Recall some basic facts about eigenspaces — we say that Eα is an eigenspace of a matrixM of dimension d if the dimension of the nullspace of αI −M has dimension d. Moreover, by the theory covered in class, we must have that the polynomial (λ− α) divides the characteristic polynomial of M , which is defined to be det(λI −M). We first calculate the eigenspaces of Bn. Observe that if α = 1, then w...

Problem 1 (a) We will compute theH2 norm ofG using state-space methods (see Section 2.6 in DFT). We begin by finding a minimal state-space representation of the system with transfer function G(s). Assuming u(0) = y(0) = 0 as usual, we have: sy(s) + 2sy(s) + sy(s) + y(s) = u(s) ⇔ ... y (t) + 2ÿ(t) + ẏ(t) + y(t) = u(t) Defining x1 = y, x2 = ẏ, x3 = ÿ, the corresponding state-space representation ...

This is a closed-book exam. The exam lasts for 180 minutes. Please write clearly and legibly. Be especially careful in the definition of the game, the payoff function and the equilibrium notions. The allocated points are also a good indicator for your time budget. Please record the answer for each question in a separate bluebook.

Bonus: This is the unique function on matrices that satisfies multilinearity, alternation, and such that det(I) = 1. Here multilinearity means that the function is linear in each column separately, i.e., it is multiadditive and multihomogeneous: multiadditive means that, if A and B are two matrices which have the same columns except for the k-th column, and C is the matrix which has the same co...

Phylogenetics is the study of evolutionary relations between different organisms. Phylogenetic trees are the representations of these relations. Researchers have been working on finding fast and systematic approaches to reconstruct phylogenetic trees from observed data for over 40 years. It has been shown that, given a certain criterion to evaluate each tree, finding the best fitted phylogeneti...

A number of different types of final programming exams used or considered at the Department of Computer Science, Aalborg University, are identified and analyzed. Based on this analysis, a new type of programming exam is introduced called a Mini Project Programming (MIP) exam. MIP is a group-based programming assignment that is intended as a medium-scale programming effort followed by a final or...