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macro mathcamp theorems
Flashcards do conjunto (6)
- 1characteristic polynomial of a \( 2 \times 2 \) matrix \( A \)The characteristic polynomial is given by: \[ p_A \left( \lambda \right) = \operatorname{det} \left( A - \lambda I_n \right) = \lambda^2 - \operatorname{tr} \left( A \right) \lambda + \operatorname{det} \left( A \right). \]
- 2Hyperbolic Steady StateA steady state \( \bar{x} \) of system \[ \dot{x} \left( t \right) = G \left( x \left( t \right) \right) \] is hyperbolic if the matrix \( DG \left( \bar{x} \right) \) does not have eigenvalues with zero real parts.
- 3Let \( \bar{x} \) be a hyperbolic steady state of system \( \dot{x} \left( t \right) = G \left( x \left( t \right) \right) \), where \( G: \mathbb{R}^n \rightarrow \mathbb{R}^n \) is a continuous differentiable mapping. Then ...there exists a neighborhood \( U \) of \( \bar{x} \) such that the system is topologically equivalent in \( U \) to the linear system \[ \dot{x} \left( t \right) = DG \left( \bar{x} \right) \left( x \left( t \right) - \bar{x} \right) \] by the Grobman-Hartman Theorem.
- 4properties of the eigenvalues of a matrix \( A \)The determinant of \( A \) equals the product of its eigenvalues, \[ \det \left( A \right) = \lambda_{1}^{m_1} \ldots \lambda_{k}^{m_k}. \] The trace of \(A\), \( \operatorname{tr} \left( A \right) \), equals the sum of its eigenvalues.
Let \( p_A \left( \lambda \right) = \lambda^n + c_1 \lambda^{n-1} + \ldots + c_n \) be the characteristic polynomial of \( A \). Then \( c_1 = - \operatorname{tr} \left( A \right) \) and \(c_n = \left(-1\right)^n \det \left( A \right) \). - 5separable differential equationa differential equation \[ \dot{x} \left( t \right) = g \left( x \left( t \right), \, t \right) \] is separable if \( g \) can be written \[ g \left( x, \, t \right) \equiv f \left( x \right) h \left( t \right). \] In other words, the part of \( g \) that depends on \( x \) is separate from the one that depends on \( t \).
- 6exact differential equationa differential equation \[ \dot{x} \left( t \right) = g \left( x \left( t \right), \, t \right) \] is exact if \( g \) can be written as \[ g \left( x \left( t \right), \, t \right) \equiv \frac{G_1 \left( x \left( t \right), \, t \right)}{G_2 \left( x \left( t \right), \, t \right)} \] where \[ G_1 \left( x \left( t \right), \, t \right) \equiv \frac{\partial F \left( x \left ( t \right), \, t \right)}{\partial t} \text{ and } G_2 \left( x \left( t \right), \, t \right) \equiv \frac{\partial F \left( x \left ( t \right), \, t \right)}{\partial x} \] for some function \( F \left( \cdot, \, \cdot \right) \).
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- FlashCards sobre macro mathcamp theorems, criado por Eric Andersen em 01-09-2015.
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- Eric Andersen
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