MATHS 7027 Mathematical Foundations of
Data Science
Assignment 4
(PDF only)
CHECKLIST
Have you shown all of your working for every mathematical question?
Are your questions submitted in order?
Is the writing in your submission legible and clearly structured? Is your submission a single, correctly oriented, pdf file? If after the deadline, but within 24 hours, have you contacted us via email (if applying for an extension) and submitted your assignment online via Canvas?
Have you checked that the assignment submitted is the correct one, as we cannot accept other submissions after the due date?
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1. Find the eigenvalues and eigenvectors of the following matrices. Make sure you check your answer, this will be worth one mark.
(a)
(b)
2. 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.
3. 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.)
4. At a company board meeting the five directors sit at the front facing the employees.
(a) How many different seating arrangements are possible for the directors?
(b) 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?
(c) 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?
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