Foundations of Data Science

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Exercise 12.27 Given that is the generating function for the sequence 1, 1, . . ., for
1−x
1
what sequence is the generating function?
1−2x
Exercise 12.28 Find a closed form for the exponential generating function for the infinite
sequence of prefect squares 1, 4, 9, 16, 25, . . .
Exercise 12.29 Prove that the L 2 norm of (a 1 , a 2 , . . . , a n ) is less than or equal to the L 1
norm of (a 1 , a 2 , . . . , a n ).
Exercise 12.30 Prove that there exists a y, 0 ≤ y ≤ x, such that f(x) = f(0) + f ′ (y)x.
Exercise 12.31 Show that the eigenvectors of a matrix A are not a continuous function
of changes to the matrix.
Exercise 12.32 What are the eigenvalues of the two graphs shown below? What does
this say about using eigenvalues to determine if two graphs are isomorphic.
Exercise 12.33 Let A be the adjacency matrix of an undirected graph G.
eigenvalue λ 1 of A is at least the average degree of G.
Prove that
Exercise 12.34 Show that if A is a symmetric matrix and λ 1 and λ 2 are distinct eigenvalues
then their corresponding eigenvectors x 1 and x 2 are orthogonal.
Hint:
Exercise 12.35 Show that a matrix is rank k if and only if it has k nonzero eigenvalues
and eigenvalue 0 of rank n-k.
Exercise 12.36 Prove that maximizing xT Ax
x T x
to the condition that x be of unit length.
is equivalent to maximizing xT Ax subject
Exercise 12.37 Let A be a symmetric matrix with smallest eigenvalue λ min .
bound on the largest element of A −1 .
Give a
Exercise 12.38 Let A be the adjacency matrix of an n vertex clique with no self loops.
Thus, each row of A is all ones except for the diagonal entry which is zero. What is the
spectrum of A.
Exercise 12.39 Let A be the adjacency matrix of an undirect graph G. Prove that the
eigenvalue λ 1 of A is at least the average degree of G.
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