Collection of Maths Problems

Jordan form - diagonalizable matrices

Task number: 2663

• Variant 1

  \( \begin{pmatrix} -11 & 30 \\ -10 & 24 \\ \end{pmatrix} \)

• Hint

  When the multiplicity of each eigenvalue is one, the Jordan's factorization \(\mathbf R\mathbf D\mathbf R^{-1}\) can be constructed from a diagonal matrix \(\mathbf D\) with eigenvalues on the diagonal and \(\mathbf R\) with the eigenvectors as columns.

• Solution

  \(\begin{vmatrix} -11-t & 30 \\ -10 & 24-t \\ \end{vmatrix} =t^2-13t+36=(t-9)(t-4)\) hence \(\lambda_1=9\), \(\lambda_2=4\).

  For \(\lambda_1=9\) we have \(-20x_1+30x_2=0\) with a solution \(\mathbf x=(x_1, x_2)^T=(3, 2)^T\).

  For \(\lambda_2=4\) we have \(-15x_1+30x_2=0\) with a solution \(\mathbf x=(2, 1)^T\).

• Result

  \(\mathbf R=\begin{pmatrix} 3 & 2 \\ 2 & 1 \\ \end{pmatrix} \), \(\mathbf D=\begin{pmatrix} 9 & 0 \\ 0 & 4 \\ \end{pmatrix} \), \(\mathbf R^{-1}=\begin{pmatrix} -1 & 2 \\ 2 & -3 \\ \end{pmatrix} \)

• Variant 2

  \( \begin{pmatrix} 0 & 2 & -2 \\ 1 & -1 & 5 \\ 2 & -4 & 8 \\ \end{pmatrix} \)

• Result

  \(\mathbf R=\begin{pmatrix} 0 & 1 & 2 \\ 1 & 2 & 1 \\ 1 & 1 & 0 \\ \end{pmatrix}\), \(\mathbf D=\begin{pmatrix} 4 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix}\), \(\mathbf R^{-1}=\begin{pmatrix} 1 & -2 & 3 \\ -1 & 2 & -2 \\ 1 & -1 & 1 \\ \end{pmatrix}\)

• Variant 3

  \( \begin{pmatrix} 2 & 0 & 0 \\ -4 & 1 & 3 \\ -4 & 0 & 4 \\ \end{pmatrix} \)

• Result

  \(\mathbf R=\begin{pmatrix} 0 & 1 & 0 \\ 1 & 2 & 1 \\ 1 & 2 & 0 \\ \end{pmatrix}\), \(\mathbf D=\begin{pmatrix} 4 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix}\), \(\mathbf R^{-1}=\begin{pmatrix} -2 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & -1 \\ \end{pmatrix}\)

• Variant 4

  \( \begin{pmatrix} 4 & -2 & 0 \\ 0 & 2 & 0 \\ 6 & -5 & 1 \\ \end{pmatrix} \)

• Result

  \(\mathbf R=\begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 2 & 1 & 1 \\ \end{pmatrix}\), \(\mathbf D=\begin{pmatrix} 4 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix}\), \(\mathbf R^{-1}=\begin{pmatrix} 1 & -1 & 0 \\ 0 & 1 & 0 \\ -2 & 1 & 1 \\ \end{pmatrix}\)