Exercises – Short Day

Exercises – Short Day#

Exercise 1: Diagonalization of Matrix#

A linear map \(\,f: \mathbb{R}^3\rightarrow\ \mathbb{R}^3\,\) has with respect to the standard ordered basis in \(\, \mathbb{R}^3 \,\) the mapping matrix:

\[\begin{split}{\mathbf {}_e [f]_e}=\left[\begin{array}{cc} 1 & 0 & 0\\ 1 & 1 & 1\\ 1 & 0 & 2 \end{array}\right] \in \mathbb{R}^{3 \times 3}.\end{split}\]

Question a#

Provide an ordered basis \(\,v\,\) for \(\,\mathbb{R}^3\,\) with respect to which the mapping matrix of \(\,f\,\) becomes a diagonal matrix, and determine the corresponding change-of-basis matrix \(\, \mathbf {}_e [\mathrm{id}_{\mathbb{R}^3}]_v \,\) that switches from \(v\)-coordinates to \(e\)-coordinates.

Hint

\(\,{\mathbf {}_e [f]_e}\,\) has the eigenvalues \(1\) and \(2\), where \(\mathrm{am}(1)=2\) and \(\mathrm{am}(2)=1\). The associated eigenspaces are

\[\begin{split}\,E_1=\mathrm{span}_\mathbb{R}\left(\left[\begin{array}{cc} 0 \\ 1 \\ 0 \end{array}\right], \left[\begin{array}{cc} -1 \\ 0 \\ 1 \end{array}\right]\right)\,\,\,\,\mathrm{and}\,\,\,\,\, E_2=\mathrm{span}_\mathbb{R}\left(\left[\begin{array}{cc} 0 \\ 1 \\ 1 \end{array}\right]\right).\end{split}\]

How can one without further computations see and argue that the three shown vectors are linearly independent? And that they, when merged as columns into a \(\,3\times 3\) matrix, constitute the wanted change-of-basis matrix?

Question b#

State an invertible matrix \(\, \mathbf V\,\) and a diagonal matrix \(\, \mathbf{\Lambda}\, \) such that

\[\mathbf{\Lambda} = {\mathbf V}^{-1} \cdot {}_e [f]_e \cdot {\mathbf V}.\]

Hint

A matrix \(\, \mathbf V\,\) consisting of linearly independent egenvectors as columns has the wanted properties.

Answer

By choosing \(\,\mathbf V=\left[\begin{array}{cc} 0 & 0 & -1\\ 1 & 1 & 0\\ 1 & 0 & 1 \end{array}\right]\) we achieve that \(\mathbf{\Lambda} = {\mathbf V}^{-1} \cdot {}_e [f]_e \cdot {\mathbf V}\), where \(\,\mathbf{\Lambda}=\left[\begin{array}{cc} 2 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{array}\right]\,.\)

Exercise 2: Diagonalization#

This exercise is to be solved purely by hand.

Question a#

We are given the matrix:

\[\begin{split}{\mathbf A}=\left[\begin{array}{cc} 9 & -6\\ 8 & -7 \end{array}\right] \in \mathbb{R}^{2 \times 2}.\end{split}\]

Investigate whether \(\,\mathbf A\,\) can be diagonalized, and if so then state an invertible matrix \(\,\mathbf V\,\) and a diagonal matrix \(\,\mathbf{\Lambda}\,\), such that

\[\,\mathbf{\Lambda}={\mathbf V}^{-1}\cdot\mathbf A\cdot\mathbf V.\]

Answer

Diagonalization is possible since two linearly independent eigenvectors of \(\,\mathbf A\,\) exist.

There are several correct choices – here is one:

\[\begin{split}\,\mathbf V =\left[\begin{array}{cc} 1 & 3\\ 2 & 2 \end{array}\right] \,\,\,\,\mathrm{and}\,\,\,\, \mathbf{\Lambda}=\left[\begin{array}{cc} -3 & 0\\ 0 & 5 \end{array}\right].\end{split}\]

Question b#

We are given the matrix

\[\begin{split}{\mathbf B}=\left[\begin{array}{cc} 3 & 1 & -2 \\ 1 & 3 & 1 \\0 & 0 & 2 \end{array}\right] \in \mathbb{R}^{3 \times 3}.\end{split}\]

Investigate whether \(\,\mathbf B \,\) can be diagonalized, and if so then state an invertible matrix \(\,\mathbf V \,\) and a diagonal matrix \(\,\mathbf{\Lambda}\,,\) such that

\[\mathbf{\Lambda}={\mathbf V}^{-1}\cdot\mathbf B\cdot\mathbf V.\]

Hint

Find all eigenvalues of \(\,\mathbf B\,\) and determine their algebraic and geometric multiplicities.

Answer

\(\,\mathbf B\,\) cannot be diagonalized.

Question c#

We are given the matrix

\[\begin{split}{\mathbf C}=\left[\begin{array}{cc} 2 & 0 & 0 \\ 1 & 1 & 1 \\-1 & 1 & 1 \end{array}\right] \in \mathbb{R}^{3 \times 3}.\end{split}\]

Investigate whether \(\,\mathbf C\,\) can be diagonalized, and if so then state an invertible matrix \(\,\mathbf V\,\) and a diagonal matrix \(\,\mathbf{\Lambda}\,,\) such that

\[\mathbf{\Lambda}={\mathbf V}^{-1}\cdot\mathbf C\cdot\mathbf V.\]

Answer

Note that \(0\) is one of the eigenvalues.

Diagonalization is possible since three linearly independent eigenvectors of \(\mathbf C\,\) exist.

There are several correct choices – here is one:

\[\begin{split}\mathbf V=\left[\begin{array}{cc} 1 & 0 & 1 \\ 1 & -1 & 0 \\ 0 & 1 & -1 \end{array}\right] \,\,\,\,\mathrm{and}\,\,\,\,\mathbf{\Lambda}=\left[\begin{array}{cc} 2 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 2 \end{array}\right].\end{split}\]

Exercise 3: Complex Diagonalization#

We are given the matrix

\[\begin{split}{\mathbf M}=\left[\begin{array}{cc} 3 & 0 & 0 \\ 0 & 1 & -1 \\ 5 & 1 & 1 \end{array}\right] \in \mathbb{C}^{3 \times 3}.\end{split}\]

Question a#

Find eigenvalues and associated complex eigenspaces of \(\,\mathbf M\,.\)

Answer

The eigenvalues are \(1+i\), \(1-i\) and \(3\). They all have an algebraic multiplicity of \(1\).

The eigenvectors corresponding to \(\lambda=1+i\) are \(\mathbf x=t_1\cdot \left[\begin{array}{cc} 0 \\ i \\ 1 \end{array}\right]\), where \(t_1\in\mathbb{C}\),

the eigenvectors corresponding to \(\lambda=1-i\) are \(\mathbf x=t_2\cdot \left[\begin{array}{cc} 0 \\ -i \\ 1 \end{array}\right]\), where \(t_2\in\mathbb{C}\), and

the eigenvectors corresponding to \(\lambda=3\) are \(\mathbf x=t_3\cdot \left[\begin{array}{cc} 1 \\ -1 \\ 2 \end{array}\right]\), where \(t_3\in\mathbb{C}\).

Question b#

Diagonalize \(\,\mathbf M \,\), meaning determine matrices \(\mathbf Q\) and \(\mathbf{\Lambda}\) such that:

\[\mathbf{\Lambda}={\mathbf Q}^{-1}\,\mathbf M\,\mathbf Q.\]

Answer

If we choose \(\,\mathbf Q=\left[\begin{array}{cc} 0&0&1\\ i&-i&-1\\ 1&1&2 \end{array}\right] \,\) and \(\,\mathbf{\Lambda}=\left[\begin{array}{cc} 1+i&0&0\\ 0&1-i&0\\ 0&0&3 \end{array}\right]\,\) then we achieve as wanted that

\[\mathbf{\Lambda}={\mathbf Q}^{-1}\,\mathbf M\,\mathbf Q.\]

Exercise 4: Diagonalization of Matrix with SymPy#

Question a#

Use SymPy’s eigenvects() command to determine all eigenvalues and associated eigenspaces of the following matrix:

\[\begin{split}{\mathbf A}=\left[\begin{array}{cc} -1 & -1 & -6 & 3\\ 1 & -2 & -3 & 0\\ -1 & 1 & 0 & 1 \\ -1 & -1 & -5 & 3 \end{array}\right] \in \mathbb{R}^{4 \times 4}.\end{split}\]

Answer

The eigenvalues are \(1\) with \(\mathrm{gm}(1) = \mathrm{am}(1) = 1\), \(0\) with \(\mathrm{gm}(0) =1 < 2 = \mathrm{am}(0)\), and \(-1\) with \(\mathrm{gm}(-1) = \mathrm{am}(-1) = 1\).

The associated eigenspaces are:

\[\begin{split} E_1=\mathrm{span}_\mathbb{R}\left(\left[\begin{array}{cc} 3 \\ 0 \\ 1 \\ 4 \end{array}\right]\right)\,,\,\, E_0 = \mathrm{span}_\mathbb{R}\left(\left[\begin{array}{cc} 2 \\ 1 \\ 0 \\ 1\end{array}\right]\right) \,,\,\,\,\mathrm{and}\,\,\,\, E_{-1}= \mathrm{span}_\mathbb{R}\left(\left[\begin{array}{cc} 3 \\ 0 \\ 1 \\ 2 \end{array}\right]\right).\end{split}\]

Question b#

Investigate whether an invertible matrix \(\, \mathbf V\,\) and a diagonal matrix \(\,\mathbf{\Lambda}\,\) exist such that

\[\mathbf{\Lambda}={\mathbf V}^{-1}\cdot \mathbf A\cdot{\mathbf V}.\]

Exercise 5: Similar Matrices#

We are given the matrices

\[\begin{split}{\mathbf A}=\left[\begin{array}{cc} 0&1\\ -1&0 \end{array}\right] \,\,\, , \,\,\, {\mathbf B}=\left[\begin{array}{cc} 0&-1\\ 1&0 \end{array}\right] \in \mathbb{C}^{2 \times 2}.\end{split}\]

Question a#

Justify that \(\mathbf A\) and \(\mathbf B\) are similar.

Hint

Show that \(\mathbf A\) and \(\mathbf B\) are similar to the same diagonal matrix.

Hint

If we let \(\,\mathbf V=\left[\begin{array}{cc} -i&i\\ 1&1 \end{array}\right] \,,\) \(\,\mathbf U=\left[\begin{array}{cc} i&-i\\ 1&1 \end{array}\right] \,,\) and \(\,\mathbf{\Lambda}=\left[\begin{array}{cc} i&0\\ 0&-i \end{array}\right] \,,\) then it applies that

\[\mathbf{\Lambda}={\mathbf V}^{-1}\cdot\mathbf A\cdot\mathbf V={\mathbf U}^{-1}\cdot\mathbf B\cdot\mathbf U \,.\]

Hint

Now determine an invertible matrix \(\, \mathbf M \,\) that fulfills:

\[\mathbf B={\mathbf M}^{-1}\,\mathbf A\,\mathbf M\,.\]

Answer

\[\begin{split}\mathbf M =\mathbf V \cdot {\mathbf U}^{-1} = \left[\begin{array}{cc} -1&0\\ 0&1 \end{array}\right].\end{split}\]