## 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 $\Gamma$.