let-give-A-1-1-2-1-find-e-A-and-e-tA-

Question Number 28258 by abdo imad last updated on 22/Jan/18

l e t g i v e A = (1 1)

(2 - 1)

f i n d e^{A} a n d e^{- t A} .

Commented by abdo imad last updated on 23/Jan/18

e^{A} = \sum_{n = 0}^{\propto} \frac{A^{n}}{n!} l e t f i n d A^{n} t h e c a r s c t e r i s t i c p o l y n o m i a l

o f A i s P_{c} (A) = d e t (A - X I) = | \begin{matrix} 1 - X 1 \\ 2 - 1 - X \end{matrix} |

= - (1 - X^{2}) - 2 = X^{2} - 3 = (X - \sqrt{3}) (X + \sqrt{3)}

s o t h e p r o p e r v a l u e s o f A ? a r e λ_{1} = \sqrt{3} a n d λ_{2} = - \sqrt{3}

V (λ_{1}) = k e r (A - λ_{1} I)

u (x, y) \in V (λ_{1}) \Leftrightarrow (A - λ_{1} I) u = 0

(\begin{matrix} 1 - \sqrt{3} 1 \\ 2 - 2 - \sqrt{3} \end{matrix}) (\begin{matrix} x \\ y \end{matrix}) = (\begin{matrix} o \\ o \end{matrix}) a f t e r c s l c u l u s w e f i n d

V (λ_{1}) = D_{e_{1}} w i t h e_{1} (\begin{matrix} 1 \\ - (1 - \sqrt{3}) \end{matrix})

V (λ_{2}) = k e r (A - λ_{2} I)

u (x, y) \in V (λ_{2}) \Leftrightarrow (A - λ_{2} I) u = 0

\Leftrightarrow (\begin{matrix} 1 + \sqrt{3} 1 \\ 2 - 1 + \sqrt{3} \end{matrix}) (\begin{matrix} x \\ y \end{matrix}) = (\begin{matrix} 0 \\ 0 \end{matrix}) a f t e r c a l c u l u s w e f i n d

V (λ_{2}) = D_{e_{2}} w i t h e_{2} (\begin{matrix} 1 \\ - (1 + \sqrt{3)} \end{matrix}) s o t h e d i a g o n a l

m s t r i x i s D = (\begin{matrix} \sqrt{3} 0 \\ 0 - \sqrt{3_{}} \end{matrix}) a n d t h e i n v e r s i b l e

m a t r i x i d P = (\begin{matrix} 1 1 \\ - (1 - \sqrt{3} - (1 + \sqrt{3}) \end{matrix}) w e h a v e

P^{- 1} = \frac{t r a n (c o m P)}{d e t P} . c o m (P) = {(- 1)}^{i + j} a_{i j} w e f i n d

P^{- 1} = \frac{1}{2 \sqrt{3}} (\begin{matrix} 1 + \sqrt{3} 1 \\ - (1 - \sqrt{3)} - 1 \end{matrix}) w e h a v e A = {P D P}^{- 1}

\Rightarrow A^{n} = P D^{n} P^{- 1} a f t e r a l l c a l c u l u s w e f i n d

A^{n} = \frac{1}{2 \sqrt{3}} ({(\sqrt{3})}^{n} (1 + \sqrt{3}) - {(- \sqrt{3})}^{n} (1 - \sqrt{3}) {(\sqrt{3})}^{n} - {(- \sqrt{3})}^{n})

(2 {(\sqrt{3})}^{n} - 2 {(- \sqrt{3})}^{n} - (1 - \sqrt{3}) {(\sqrt{3})}^{n} + (1 + \sqrt{3}) {(- \sqrt{3})}^{n}) .

w e h a v e e^{A} = \sum_{n = 0}^{\propto} \frac{A^{n}}{n!} a n d e^{t A} = \sum_{n = 0}^{\propto} A^{n} \frac{t^{n}}{n!} .