227A-HW1

15 flashcards · Mathematics · by John Viljoen

1. how do we compute the eigenvalues of a matrix

   det(lambda * I -A) = 0

2. how do we compute the eigenvectors of a matrix

   first compute the eigenvalues, then substitute each in the equation: A* v = lambda * v

3. what makes a matrix diagonalisable?

   if it has distinct eigenvalues or if it is symmetric (over the reals)

4. in what order are the eigenvectors placed in the T matrices of a diagonalisation decomposition T J T^(-1)

   [v1, v2] * [lambda1, 0; 0, lambda2] * [v1, v2].T, where v1, v2 corresponds to lambda 1, 2 respectively

5. what are another names for the diagonalisation decomposition?

   eigendecomposition, or eigenvalue decomposition

6. prove A^k, where A^k is a diagonalisable matrix, is = T J T^(-1), where T J T^(-1) is the eigendecomposition of A

7. how do you calculate the singular values of a general matrix A

   singular value i is equal to the square root of the ith eigenvalue of A.T @ A

8. how is the determinant of a transformation matrix related to the change of area of a square to a parallelogram?

   the area of the resulting parallelogram is = to the absolute value of the determinant:

9. what is the solution to least squares?

   (A.T A)^(-1) A.T b

10. derive the solution to the least squares problem:

    assume x1 != x2 by orthogonality principle: (Xw - y is orthogonal to range(X)) rearranging we get the standard least squares solution:

11. what is the orthogonality principle?

    the minimisation of a linear estimator problem: occurs when is orthogonal to the range of X. Therefore:

12. what is the rank of a matrix?

    the dimension of its range, also called its span, and is denoted:

13. find the rank of the dyad:

14. show that:

15. show that the rank of A is less than or equal to k remember this decomposition can be written as a sum of dyads

    we know that the rank of a dyad is 1 or 0. we also know that rank(A+B) <= rank(A) + rank(B) for any arbritrary A, B putting together these two facts: as desired