403 – Homework 3

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.
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This entry was posted on Monday, April 15th, 2013 at 7:27 am and is filed under 403/503: Linear Algebra II. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

One Response to 403 – Homework 3

  1. andrescaicedo says:
    May 17, 2013 at 9:44 am

    Thanks to Kyle Beserra for identifying the source of problem 4.

    Reply

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