Exercises

Exercise 6.1

(See Solution 6.1.) Is the following matrix diagonalizable?

\[ A = \left ( \begin{array}{ccc} 2 & 1 & 1 \\ 0 & 1 & 0 \\ 1 & -1 & 2 \end{array} \right ). \]

Hint: you will find that the eigenvalues of \(A\) are among the numbers \(0, 1, 2, 3\). You will be able to choose basis vectors of the eigenspaces all of whose coordinates are \(-1, 0, 1\).

Exercise 6.2

(See Solution 6.2.) Let \(A = \left ( \begin{array}{cc} a & b \\ c & d \end{array} \right )\). Show that:

  • \(\chi_A(t) = t^2 - \mathrm {tr} (A) t + \det A = t^2 - (a+d) t + (ad-bc)\). Here \(\mathrm {tr} (A)\) is the trace of \(A\), cf. Exercise 4.24.

  • The eigenvalues of \(A\) are

\[ \lambda_{1/2} = \frac{a+d}2 \pm \sqrt{\frac{(a-b)^2}4+4bc}. \]

Exercise 6.3

(See Solution 6.3.) For each of the following matrices, compute \(\chi_A(t)\), the eigenvalues of \(A\), the eigenspaces for these eigenvalues. Also decide whether \(A\) is diagonalizable and compute an eigenbasis if one exists.

  1. \(A = \left ( \begin{array}{cc} 3 & 5 \\ 1 & -1 \end{array} \right )\)

  2. \(A = \left ( \begin{array}{ccc} 0 & 1 & 0 \\ 3 & 0 & 1 \\ 2 & 0 & 0 \end{array} \right )\)

  3. \(A = \left ( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 & \frac 32 \\ 0 & 0 & 1 \end{array} \right )\)

  4. \(A = \left ( \begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array} \right )\)

Exercise 6.4

(See Solution 6.4.) Consider the matrix \(A = \left ( \begin{array}{ccc} 1 & 0 & 0 \\ 1 & 1 & 2 \\ 1 & 0 & 1 \end{array} \right )\). Compute its characteristic polynomial, its eigenvalues and its eigenspaces. Is \(A\) diagonalizable? If so, find a basis of \({\bf R}^3\) such that the associated matrix is a diagonal matrix, as in Definition 6.14.

Exercise 6.5

(See Solution 6.5.) Let

\[ f: {\bf R}^3 \to {\bf R}^3 \]

be the linear map such that \(f(1,0,1)=(2,0,2)\), \(\ker f = L((1,1,1))\) and \(f(2,0,-3)=(-2,0,3)\). Compute the matrix of \(f\) with respect to the standard basis.

Exercise 6.6

(See Solution 6.6.) For which \(a \in {\bf R}\) is the matrix

\[ A_a = \left ( \begin{array}{ccc} a & 0 & 0 \\ a-2 & 1 & 1 \\ 0 & 1 & 1 \end{array} \right ) \]

diagonalizable?

Exercise 6.7

(See Solution 6.7.) For a parameter \(a \in {\bf R}\), let

\[ A_a = \left ( \begin{array}{ccc} 4 & 0 & 4 \\ a & 2 & a \\ -2 & 0 & -2 \end{array} \right ). \]

  1. Compute the characteristic polynomial and the eigenvalues of \(A_a\), for all \(a \in {\bf R}\).

  2. Compute the values of \(a\) for which \(A_a\) is diagonalizable. For these \(a\), find an invertible matrix \(P\) such that \(P^{-1} A_a P\) is a diagonal matrix.

Exercise 6.8

(See Solution 6.8.) Consider the matrices

\[ A = \left ( \begin{array}{ccc} 1 & 1 & 1 \\ 0 & 2 & 0 \\ 1 & -1 & 1 \end{array} \right ) \ \text{ and } \ B = \left ( \begin{array}{ccc} 2 & 1 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 0 \end{array} \right ). \]

  1. Compute the eigenvalues and eigenvectors of \(A\) and show \(A\) is diagonalizable.

  2. Show that the characteristic polynomials of \(A\) and \(B\) are the same. Compute the eigenvalues and eigenspaces of \(B\). Explain why \(A\) and \(B\) do not represent the same linear map with respect to different bases!

Exercise 6.9

(See Solution 6.9.) For a parameter \(t \in {\bf R}\), consider the matrix

\[ A_t = \left ( \begin{array}{ccc} -1 & 2 & t \\ 2 & 0 & -2 \\ t & -2 & -1 \end{array} \right ). \]

  1. For which values of \(t\) does \(A_t\) have 0 as an eigenvalue?

  2. Compute the eigenvalues and eigenspaces of \(A_t\) for those values of \(t\) obtained in the previous part.

Exercise 6.10

(See Solution 6.10.) Consider the space \({\mathrm {Mat}}_{2 \times 2}\) of \(2 \times 2\)-vector spaces. Consider \(A = \left ( \begin{array}{cc} -4 & 8 \\ 1 & -2 \end{array} \right )\) and the map

\[ F: {\mathrm {Mat}}_{2 \times 2} \to {\mathrm {Mat}}_{2 \times 2}, X \mapsto AX. \]

  1. Is \(F\) a linear map? Justify your answer.

  2. Compute the \(4 \times 4\)-matrix of \(F\) with respect to the standard basis of \({\mathrm {Mat}}_{2 \times 2}\), i.e., the matrices

\[ E_{11} = \left ( \begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array} \right ), E_{12} = \left ( \begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array} \right ), E_{21} = \left ( \begin{array}{cc} 0 & 0 \\ 1 & 0 \end{array} \right ), E_{22} = \left ( \begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array} \right ). \]

  1. Compute a basis of \(\ker F\) and \({\operatorname{im}\ } F\).

  2. Compute the eigenvalues and eigenspaces of \(F\).

Remark 6.26

The linearity of \(F\) is a consequence of Lemma 4.60. It is also very similar to Proposition 4.19.

Exercise 6.11

(See Solution 6.11.) Consider the two matrices

\[ A = \left ( \begin{array}{ccc} 1 & 0 & 2 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{array} \right ) \text{ and } B = \left ( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 3 & 2 \\ 0 & 0 & 3 \end{array} \right ). \]

Do they represent the same linear map \(f : {\bf R}^3 \to {\bf R}^3\) (with respect to different bases)?

Exercise 6.12

(See Solution 6.12.) Let \(A = \left ( \begin{array}{ccc} 1 & 1 & 1 \\ 0 & 2 & 1 \\ 0 & 0 & 2 \end{array} \right )\). Determine the eigenvalues of \(A\) and the corresponding eigenspaces. Is \(A\) diagonalizable? Is \(A^2\) similar to \(A\)? I.e., does \(A^2\) represent the same linear map \({\bf R}^3 \to {\bf R}^3\) as \(A\)?

Exercise 6.13

(See Solution 6.13.) Consider the vectors \(v_1 = (1,0,1)\), \(v_2 = (1,1,1)\) and \(v_3=(1,1,2)\).

  1. Explain why there is a unique linear map \(f : {\bf R}^3 \to {\bf R}^3\) such that \(f(v_1)=(0,0,0)\), \(f(v_2)=(1,0,3)\) and \(v_3\) is an eigenvector of eigenvalue 4.

  2. Compute the matrix \(A\) of \(f\) with respect to the basis \(v_1, v_2, v_3\) (both on the domain and on the codomain).

  3. Compute the matrix \(B\) of \(f\) with respect to the standard basis (both for the domain and the codomain).

  4. For \(t \in {\bf R}\), consider the vector \(v_t = (2,t,5)\). For which values of \(t\) is \(v_t \in {\operatorname{im}\ } f\)?

Exercise 6.14

(See Solution 6.14.) Consider the matrix

\[ A = \begin{pmatrix} 0 & 2 & t \\ -3 & -5 & 6 \\ -2 & -2 & 5 \end{pmatrix} \]

  1. Determine the value of \(t\) for which \(A\) is not invertible.

  2. We now put \(t = 2\) for the remainder of this exercise. Determine the value of \(a\) for which the vector \(v = (2,0,a)\) is an eigenvector of \(A\). What is the corresponding eigenvalue?

  3. Determine all the eigenvalues of \(A\) and decide whether \(A\) is diagonalizable.

  4. Decide whether \(A\) is similar to the matrix \(A^2\) (justify your response).

Exercise 6.15

(See Solution 6.15.) Consider the matrix

\[ A = \begin{pmatrix} 2 & 0 & -1 \\ 1 & 1 & 2 \\ -1 & 1 & t \end{pmatrix} \]

  1. Compute the value of \(t\) for which the kernel of \(A\) is different from \(\{0\}\).

  2. For the remainder of the exercise we put \(t\) to be equal to the value computed in part (a). Compute the characteristic polynomial and the eigenvalues of \(A\).

  3. Find an invertible matrix \(P\) such that \(P^{-1}AP\) is a diagonal matrix.

  4. Explain why any \(3 \times 3\)-matrix \(B\) such that \(\chi_A(t) = \chi_B(t)\) is diagonalizable.

Exercise 6.16

(See Solution 6.16.) Consider the matrix \(A = \begin{pmatrix} 1 & 0 & t \\ 1 & 2 & 1 \\ 2 & 0 & -1 \end{pmatrix}\)

  1. For what value of \(t \in {\bf R}\) is the matrix \(A\) non-invertible?

  2. For each \(t \in {\bf R}\), determine the eigenvalues of \(A\). Specify for which values \(t \in {\bf R}\) all the eigenvalues of \(A\) are real numbers.

  3. Determine for which \(t \in {\bf R}\) there are eigenvalues with multiplicity \(> 1\).

  4. For the value of \(t\) found in the first part of the exercise: compute a basis of all eigenspaces and decide whether \(A\) is diagonalizable.