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Problem 293

Let the linear operator \(\mathrm{T}\) be nilpotent of degree 4 on \(\mathrm{C}^{6}\), \(\mathrm{T} \neq 0^{\rightarrow}\). Find the Jordan canonical form of \(\mathrm{T}\).

Expert verified

The Jordan canonical form of the nilpotent linear operator T is:
\[
J_T = \begin{pmatrix}
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix}.
\]

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Chapter 12

Find the Jordan Canonical form of $\begin{array}{rrrrrrrrrrrr}15 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 01 \\\ 10 & 5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 01 \\ 11 & 0 & 5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 01 \\ 10 & 1 & 0 & 5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 01 \\\ 10 & 0 & 1 & 0 & 5 & 0 & 0 & 0 & 0 & 0 & 0 & 01 \\ 10 & 0 & 0 & 1 & 0 & 5 & 0 & 0 & 0 & 0 & 0 & 01 \\ A=10 & 0 & 0 & 0 & 1 & 0 & 5 & 0 & 0 & 0 & 0 & 01 \\\ 10 & 0 & 0 & 0 & 0 & 1 & 0 & 5 & 0 & 0 & 0 & 01 \\ 10 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 2 & 0 & 0 & 01 \\ 10 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 2 & 0 & 01 \\\ 10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 2 & 01 \\ 10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 21\end{array}$

Chapter 12

Determine all possible Jordan canonical forms for a linear operator \(\mathrm{T}: \mathrm{V} \rightarrow \mathrm{V}\) whose characteristic polynomial is \(\mathrm{f}(\lambda)=(\lambda-2)^{3}(\lambda-5)^{2}\)

Chapter 12

Define Jordan block and Jordan form matrix.

Chapter 12

State the primary decomposition theorem. Verify that it is true by adducing an example.

Chapter 12

Find the Jordan matrix of $$ A=\begin{array}{lll} 13 & 1 & -3 \mid \\ \mid-7 & -2 & 9 \mid \\ 1-2 & -1 & 4 \mid \end{array} $$

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