Application: Find e^{At}


    
        
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A = matrix(2,2,[2, -12, 1, -5])   
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f = A.characteristic_polynomial()
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show(f)
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f.factor()

    
    
    
    
        
            
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Find the corresponding eigenvectors. Thus we find a diagonal matrix

D

and an invertible matrix

P

such that

P^{- 1} A P = D or A = P D P^{- 1} .


    
        
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A.eigenvectors_right()
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D, P = A.eigenmatrix_right()

    
    
    
    
        
            
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Therefore,

\begin{aligned} e^{A} & = e^{P D P^{- 1}} \\ = I + P D P^{- 1} + \frac{(P D P^{- 1}) (P D P^{- 1})}{2!} + \dots \\ = P P^{- 1} + P D P^{- 1} + \frac{P D^{2} P^{- 1}}{2!} + \dots \\ = P (I + D + \frac{D^{2}}{2!} + \dots) P^{- 1} \\ = P ((\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}) + (\begin{array}{c} - 1 & 0 \\ 0 & - 2 \end{array}) + \frac{1}{2!} (\begin{array}{c} (- 1)^{2} & 0 \\ 0 & (- 2)^{2} \end{array}) + \dots) P^{- 1} \\ = P (\begin{array}{c} 1 + (- 1) + \frac{(- 1)^{2}}{2!} + \dots & 0 \\ 0 & 1 + (- 2) + \frac{(- 2)^{2}}{2!} + \dots \end{array}) P^{- 1} \\ = P (\begin{array}{c} e^{- 1} & 0 \\ 0 & e^{- 2} \end{array}) P^{- 1} \\ = (\begin{array}{c} \frac{4}{e} - \frac{3}{e^{2}} & \frac{12}{e^{2}} - \frac{12}{e} \\ \frac{1}{e} - \frac{1}{e^{2}} & \frac{4}{e^{2}} - \frac{3}{e} \end{array}) \end{aligned}


    
        
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B = matrix(RR,2,2,[e^(-1),0,0,e^(-2)])
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C = P*B*P.inverse() 
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pretty_print(C)
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P, P.inverse()
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(4/e-3/(e^(2))).n()

    
    
    
    
        
            
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🔗

Activity 8.4.1.

🔗

Supose that

A

is diagonalizable. How to compute

e^{A t} ?

🔗

Solution.

Since

A

is diagonalizable, there exists an invertible matrix

P

and a diagonal matrix

D = (\begin{matrix} λ_{1} \\ ⋱ \\ λ_{n} \end{matrix})

such that

P^{- 1} A P = D .

It is easy to see that

P^{- 1} (A t) P = P^{- 1} (A (t I_{n})) P = P^{- 1} A P (t I_{n}) = D (t I_{n}) = t D .

That is,

e^{A t} = P (\begin{matrix} e^{λ_{1}} \\ ⋱ \\ e^{λ_{n}} \end{matrix}) P^{- 1}

🔗

Exercise: compute

e^{A t},

where

A = (\begin{matrix} 1 & 2 & - 2 \\ - 2 & 5 & - 2 \\ - 6 & 6 & - 3 \end{matrix})

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Section 8.4 Application: Find eAt

Activity 8.4.1.

Section 8.4 Application: Find $e^{A t}$