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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 $$\left[ \begin{array} {ccc} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \end{array} \right]$$