Description
Int. to Statistical Machine Learning
Questions
1) Write a program using Python that does the following:
– Takes two matrices of any size as the input
– Returns their dot product as the output
Note: You cannot use pre-packaged algorithms for matrix operations for this question.
You can use numpy or pandas to store your data (not for calculations).
Please do the following:
a. Please test the following matrix multiplications using your hand-written code
and report the result:
b. Compare the result to the packaged dot product numpy.dot. Are they same?
c. Please add your code to your .pdf file and also save it as an .ipynb file
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2) Assume that we have two (2) d-dimensional real vectors x and y. And denote by xi (or yi)
the value in the i-th coordinate of x (or y). Prove or disprove the following statements by
checking non-negativity, definiteness, homogeneity, and triangle inequality.
a. The following distance function is a metric. (5 points)
b. The following distance function is a metric. (5 points)
c. The following distance function is a metric. (10 points)
3) Calculating by hand, find the characteristic polynomial, eigenvalues and the eigenvectors
of the following matrix:
4) Provide a proof for the following: Let A, B, and C be any n x n matrices:
a. Show that trace(ABC) = trace(CAB) = trace(BCA) (10 points)
b. trace(ABC) = trace(BAC). Provide a proof or a counterexample (10 points)
5) Let A and B be n x n matrices with AB = 0.
Each question below is 5 points. Provide a proof or counterexample for each of the
following:
a) BA = 0
b) Either A = 0 or B = 0 (or both)
c) If det(A) = -3, then B = 0
d) There is a vector v ≠ 0 such that BAv = 0
DSCC46
