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MAT208 Linear Algebra

MAT208 Linear Algebra. Eigenvalues and Eigenvectors. Real Eigenvalues. Consider the linear homogeneous system: To find eigenvalues, we consider the characteristic equation:. Distinct Eigenvalues. If the eigenvalues are distinct, we know that the general solution may be represented as:

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MAT208 Linear Algebra

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  1. MAT208Linear Algebra Eigenvalues and Eigenvectors

  2. Real Eigenvalues Consider the linear homogeneous system: To find eigenvalues, we consider the characteristic equation:

  3. Distinct Eigenvalues • If the eigenvalues are distinct, we know that the general solution may be represented as: Keep in mind that and are two constant vectors. • We also know that, since the eigenvalues are distinct, we may assume that

  4. Distinct Eigenvalues (cont) • Let us discuss the behavior of the solutions when (meaning the future) and when (meaning the past). • Since the two eigenvalues are real numbers, we have three cases to consider depending on their signs. We will look at each of these in turn.

  5. Three Cases: Case 1 Both positive For this case, This means that the solutions emanate from the origin. (Think of it this way--if you go to the past, you will die at the origin). When ,Y(t) explodes.

  6. Eigenvalues Both Positive • Graphical representation: the origin as a source. The origin is an equilibrium point. (Graph courtesy of SOS Math)

  7. Three Cases: Case 2 Both negative: For this case, This means that in the future, solutions will die at the origin. When ,Y(t) explodes.

  8. Eigenvalues: Both Negative • Graphical representation: the origin as a sink. The origin is an equilibrium point. (Graph courtesy of SOS Math)

  9. Three Cases: Case Three • Eigenvalues have different signs: • In this case, the origin behaves like a saddle: (Graph courtesy of SOS Math)

  10. One More Case The case where one of the eigenvalues is zero will be discussed separately.

  11. Repeated Eigenvalues • The general solution will be of the system looks like

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