Essential Mathematics: Eigenvalues & Eigenvectors

21 flashcards · Mathematics · by lakesare

  1. When do we say that a line is an invariant line of a linear transformation? E.g. y = x and y = -x are invariant under the reflection in the line y = x.

    f(line) = line or a point on this line

  2. What's eigenvalue and eigenvector?

    λ - eigenvalue x - eigenvector

  3. Why can't an eigenvector x be 0 (...Ax = λx for some non-zero vector x)?

    Because then the eigenvector equation Ax = λx works for any A and any λ.

  4. What's the form of the eigenvector equation?

  5. What's a characteristic equation of a square matrix? Also called characteristic polynomial.

  6. What's a trace of this matrix tr(A)?

    Tr(A) = a + d

  7. How many eigenvalues does a 2x2 matrix have?

    Either 1 or 2 eigenvalues.

  8. Explain the pipeline from: matrix A to eigenvalues λ to eigenvectors x.

  9. How would you show that some vector is an eigenvector of a matrix?

    Show that the Ax = λx holds for some λ.

  10. What form do each of these matrices have?

    upper triangular lower triangular diagonal matrix - BOTH upper AND lower triangular

  11. [Systems of differential equations] What's a matrix differential equation?

    'Matrix differential equation' is a system of ODEs in this abcd form:

  12. [Systems of differential equations] Expand/clarify the matrix differential notation.

  13. [Systems of differential equations] What's considered to be a solution to the system of ODEs?

    x(t) = ...t... y(t) = ...t... E.g.: A solution of a system of differential equations consists of an equation for x in terms of t and an equation for y in terms of t that simultaneously satisfy the system.

  14. [Systems of differential equations] What's meant by decoupled VS coupled matrix differential equation?

    Decoupled systems have a diagonal matrix Easy to solve - just solve both differential equations one by one.Coupled systems - everything else Harder to solve - use matrix diagonalisation.

  15. What does this formula come down to?

    That Av (applying transformation A to some vector v) gives the same result as scaling the eigenvector v by some λ.

  16. What does the determinant of the transformation represent physically?

    The area by which the figure is scaled.

  17. [diagonalising matrices] What does it mean to diagonalise a matrix?

    To write it in the A = PDP-1 form, where D is a diagonal matrix.

  18. [diagonalising matrices] What does A = PDP-1 consist of?

    P - eigenvectors x1, x2 D - eigenvalues λ1, λ2 P-1 - inverse of P

  19. [diagonalising matrices] What's the formula for powers of the diagonal matrix?

  20. [diagonalising matrices] How is diagonalising a matrix A = PDP-1 helpful in finding A5?

  21. What 4 things do eigenvectors&values let us do (as we learned in this unit)?

    we can find invariant lines of the transformation easy way to calculate matrix powers A5 we can solve abcd systems of differential equations we can solve some simplified predator-prey systems [This is about matrix powers A5 too]