(\mathrm{2}{p}+\mathrm{1}\right)!}\:=−\frac{\mathrm{1}}{\mathrm{2}}{sin}\left(\mathrm{1}\right)\:\Rightarrow \\ $$$${cosA}\:=\:\begin{pmatrix}{{cos}\left(\mathrm{1}\right)\:\:\:\:\:\:\:\:\:\:\:\:{sin}\left(\mathrm{1}\right)}\\{\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{cos}\left(\mathrm{1}\right)}\end{pmatrix} \\ $$

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