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I tried to solve the following question:

  1. Find 2 matrices A and B in M2(C) such that A is similar to B but not congruent.
  2. Find 2 matrices A and B in M2(C) such that A is congruent to B but not similar.

What is the best strategy to find such matrices? I have tried guessing matrices but it doesn't seem to be a good strategy.

arm46
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1 Answers1

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First, notice that a symmetric matrix can only be congruent to other symmetric matrices. See if you can find two similar matrices, one of which is symmetric while the other is not.

Next, notice that $A=I$ is only similar to itself, so given any non-orthogonal, invertible matrix $P$, let $B=P^{\intercal}P$ to find two matrices that are congruent but not similar.

Jared
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