Homework #7 for Math H110, fall '02
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1. Given M a matrix in Jordan canonical form, how can you quickly
compute the JCF of M^2?

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2a. Find a degree 2 polynomial p(x,y,z,w) in four variables,
such that for every matrix
<table border=1>
<tr><td>a <td>b
<tr><td>c <td>d
</table>
that is not diagonalizable, we have p(a,b,c,d)=0.
(Hint: if the matrix has 2 distinct eigenvalues, then it's
diagonalizable.)
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b. If p(a,b,c,d)=0, does that imply the matrix is not diagonalizable?

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3. Let G be a diagonal matrix with s 1's then t -1's down the diagonal.
Describe the matrices M such that M G M^T = G.
That set is called O(G).

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4. Say V is the real vector space R^{s+t}, and G and O(G) are as in #3.
Say there is a constant B such that for all M in O(G), 
each matrix entry m_ij of M has | m_ij | < B.
What can you say about s, t?
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If you're stuck, consider the three cases when V is 2-dimensional.