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- Homework Statement
- Given the matrix A = {{1, 0}, {1, 0}}, find the matrix exponential e^(At)

- Relevant Equations
- e^(Mt) = I + Mt + (Mt)^2/2! + ... + (Mt)^k/k!

e^t = 1 + t + t^2/2! + t^3/3! + ... + t^k/k!

I've attempted to solve this by separating A into a diagonal matrix D and nilpotent matrix N:

D = {{1, 0}, {0, 0}}

N = {{0, 0}, {1, 0}}

e^(At) = e^((D + N)t) = e^(Dt) * e^(Nt)

When N is raised to the second power, it becomes the zero matrix. Therefore,

e^(Nt) = I + Nt = {{1, 0}, {t, 1}}

Note that D^2 = D

e^(Dt) = I + Dt + Dt^2/2! + ... + Dt^k/k! = {{1, 0}, {0, 1}} + {{1, 0}, {0, 0}}t + {{1, 0}, {0, 0}}t^2/2! + ... + {{1, 0}, {0, 0}}t^k/k!

e^(Dt) = {{1 + t + t^2/2! + ...+ t^k/k!, 0}, {0, 1}} = {{e^t, 0}, {0, 1}}

e^(Dt) * e^(Nt) = {{e^t, 0}, {0, 1}} * {{1, 0}, {t, 1}} = {{e^t, 0}, {t, 1}}

e^(At) = {{e^t, 0}, {t, 1}}

This answer is incorrect. The correct answer is

e^(At) = {{e^t, 0}, {e^t-1, 1}}, which can be obtained without separating A into D and N by noting that A^2 = A and substituting the series representation of e^t. I can see how this is done, but what I can't see is why the method I used failed, since the textbook itself used this method to find the matrix exponential of B = {{2, 1}, {0, 2}} by separating it into a diagonal and nilpotent matrix. If someone could teach me why this method does not work for A, or if it does, where I made a mistake, I would appreciate it a lot. Thanks in advance!

D = {{1, 0}, {0, 0}}

N = {{0, 0}, {1, 0}}

e^(At) = e^((D + N)t) = e^(Dt) * e^(Nt)

When N is raised to the second power, it becomes the zero matrix. Therefore,

e^(Nt) = I + Nt = {{1, 0}, {t, 1}}

Note that D^2 = D

e^(Dt) = I + Dt + Dt^2/2! + ... + Dt^k/k! = {{1, 0}, {0, 1}} + {{1, 0}, {0, 0}}t + {{1, 0}, {0, 0}}t^2/2! + ... + {{1, 0}, {0, 0}}t^k/k!

e^(Dt) = {{1 + t + t^2/2! + ...+ t^k/k!, 0}, {0, 1}} = {{e^t, 0}, {0, 1}}

e^(Dt) * e^(Nt) = {{e^t, 0}, {0, 1}} * {{1, 0}, {t, 1}} = {{e^t, 0}, {t, 1}}

e^(At) = {{e^t, 0}, {t, 1}}

This answer is incorrect. The correct answer is

e^(At) = {{e^t, 0}, {e^t-1, 1}}, which can be obtained without separating A into D and N by noting that A^2 = A and substituting the series representation of e^t. I can see how this is done, but what I can't see is why the method I used failed, since the textbook itself used this method to find the matrix exponential of B = {{2, 1}, {0, 2}} by separating it into a diagonal and nilpotent matrix. If someone could teach me why this method does not work for A, or if it does, where I made a mistake, I would appreciate it a lot. Thanks in advance!