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Project: edm
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var('t') assume(t,'real') M=matrix([[1+4*t,1+8*t+2*t*t,-1+6*t+2*t*t],[2,1+2*t,2*t],[0,2,2]]) W=e^t*M W
t [ (4*t + 1)*e^t (2*t^2 + 8*t + 1)*e^t (2*t^2 + 6*t - 1)*e^t] [ 2*e^t (2*t + 1)*e^t 2*t*e^t] [ 0 2*e^t 2*e^t]
I=W.inverse() I
[e^(-t)/(4*t + 1) + 2*((2*t^2 + 6*t - 1)/(4*t + 1) - 2*(2*t^2 + 8*t + 1)*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))/(((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1))*(4*t + 1)))*e^t/(((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1))*(4*t + 1)*(2*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))*e^t/((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1)) - e^t)) + 2*(2*t^2 + 8*t + 1)/(((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1))*(4*t + 1)^2) -((2*t^2 + 6*t - 1)/(4*t + 1) - 2*(2*t^2 + 8*t + 1)*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))/(((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1))*(4*t + 1)))*e^t/(((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1))*(2*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))*e^t/((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1)) - e^t)) - (2*t^2 + 8*t + 1)/(((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1))*(4*t + 1)) 1/2*((2*t^2 + 6*t - 1)/(4*t + 1) - 2*(2*t^2 + 8*t + 1)*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))/(((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1))*(4*t + 1)))/(2*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))*e^t/((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1)) - e^t)] [ -2/(((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1))*(4*t + 1)) + 4*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))*e^t/(((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1))^2*(4*t + 1)*(2*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))*e^t/((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1)) - e^t)) 1/((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1)) - 2*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))*e^t/(((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1))^2*(2*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))*e^t/((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1)) - e^t)) (t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))/(((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1))*(2*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))*e^t/((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1)) - e^t))] [ -2*e^t/(((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1))*(4*t + 1)*(2*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))*e^t/((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1)) - e^t)) e^t/(((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1))*(2*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))*e^t/((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1)) - e^t)) -1/2/(2*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))*e^t/((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1)) - e^t)]
D=diff(W) D
[ (4*t + 1)*e^t + 4*e^t (2*t^2 + 8*t + 1)*e^t + 4*(t + 2)*e^t (2*t^2 + 6*t - 1)*e^t + 2*(2*t + 3)*e^t] [ 2*e^t (2*t + 1)*e^t + 2*e^t 2*t*e^t + 2*e^t] [ 0 2*e^t 2*e^t]
A=D*I A
[((4*t + 1)*e^t + 4*e^t)*(e^(-t)/(4*t + 1) + 2*((2*t^2 + 6*t - 1)/(4*t + 1) - 2*(2*t^2 + 8*t + 1)*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))/(((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1))*(4*t + 1)))*e^t/(((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1))*(4*t + 1)*(2*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))*e^t/((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1)) - e^t)) + 2*(2*t^2 + 8*t + 1)/(((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1))*(4*t + 1)^2)) - 2*((2*t^2 + 8*t + 1)*e^t + 4*(t + 2)*e^t)*(1/(((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1))*(4*t + 1)) - 2*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))*e^t/(((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1))^2*(4*t + 1)*(2*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))*e^t/((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1)) - e^t))) - 2*((2*t^2 + 6*t - 1)*e^t + 2*(2*t + 3)*e^t)*e^t/(((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1))*(4*t + 1)*(2*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))*e^t/((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1)) - e^t)) -((4*t + 1)*e^t + 4*e^t)*(((2*t^2 + 6*t - 1)/(4*t + 1) - 2*(2*t^2 + 8*t + 1)*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))/(((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1))*(4*t + 1)))*e^t/(((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1))*(2*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))*e^t/((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1)) - e^t)) + (2*t^2 + 8*t + 1)/(((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1))*(4*t + 1))) + ((2*t^2 + 8*t + 1)*e^t + 4*(t + 2)*e^t)*(1/((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1)) - 2*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))*e^t/(((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1))^2*(2*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))*e^t/((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1)) - e^t))) + ((2*t^2 + 6*t - 1)*e^t + 2*(2*t + 3)*e^t)*e^t/(((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1))*(2*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))*e^t/((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1)) - e^t)) 1/2*((4*t + 1)*e^t + 4*e^t)*((2*t^2 + 6*t - 1)/(4*t + 1) - 2*(2*t^2 + 8*t + 1)*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))/(((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1))*(4*t + 1)))/(2*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))*e^t/((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1)) - e^t) - 1/2*((2*t^2 + 6*t - 1)*e^t + 2*(2*t + 3)*e^t)/(2*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))*e^t/((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1)) - e^t) + ((2*t^2 + 8*t + 1)*e^t + 4*(t + 2)*e^t)*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))/(((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1))*(2*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))*e^t/((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1)) - e^t))] [ -2*((2*t + 1)*e^t + 2*e^t)*(1/(((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1))*(4*t + 1)) - 2*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))*e^t/(((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1))^2*(4*t + 1)*(2*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))*e^t/((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1)) - e^t))) + 2*(e^(-t)/(4*t + 1) + 2*((2*t^2 + 6*t - 1)/(4*t + 1) - 2*(2*t^2 + 8*t + 1)*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))/(((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1))*(4*t + 1)))*e^t/(((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1))*(4*t + 1)*(2*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))*e^t/((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1)) - e^t)) + 2*(2*t^2 + 8*t + 1)/(((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1))*(4*t + 1)^2))*e^t - 4*(t*e^t + e^t)*e^t/(((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1))*(4*t + 1)*(2*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))*e^t/((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1)) - e^t)) ((2*t + 1)*e^t + 2*e^t)*(1/((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1)) - 2*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))*e^t/(((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1))^2*(2*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))*e^t/((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1)) - e^t))) - 2*(((2*t^2 + 6*t - 1)/(4*t + 1) - 2*(2*t^2 + 8*t + 1)*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))/(((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1))*(4*t + 1)))*e^t/(((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1))*(2*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))*e^t/((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1)) - e^t)) + (2*t^2 + 8*t + 1)/(((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1))*(4*t + 1)))*e^t + 2*(t*e^t + e^t)*e^t/(((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1))*(2*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))*e^t/((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1)) - e^t)) ((2*t^2 + 6*t - 1)/(4*t + 1) - 2*(2*t^2 + 8*t + 1)*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))/(((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1))*(4*t + 1)))*e^t/(2*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))*e^t/((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1)) - e^t) + ((2*t + 1)*e^t + 2*e^t)*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))/(((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1))*(2*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))*e^t/((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1)) - e^t)) - (t*e^t + e^t)/(2*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))*e^t/((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1)) - e^t)] [ -4*(1/(((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1))*(4*t + 1)) - 2*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))*e^t/(((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1))^2*(4*t + 1)*(2*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))*e^t/((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1)) - e^t)))*e^t - 4*e^(2*t)/(((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1))*(4*t + 1)*(2*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))*e^t/((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1)) - e^t)) 2*(1/((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1)) - 2*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))*e^t/(((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1))^2*(2*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))*e^t/((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1)) - e^t)))*e^t + 2*e^(2*t)/(((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1))*(2*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))*e^t/((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1)) - e^t)) 2*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))*e^t/(((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1))*(2*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))*e^t/((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1)) - e^t)) - e^t/(2*(t*e^t - (2*t^2 + 6*t - 1)*e^t/(4*t + 1))*e^t/((2*t + 1)*e^t - 2*(2*t^2 + 8*t + 1)*e^t/(4*t + 1)) - e^t)]
y=A.simplify_full() y
[1 2 3] [0 1 1] [0 0 1]
D==A*W D==y*W
True True