3.3E: Eigenvalues and Eigenvectors Exercises

Exercise \(\PageIndex{1}\)

In each case find the characteristic polynomial, eigenvalues, eigenvectors, and (if possible) an invertible matrix \(P\) such that \(P^{-1}AP\) is diagonal.

1. \(A = \left[ \begin{array}{rr} 1 & 2 \\ 3 & 2 \end{array}\right]\)
2. \(A = \left[ \begin{array}{rr} 2 & -4 \\ -1 & -1 \end{array}\right]\)
3. \(A = \left[ \begin{array}{rrr} 7 & 0 & -4 \\ 0 & 5 & 0 \\ 5 & 0 & -2 \end{array}\right]\)
4. \(A = \left[ \begin{array}{rrr} 1 & 1 & -3 \\ 2 & 0 & 6 \\ 1 & -1 & 5 \end{array}\right]\)
5. \(A = \left[ \begin{array}{rrr} 1 & -2 & 3 \\ 2 & 6 &-6 \\ 1 & 2 & -1 \end{array}\right]\)
6. \(A = \left[ \begin{array}{rrr} 0 & 1 & 0 \\ 3 & 0 & 1 \\ 2 & 0 & 0 \end{array}\right]\)
7. \(A = \left[ \begin{array}{rrr} 3 & 1 & 1 \\ -4 & -2 & -5 \\ 2 & 2 & 5 \end{array}\right]\)
8. \(A = \left[ \begin{array}{rrr} 2 & 1 & 1 \\ 0 & 1 & 0 \\ 1 & -1 & 2 \end{array}\right]\)
9. \(A = \left[ \begin{array}{rrr} \lambda & 0 & 0 \\ 0 & \lambda & 0 \\ 0 & 0 & \mu \end{array}\right]\), \(\lambda \neq \mu\)

Answer

2. \((x-3)(x+2); 3; -2\); \(\left[ \begin{array}{r} 4 \\ -1 \end{array} \right], \left[ \begin{array}{r} 1 \\ 1 \end{array} \right]\); \(P = \left[ \begin{array}{rr} 4 & 1 \\ -1 & 1 \end{array} \right]\); \(P^{-1}AP = \left[ \begin{array}{rr} 3 & 0 \\ 0 & -2 \end{array}\right].\)
3. \((x-2)^3 ; 2 ; \left[ \begin{array}{c} 1 \\ 1 \\ 0 \end{array}\right], \left[ \begin{array}{r} -3 \\ 0 \\ 1 \end{array}\right]\); No such \(P\); Not diagonalizable.
4. \((x+1)^2(x-2) ; -1, -2; \left[ \begin{array}{r} -1 \\ 1 \\ 2 \end{array}\right], \left[ \begin{array}{r} 1 \\ 2 \\ 1 \end{array}\right]\); No such \(P\); Not diagonalizable. Note that this matrix and the matrix in Example 3.4.2 have the same characteristic polynomial, but that matrix is diagonalizable.
5. \((x-1)^2(x-3) ; 1, 3; \left[ \begin{array}{r} -1 \\ 0 \\ 1 \end{array}\right], \left[ \begin{array}{r} 1 \\ 0 \\ 1 \end{array}\right]\) No such \(P\); Not diagonalizable.

Exercise \(\PageIndex{2}\)

Consider a linear dynamical system \(\mathbf{v}_{k+1} = A\mathbf{v}_{k}\) for \(k \geq 0\). In each case approximate \(\mathbf{v}_{k}\) using Theorem 3.5.1.

1. \(A = \left[ \begin{array}{rr} 2 & 1 \\ 4 & -1 \end{array}\right], \mathbf{v}_0 = \left[ \begin{array}{r} 1 \\ 2 \end{array}\right]\)
2. \(A = \left[ \begin{array}{rr} 3 & -2 \\ 2 & -2 \end{array}\right], \mathbf{v}_0 = \left[ \begin{array}{r} 3 \\ -1 \end{array}\right]\)
3. \(A = \left[ \begin{array}{rrr} 1 & 0 & 0 \\ 1 & 2 & 3 \\ 1 & 4 & 1 \end{array}\right], \mathbf{v}_0 = \left[ \begin{array}{r} 1 \\ 1 \\ 1 \end{array}\right]\)
4. \(A = \left[ \begin{array}{rrr} 1 & 3 & 2 \\ -1 & 2 & 1 \\ 4 & -1 & -1 \end{array}\right], \mathbf{v}_0 = \left[ \begin{array}{r} 2 \\ 0 \\ 1 \end{array}\right]\)

Answer

2. \(V_k = \frac{7}{3} 2^k \left[ \begin{array}{r} 2 \\ 1 \end{array}\right]\)
3. \(V_k = \frac{3}{2} 3^k \left[ \begin{array}{r} 1 \\ 0 \\ 1 \end{array}\right]\)

Exercise \(\PageIndex{3}\)

Show that \(A\) has \(\lambda = 0\) as an eigenvalue if and only if \(A\) is not invertible.

Exercise \(\PageIndex{4}\)

Let \(A\) denote an \(n \times n\) matrix and put \(A_{1} = A - \alpha I\), \(\alpha\) in \(\mathbb{R}\). Show that \(\lambda\) is an eigenvalue of \(A\) if and only if \(\lambda -\alpha\) is an eigenvalue of \(A_{1}\). (Hence, the eigenvalues of \(A_{1}\) are just those of \(A\) “shifted” by \(\alpha\).) How do the eigenvectors compare?

Answer

\(A\mathbf{x} = \lambda\mathbf{x}\) if and only if \((A - \alpha I)\mathbf{x} = (\lambda - \alpha)\mathbf{x}\). Same eigenvectors.

Exercise \(\PageIndex{5}\)

Show that the eigenvalues of \(\left[ \begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array} \right]\) are \(e^{i\theta}\) and \(e^{-i\theta}\). (See Appendix A)

Exercise \(\PageIndex{6}\)

Find the characteristic polynomial of the \(n \times n\) identity matrix \(I\). Show that \(I\) has exactly one eigenvalue and find the eigenvectors.

Exercise \(\PageIndex{7}\)

Given \(A = \left[ \begin{array}{rr} a & b \\ c & d \end{array} \right]\) show that:

1. \(c_{A}(x) = x^{2} - tr \; Ax + \det A\), where \(tr \; A = a + d\) is called the trace of \(A\).
2. The eigenvalues are \(\frac{1}{2} \left[ (a+d) \pm \sqrt{(a-b)^2 + 4bc}\right]\).

Exercise \(\PageIndex{8}\)

In each case, find \(P^{-1}AP\) and then compute \(A^{n}\).

1. \(A = \left[ \begin{array}{rr} 6 & -5 \\ 2 & -1 \end{array}\right], P = \left[ \begin{array}{rr} 1 & 5 \\ 1 & 2 \end{array}\right]\)
2. \(A = \left[ \begin{array}{rr} -7 & -12 \\ 6 & -10 \end{array}\right], P = \left[ \begin{array}{rr} -3 & 4 \\ 2 & -3 \end{array}\right]\)

[Hint: \((PDP^{-1})^{n} = PD^{n}P^{-1}\) for each \(n = 1, 2, \dots\).]

Answer

2. \(P^{-1}AP = \left[ \begin{array}{rr} 1 & 0 \\ 0 & 2 \end{array}\right]\), so \(A^n = P \left[ \begin{array}{rr} 1 & 0 \\ 0 & 2^n \end{array}\right] P^{-1} = \left[ \begin{array}{cc} 9 - 8 \cdot 2^n & 12(1-2^n) \\ 6(2^n-1) & 9\cdot 2^n - 8 \end{array}\right]\)