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
Spring 2022: Int. to Statistical Machine Learning University of Rochester
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
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