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# hw4_scribe - 1 Show that if A is diagonalizable then det Λ...

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Unformatted text preview: 1. Show that if A is diagonalizable, then det . Λ det det det det … det det Theorem: So, det 2. Use polar coordinates to solve ∑ . λi Plugging the last four expressions into the first two gives: (1) (2) Multiply (2) by and subtract (1) by Multiply (1) by and add (2) by Thus, we have 3. For ,0 1, use local existence and uniqueness to estimate the time of existence. Solve the equation direction and find the actual time of existence. We first show that is Lipschitz on an open ball about the initial condition 1 ,1 | || | | 2 2| 2 2 , Lipschitz satisfied. ,1 and find a bount, , 1 such that Now, take 1 1 max To estimate the time of existence, we set, / with the maximal value of alpha for / 1, / 1 max 1 1 0 , . The best estimate will be 0. 1 1 1 1 1 0 1 1/4 1 Solving the ODE directly we find This blows up at 1 1 1, so the actual time of existence is 1. ...
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