403 – Homework 3

April 15, 2013

This set is due Monday, April 29, at the beginning of lecture.

  1. Suppose A is an n\times n matrix with complex entries such that A^2=I. Verify that e^{iAx}=\cos(x)I+i\sin(x)A for x real.
  2. Suppose that N is nilpotent. Verify that I+N is invertible.
  3. Use linear algebra to find a closed form expression for the terms of the sequence \{a_n\}_{n\ge0} given by the recurrence relation a_0=1, a_1=-1, a_{n+2}=3a_{n+1}+10a_n for all n\ge 0.
  4. (Norman Biggs, Algebraic Graph Theory, Proposition 2.3) Let \Gamma be a (finite, simple) graph with n vertices, and let A be its adjacency matrix. Suppose that the characteristic polynomial of A is {\rm char}_A(x)=x^n+c_1x^{n-1}+c_2x^{n-2}+\dots+c_n. Verify that:
    1. c_1=0.
    2. -c_2 is the number of edges in \Gamma.
    3. -c_3 is twice the number of triangles in \Gamma.