Description
- Find the eigenvalues and eigenvectors of the following matrices. Make sure you check your answer, this will be worth one mark.
(a)
(b)
- If possible, diagonalise the matrix
Make sure to check your answer (e.g. by verifying that P−1AP = D), this will be worth one mark.
- Consider X ∈Rn×p for which we have performed a principal component analysis (PCA), that is let
X −1n×1x¯ and as usual. Suppose we are given another row of data which happens to be equal to x¯ so that we now have the data Y ∈R(n+1)×p which we can write as
.
1
How does the PCA of Y differ from that of X?
(Note: A related problem from the week 8 Practice Questions may be helpful here.)
- At a company board meeting the five directors sit at the front facing the employees.
- How many different seating arrangements are possible for the directors?
- Suppose one of the five directors was nominated in advance to chair the meeting and must sit in the middle, how many different seating arrangements are there now?
- Two of the directors, neither of which is the chair person, don’t get along and will sit on opposite sides of the chair. How many seating arrangements are there now?
