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June 30, 2016

Jordan form of a matrix - Problem Discussion

Problem (Artin, chapter 4, 7.1) Determine the Jordan form of a matrix $$ \left[ \begin{array}{ccc} 1 & 1 & 0 \ 0 & 1 & 0 \ 0 & 1 & 1 \end{array} \right] $$

Discussion According to the Jordan form of a matrix, we first determine the characteristic polynomial of the above matrix.

To do that, first we subtract $$ \lambda $$ from each of the diagonal entries of the matrix and then the matrix looks like $$ \left[ \begin{array} {ccc} 1-\lambda & 1& 0\ 0 & 1-\lambda & 0\ 0 & 1 & 1-\lambda \end{array} \right] $$

Now the determinant of this second matrix will give us the desired eigenvalues so the determinant is $$ (1-\lambda)^3 $$

Equating the determinant value = 0 we get that the only eigenvalue of the matrix is 1 and it is a repeated eigenvalue.

So now the Jordan form of the matrix will be of the form $$ \left[ \begin{array} {ccc} 1 & 0 & 0 \ 1 & 1 & 0 \ 0 & 1 & 1 \end{array} \right] $$

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