3.4E: Diagonalization Exercises

Exercise \(\PageIndex{1}\)

2. If \(A = \left[ \begin{array}{rr} 1 & 3 \\ 0 & 2 \end{array} \right]\) and \(B = \left[ \begin{array}{rr} 2 & 0 \\ 0 & 1 \end{array}\right]\) verify that \(A\) and \(B\) are diagonalizable, but \(AB\) is not.
3. If \(D = \left[ \begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array}\right]\) find a diagonalizable matrix \(A\) such that \(D + A\) is not diagonalizable.

Answer

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

Exercise \(\PageIndex{2}\)

If \(A\) is an \(n \times n\) matrix, show that \(A\) is diagonalizable if and only if \(A^{T}\) is diagonalizable.

Exercise \(\PageIndex{3}\)

If \(A\) is diagonalizable, show that each of the following is also diagonalizable.

1. \(A^{n}\), \(n \geq 1\)
2. \(kA\), \(k\) any scalar.
3. \(p(A)\), \(p(x)\) any polynomial (Theorem 3.3.1)
4. \(U^{-1}AU\) for any invertible matrix \(U\).
5. \(kI + A\) for any scalar \(k\).

Answer

2. \(PAP^{-1} = D\) is diagonal
3. \(P^{-1}(kA)P = kD\) is diagonal
4. \(Q(U^{-1}AU)Q = D\) where \(Q = PU\).

Exercise \(\PageIndex{4}\)

Give an example of two diagonalizable matrices \(A\) and \(B\) whose sum \(A + B\) is not diagonalizable.

Answer

\(\left[ \begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array}\right]\) is not diagonalizable by Example 3.4.1. But \(\left[ \begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right] = \left[ \begin{array}{rr} 2 & 1 \\ 0 & -1 \end{array}\right] + \left[ \begin{array}{rr} -1 & 0 \\ 0 & 2 \end{array}\right]\) where \(\left[ \begin{array}{rr} 2 & 1 \\ 0 & -1 \end{array}\right]\) has diagonalizing matrix \(P = \left[ \begin{array}{rr} 1 & -1 \\ 0 & 3 \end{array}\right]\) and \(\left[ \begin{array}{rr} -1 & 0 \\ 0 & 2 \end{array}\right]\) is already diagonal.

Exercise \(\PageIndex{5}\)

If \(A\) is diagonalizable and \(1\) and \(-1\) are the only eigenvalues, show that \(A^{-1} = A\).

Exercise \(\PageIndex{6}\)

If \(A\) is diagonalizable and \(0\) and \(1\) are the only eigenvalues, show that \(A^{2} = A\).

Answer

We have \(\lambda^{2} = \lambda\) for every eigenvalue \(\lambda\) (as \(\lambda = 0, 1\)) so \(D^{2} = D\), and so \(A^{2} = A\) as in Example 3.4.2.

Exercise \(\PageIndex{9}\)

If \(A\) is diagonalizable and \(\lambda \geq 0\) for each eigenvalue of \(A\), show that \(A = B^{2}\) for some matrix \(B\).

Exercise \(\PageIndex{10}\)

If \(P^{-1}AP\) and \(P^{-1}BP\) are both diagonal, show that \(AB = BA\). [Hint: Diagonal matrices commute.]

Exercise \(\PageIndex{11}\)

A square matrix \(A\) is called nilpotent if \(A^{n} = 0\) for some \(n \geq 1\). Find all nilpotent diagonalizable matrices. [Hint: Theorem 3.3.1.]

Exercise \(\PageIndex{12}\)

Let \(A\) be any \(n \times n\) matrix and \(r \neq 0\) a real number.

1. Show that the eigenvalues of \(rA\) are precisely the numbers \(r\lambda\), where \(\lambda\) is an eigenvalue of \(A\).
2. Show that \(c_{rA}(x) = r^n c_A\left( \frac{x}{r} \right)\).

Answer

2. \(c_{rA} (x) =\det \left[ xI - rA \right]\) \({} = r^n \det \left[ \frac{x}{r}I-A \right] = r^n c_A \left[ \frac{x}{r} \right]\)

Exercise \(\PageIndex{13}\)

1. If all rows of \(A\) have the same sum \(s\), show that \(s\) is an eigenvalue.
2. If all columns of \(A\) have the same sum \(s\), show that \(s\) is an eigenvalue.

Exercise \(\PageIndex{14}\)

Let \(A\) be an invertible \(n \times n\) matrix.

1. Show that the eigenvalues of \(A\) are nonzero.
2. Show that the eigenvalues of \(A^{-1}\) are precisely the numbers \(1/\lambda\), where \(\lambda\) is an eigenvalue of \(A\).
3. Show that \(c_{A^{-1}}(x) = \frac{(-x)^n}{\det A} c_A \left( \frac{1}{x} \right)\).

Answer

2. If \(\lambda \neq 0\), \(A\mathbf{x} = \lambda\mathbf{x}\) if and only if \(A^{-1}\mathbf{x} = \frac{1}{\lambda}\mathbf{x}\). The result follows.

Exercise \(\PageIndex{15}\)

Suppose \(\lambda\) is an eigenvalue of a square matrix \(A\) with eigenvector \(\mathbf{x} \neq \mathbf{0}\).

1. Show that \(\lambda^{2}\) is an eigenvalue of \(A^{2}\) (with the same \(\mathbf{x}\)).
2. Show that \(\lambda^{3} - 2 \lambda + 3\) is an eigenvalue of \(A^{3} - 2A + 3I\).
3. Show that \(p(\lambda)\) is an eigenvalue of \(p(A)\) for any nonzero polynomial \(p(x)\).

Answer

2. \((A^{3} - 2A - 3I)\mathbf{x} = A^{3}\mathbf{x} - 2A\mathbf{x} + 3\mathbf{x} = \lambda^{3}\mathbf{x} - 2\lambda\mathbf{x} + 3\mathbf{x} = (\lambda^{3} - 2\lambda - 3)\mathbf{x}\).

Exercise \(\PageIndex{16}\)

If \(A\) is an \(n \times n\) matrix, show that \(c_{A^2}(x^{2}) = (-1)^{n}c_{A}(x)c_{A}(-x)\).

Exercise \(\PageIndex{17}\)

An \(n \times n\) matrix \(A\) is called nilpotent if \(A^{m} = 0\) for some \(m \geq 1\).

1. Show that every triangular matrix with zeros on the main diagonal is nilpotent.
2. If \(A\) is nilpotent, show that \(\lambda = 0\) is the only eigenvalue (even complex) of \(A\).
3. Deduce that \(c_{A}(x) = x^{n}\), if \(A\) is \(n \times n\) and nilpotent.

Answer

2. If \(A^{m} = 0\) and \(A\mathbf{x} = \lambda\mathbf{x}\), \(\mathbf{x} \neq \mathbf{0}\), then \(A^{2}\mathbf{x} = A(\lambda\mathbf{x}) = \lambda A\mathbf{x} = \lambda^{2}\mathbf{x}\). In general, \(A^{k}\mathbf{x} = \lambda^{k}\mathbf{x}\) for all \(k \geq 1\). Hence, \(\lambda^{m}\mathbf{x} = A^{m}\mathbf{x} = \mathbf{0}\mathbf{x} = \mathbf{0}\), so \(\lambda = 0\) (because \(\mathbf{x} \neq \mathbf{0}\)).

Exercise \(\PageIndex{18}\)

Let \(A\) be diagonalizable with real eigenvalues and assume that \(A^{m} = I\) for some \(m \geq 1\).

1. Show that \(A^{2} = I\).
2. If \(m\) is odd, show that \(A = I\). [Hint: Theorem A.3]

Answer

1. If \(A\mathbf{x} = \lambda\mathbf{x}\), then \(A^{k}\mathbf{x} = \lambda^{k}\mathbf{x}\) for each \(k\). Hence \(\lambda^{m}\mathbf{x} = A^{m}\mathbf{x} = \mathbf{x}\), so \(\lambda^{m} = 1\). As \(\lambda\) is real, \(\lambda = \pm 1\) by the Hint. So if \(P^{-1}AP = D\) is diagonal, then \(D^{2} = I\) by Theorem 3.4.1. Hence \(A^{2} = PD^{2}P = I\).

Exercise \(\PageIndex{19}\)

Let \(A^{2} = I\), and assume that \(A \neq I\) and \(A \neq -I\).

1. Show that the only eigenvalues of \(A\) are \(\lambda = 1\) and \(\lambda = -1\).
2. Show that \(A\) is diagonalizable. [Hint: Verify that \(A(A + I) = A + I\) and \(A(A - I) = -(A - I)\), and then look at nonzero columns of \(A + I\) and of \(A - I\).]
3. If \(Q_{m} : \mathbb{R}^2 \to \mathbb{R}^2\) is reflection in the line \(y = mx\) where \(m \neq 0\), use (b) to show that the matrix of \(Q_{m}\) is diagonalizable for each \(m\).
4. Now prove (c) geometrically using Theorem 3.3.3.

Exercise \(\PageIndex{20}\)

Let \(A^{2} = I\), and assume that \(A \neq I\) and \(A \neq -I\).

Exercise \(\PageIndex{21}\)

Let \(A = \left[ \begin{array}{rrr} 2 & 3 & -3 \\ 1 & 0 & -1 \\ 1 & 1 & -2 \end{array} \right]\) and \(B = \left[ \begin{array}{rrr} 0 & 1 & 0 \\ 3 & 0 & 1 \\ 2 & 0 & 0 \end{array} \right]\). Show that \(c_{A}(x) = c_{B}(x) = (x + 1)^{2} (x - 2)\), but \(A\) is diagonalizable and \(B\) is not.

1. Show that the only diagonalizable matrix \(A\) that has only one eigenvalue \(\lambda\) is the scalar matrix \(A = \lambda I\).
2. Is \(\left[ \begin{array}{rr} 3 & -2 \\ 2 & -1 \end{array}\right]\) diagonalizable?

Answer

1. We have \(P^{-1}AP = \lambda I\) by the diagonalization algorithm, so \(A = P(\lambda I)P^{-1} = \lambda PP^{-1} = \lambda I\).
2. No. \(\lambda = 1\) is the only eigenvalue.

Exercise \(\PageIndex{22}\)

Characterize the diagonalizable \(n \times n\) matrices \(A\) such that \(A^{2} - 3A + 2I = 0\) in terms of their eigenvalues. [Hint: Theorem 3.3.1.]

Exercise \(\PageIndex{23}\)

Let \(A = \left[ \begin{array}{cc} B & 0 \\ 0 & C \end{array}\right]\) where \(B\) and \(C\) are square matrices.

1. If \(B\) and \(C\) are diagonalizable via \(Q\) and \(R\) (that is, \(Q^{-1}BQ\) and \(R^{-1}CR\) are diagonal), show that \(A\) is diagonalizable via \(\left[ \begin{array}{cc} Q & 0 \\ 0 & R \end{array}\right]\)
2. Use (a) to diagonalize \(A\) if \(B = \left[ \begin{array}{rr} 5 & 3 \\ 3 & 5 \end{array}\right]\) and \(C = \left[ \begin{array}{rr} 7 & -1 \\ -1 & 7 \end{array}\right]\).

Exercise \(\PageIndex{24}\)

Let \(A = \left[ \begin{array}{cc} B & 0 \\ 0 & C \end{array}\right]\) where \(B\) and \(C\) are square matrices.

1. Show that \(c_{A}(x) = c_{B}(x)c_{C}(x)\).
2. If \(\mathbf{x}\) and \(\mathbf{y}\) are eigenvectors of \(B\) and \(C\), respectively, show that \(\left[ \begin{array}{c} \mathbf{x} \\ 0 \end{array}\right]\) and \(\left[ \begin{array}{c} 0 \\ \mathbf{y} \end{array}\right]\) are eigenvectors of \(A\), and show how every eigenvector of \(A\) arises from such eigenvectors.