[MEEG690]

Review Problems (Exam 2)

1. Let \(A\) be a 3x3 real matrix with eigenvalues \(\lambda= -1\) of multiplicity \(2\), and \(\lambda= 3\) of multiplicity \(1\). The corresponding eigenspaces are \(V_{-1}=\text{span}(\boldsymbol{v}_1,\boldsymbol{v}_2)\) and \(V_3= \text{span}(\boldsymbol{v}_3)\) with \[ \boldsymbol{v}_1 = (1,0,-1)^T, \quad \boldsymbol{v}_2 = (0,1,-1)^T, \quad \boldsymbol{v}_3 = (1,1,1)^T \]

  1. Find matrix \(A\). Is \(A\) positive definite? Is \(A\) invertible?

  2. Is \(A^3\) diagonalizable? If so, find its eigenvalues and corresponding eigenspaces.

  3. Is \(A + I\) diagonalizable? If so, find its eigenvalues and corresponding eigenspaces.

2. Let \({\cal M}_2\) be the set of 2x2 real matrices. We consider the following linear map: \[ L: M\in {\cal M}_2\to AM-MA \in {\cal M}_2 \] where \(A\) is the 2x2 matrix \[ A= \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} \] The set \({\cal B}= (A_1,A_2,A_3,A_4)\) with \[ A_1= \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \quad A_2= \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \quad A_3= \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}, \quad A_4= \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}, \] is chosen as a standard basis of \({\cal M}_2\).

  1. Find the matrix \(T= [L]_{\cal B}\) of \(L\) in basis \({\cal B}\) by finding \(L(A_i)\) for \(i=1,2,3,4\). Examine whether \(T\) is diagonalizable (do not determine the characteristic polynomial of \(T\)). Is \(L\) invertible?

  2. It can be shown that matrix \(T\) satisfies \(T^3= 4T\) (do not verify it). Show that this imply that the eigenvalues must be in the set \(\{-2, 0, 2\}\).

  3. Find the eigenspace \(E (0,T)\) associated with the (possible) eigenvalue \(0\).

  4. Calculate \(T\boldsymbol{v}_3\) and \(T\boldsymbol{v}_4\) for \(\boldsymbol{v}_3 = (1,-1,1,-1)^T\) and \(\boldsymbol{v}_4 = (1,1,-1,-1)\). Conclude that \(T = Q D Q^{T}\) where \(D\) is a diagonal matrix. Give the expression of matrices \(D\) and \(Q\). Find a basis of \({\cal M}_2\) relative to which the matrix of \(L\) is \(D\).

3. Consider again the linear map: \[ L: M\in {\cal M}_2\to AM-MA \in {\cal M}_2 \] where now \(A\) is the 2x2 matrix \[ A= \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \]

  1. Using the fact that \(A^2 = 0\), show that \(L^3 = O\) and find the eigenvalues of \(L\).

  2. Without finding the matrix of \(L\) in basis \({\cal B}\), find the eigenspaces of \(L\). Is \(L\) diagonalizable?

  3. Find the Jordan form \(J\) of \(L\), and the basis \((E_1,E_2,E_3,E_4)\) of \({\cal M}_2\) in which \(L\) is represented by \(J\).