Exercise 3

Find the Jordan form from the given data.

• The matrix $T$ is $5\!\times\!5$ with the single eigenvalue $3$. The nullities of the powers are: $T-3I$ has nullity two, $(T-3I)^{2}$ has nullity three, $(T-3I)^{3}$ has nullity four, and $(T-3I)^{4}$ has nullity five.

• The matrix $S$ is $5\!\times\!5$ with two eigenvalues. For the eigenvalue $2$ the nullities are: $S-2I$ has nullity two, and $(S-2I)^{2}$ has nullity four. For the eigenvalue $-1$ the nullities are: $S+1I$ has nullity one.