MIT
18 06 -
Lecture 24
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Stability u t 0 means that we need e to approach zero so eigenvalues must be negative Only the real part of has to be zero the imaginary component does not matter Special case 2x2 stability The trace of the matrix is negative because the sum of the eigenvalues is negative The determinant must also be positive because the product of the eigenvalues is positive Steady state means that one eigenvalue is zero and the other eigenvalues are negative Diverges if any eigenvalue is positive S v t ASv v t S ASv v so we get v t e v 0 and u t Se S so we get our solution Lecture 24 Differential equations u t Au Example Rewrite the equation u t Au as u Sv Matrix exponentials How do we know that Exponential of a diagonal matrix Example
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