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Given any 3-dimensional vector space, I am asked to find the maximum number of pair-wise non-similar linear transformations on this space, where each transformation has the characteristic polynomial $(\lambda -1)^3$. Any hint on how to approach this question will be appreciated.

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Use the Jordan canonical form. Two linear transformations whose characteristic polynomials are equal and split are similar if and only if they have the same Jordan canonical form. –  Arturo Magidin Nov 9 '11 at 17:13
    
@ArturoMagidin. In this case they are not similar. –  smanoos Nov 9 '11 at 17:18
    
If "matrices are similar if and only if they have the same Jordan canonical form" is true, then "matrices are not similar if and only if they have different Jordan canonical forms" is true. So figure out how many different Jordan canonical forms you can have if the characteristic polynomial is $(\lambda -1)^3$. –  Arturo Magidin Nov 9 '11 at 17:48

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Two matrices with the same characteristic polynomial are similar if and only if they have the same Jordan canonical form (up to the order of the blocks).

That means that two matrices with the same characteristic polynomial are not similar if and only if they have different Jordan canonical form (up to order of the blocks).

A matrix with characteristic polynomial $(\lambda -1)^3$ has Jordan canonical form of one of the following types:

  • Three $1\times1$ Jordan blocks (i.e., diagonal);
  • One $2\times 2$ Jordan block, one $1\times 1$ Jordan block;
  • One $3\times 3$ Jordan block.

And those are the only possibilities.

So, since every matrix must be similar to one, and only one of those types...

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So there are 3 different non-similar linear transformations with that characteristic polynomial. Right? –  smanoos Nov 9 '11 at 20:16
    
@smanoos: Yes. ${}{}{}$ –  Arturo Magidin Nov 9 '11 at 20:28
    
Thank you for your help. –  smanoos Nov 9 '11 at 20:46

The information you have is that you have only the eigenvalue $\lambda=1$ with multiplicity $3$.

Given a linear transformation $T$ with a certain spectrum, you know from the general theory that you can choose a basis of the space so that $T$ appears in a particular form, called Jordan form.

Different Jordan forms belong to non-similar transformations.

Thus the question reduces to: how many Jordan forms are there with a spectrum consisting of the eigenvalue $\lambda=1$ repeated three times?

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