5.5E: Similarity and Diagonalization Exercises
- Page ID
- 132824
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solutions
2
By computing the trace, determinant, and rank, show that \(A\) and \(B\) are not similar in each case.
- \(A = \left[ \begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array} \right]\), \(B = \left[ \begin{array}{rr} 1 & 1\\ -1 & 1 \end{array} \right]\)
- \(A = \left[ \begin{array}{rr} 3 & 1 \\ 2 & -1 \end{array} \right]\), \(B = \left[ \begin{array}{rr} 1 & 1 \\ 2 & 1 \end{array} \right]\)
- \(A = \left[ \begin{array}{rr} 2 & 1 \\ 1 & -1 \end{array} \right]\), \(B = \left[ \begin{array}{rr} 3 & 0 \\ 1 & -1 \end{array} \right]\)
- \(A = \left[ \begin{array}{rr} 3 & 1 \\ -1 & 2 \end{array} \right]\), \(B = \left[ \begin{array}{rr} 2 & -1 \\ 3 & 2 \end{array} \right]\)
- \(A = \left[ \begin{array}{rrr} 2 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{array} \right]\), \(B = \left[ \begin{array}{rrr} 1 & -2 & 1 \\ -2 & 4 & -2 \\ -3 & 6 & -3 \end{array} \right]\)
- \(A = \left[ \begin{array}{rrr} 1 & 2 & -3 \\ 1 & -1 & 2 \\ 0 & 3 & -5 \end{array} \right]\), \(B = \left[ \begin{array}{rrr} -2 & 1 & 3 \\ 6 & -3 & -9 \\ 0 & 0 & 0 \end{array} \right]\)
- traces \(= 2\), ranks \(= 2\), but \(\det A = -5\), \(\det B = -1\)
- ranks \(= 2\), determinants \(= 7\), but \(\func{tr} A = 5\), \(\func{tr} B = 4\)
- traces \(= -5\), determinants \(= 0\), but \(rank \; A = 2\), \(rank \; B = 1\)
Show that \(\left[ \begin{array}{rrrr} 1 & 2 & -1 & 0 \\ 2 & 0 & 1 & 1 \\ 1 & 1 & 0 & -1 \\ 4 & 3 & 0 & 0 \end{array} \right]\) and \(\left[ \begin{array}{rrrr} 1 & -1 & 3 & 0 \\ -1 & 0 & 1 & 1 \\ 0 & -1 & 4 & 1 \\ 5 & -1 & -1 & -4 \end{array} \right]\) are not similar.
If \(A \sim B\), show that:
\(A^{T} \sim B^{T}\) \(A^{-1} \sim B^{-1}\) \(rA \sim rB\) for \(r\) in \(\mathbb{R}\) \(A^{n} \sim B^{n}\) for \(n \geq 1\)
- If \(B = P^{-1}AP\), then \(B^{-1} = P^{-1}A^{-1}(P^{-1})^{-1} = P^{-1}A^{-1}P\).
In each case, decide whether the matrix \(A\) is diagonalizable. If so, find \(P\) such that \(P^{-1}AP\) is diagonal.
\(\left[ \begin{array}{rrr} 1 & 0 & 0 \\ 1 & 2 & 1 \\ 0 & 0 & 1 \end{array} \right]\) \(\left[ \begin{array}{rrr} 3 & 0 & 6 \\ 0 & -3 & 0 \\ 5 & 0 & 2 \end{array} \right]\) \(\left[ \begin{array}{rrr} 3 & 1 & 6 \\ 2 & 1 & 0 \\ -1 & 0 & -3 \end{array} \right]\) \(\left[ \begin{array}{rrr} 4 & 0 & 0 \\ 0 & 2 & 2 \\ 2 & 3 & 1 \end{array} \right]\)
- Yes, \(P = \left[ \begin{array}{rrr} -1 & 0 & 6 \\ 0 & 1 & 0 \\ 1 & 0 & 5 \end{array} \right]\), \(P^{-1}AP = \left[ \begin{array}{rrr} -3 & 0 & 0 \\ 0 & -3 & 0 \\ 0 & 0 & 8 \end{array} \right]\)
- No, \(c_{A}(x) = (x + 1)(x - 4)^{2}\) so \(\lambda = 4\) has multiplicity 2. But \(dim \;(E_{4}) = 1\) so Theorem [thm:016250] applies.
If \(A\) is invertible, show that \(AB\) is similar to \(BA\) for all \(B\).
Show that the only matrix similar to a scalar matrix \(A = rI\), \(r\) in \(\mathbb{R}\), is \(A\) itself.
Let \(\lambda\) be an eigenvalue of \(A\) with corresponding eigenvector \(\mathbf{x}\). If \(B = P^{-1}AP\) is similar to \(A\), show that \(P^{-1}\mathbf{x}\) is an eigenvector of \(B\) corresponding to \(\lambda\).
If \(A \sim B\) and \(A\) has any of the following properties, show that \(B\) has the same property.
- Idempotent, that is \(A^{2} = A\).
- Nilpotent, that is \(A^{k} = 0\) for some \(k \geq 1\).
- Invertible.
- If \(B = P^{-1}AP\) and \(A^{k} = 0\), then \(B^{k} = (P^{-1}AP)^{k} = P^{-1}A^{k}P = P^{-1}0P = 0\).
Let \(A\) denote an \(n \times n\) upper triangular matrix.
- If all the main diagonal entries of \(A\) are distinct, show that \(A\) is diagonalizable.
- If all the main diagonal entries of \(A\) are equal, show that \(A\) is diagonalizable only if it is already diagonal.
- Show that \(\left[ \begin{array}{rrr} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \end{array} \right]\) is diagonalizable but that \(\left[ \begin{array}{rrr} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \end{array} \right]\) is not diagonalizable.
- The eigenvalues of \(A\) are all equal (they are the diagonal elements), so if \(P^{-1}AP = D\) is diagonal, then \(D = \lambda I\). Hence \(A = P^{-1}(\lambda I)P = \lambda I\).
Let \(A\) be a diagonalizable \(n \times n\) matrix with eigenvalues \(\lambda_{1}, \lambda_{2}, \dots, \lambda_{n}\) (including multiplicities). Show that:
- \(\det A = \lambda_{1}\lambda_{2}\cdots \lambda_{n}\)
- \(\func{tr} A = \lambda_{1} + \lambda_{2} + \cdots + \lambda_{n}\)
- \(A\) is similar to \(D = \func{diag}(\lambda_{1}, \lambda_{2}, \dots, \lambda_{n})\) so (Theorem [thm:016008]) \(\func{tr} A = \func{tr} D = \lambda_{1} + \lambda_{2} + \dots + \lambda_{n}\).
Given a polynomial \(p(x) = r_{0} + r_{1}x + \dots + r_{n}x^{n}\) and a square matrix \(A\), the matrix \(p(A) = r_{0}I + r_{1}A + \dots + r_{n}A^{n}\) is called the evaluation of \(p(x)\) at \(A\). Let \(B = P^{-1}AP\). Show that \(p(B) = P^{-1}p(A)P\) for all polynomials \(p(x)\).
[ex:5_5_12] Let \(P\) be an invertible \(n \times n\) matrix. If \(A\) is any \(n \times n\) matrix, write \(T_{P}(A) = P^{-1}AP\). Verify that:
\(T_{P}(I) = I\) \(T_{P}(AB) = T_{P}(A)T_{P}(B)\) \(T_{P}(A + B) = T_{P}(A) + T_{P}(B)\) \(T_{P}(rA) = rT_{P}(A)\) \(T_{P}(A^{k}) = [T_{P}(A)]^{k}\) for \(k \geq 1\) If \(A\) is invertible, \(T_{P}(A^{-1}) = [T_{P}(A)]^{-1}\). If \(Q\) is invertible, \(T_{Q}[T_{P}(A)] = T_{PQ}(A)\).
- \(T_{P}(A)T_{P}(B) = (P^{-1}AP)(P^{-1}BP) = P^{-1}(AB)P = T_{P}(AB)\).
- Show that two diagonalizable matrices are similar if and only if they have the same eigenvalues with the same multiplicities.
- If \(A\) is diagonalizable, show that \(A \sim A^{T}\).
- Show that \(A \sim A^{T}\) if \(A = \left[ \begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array} \right]\)
- If \(A\) is diagonalizable, so is \(A^{T}\), and they have the same eigenvalues. Use (a).
If \(A\) is \(2 \times 2\) and diagonalizable, show that \(C(A) = \{X \mid XA = AX\}\) has dimension \(2\) or \(4\). [Hint: If \(P^{-1}AP = D\), show that \(X\) is in \(C(A)\) if and only if \(P^{-1}XP\) is in \(C(D)\).]
If \(A\) is diagonalizable and \(p(x)\) is a polynomial such that \(p(\lambda) = 0\) for all eigenvalues \(\lambda\) of \(A\), show that \(p(A) = 0\) (see Example [exa:009262]). In particular, show \(c_{A}(A) = 0\). [Remark: \(c_{A}(A) = 0\) for all square matrices \(A\)—this is the Cayley-Hamilton theorem, see Theorem [thm:033262].]
Let \(A\) be \(n \times n\) with \(n\) distinct real eigenvalues. If \(AC = CA\), show that \(C\) is diagonalizable.
Let \(A = \left[ \begin{array}{rrr} 0 & a & b \\ a & 0 & c \\ b & c & 0 \end{array} \right]\) and \(B = \left[ \begin{array}{rrr} c & a & b \\ a & b & c \\ b & c & a \end{array} \right]\).
- Show that \(x^{3} - (a^{2} + b^{2} + c^{2})x - 2abc\) has real roots by considering \(A\).
- Show that \(a^{2} + b^{2} + c^{2} \geq ab + ac + bc\) by considering \(B\).
- \(c_{B}(x) = [x - (a + b + c)][x^{2} - k]\) where \(k = a^{2} + b^{2} + c^{2} - [ab + ac + bc]\). Use Theorem [thm:016397].
Assume the \(2 \times 2\) matrix \(A\) is similar to an upper triangular matrix. If \(\func{tr} A = 0 = \func{tr} A^{2}\), show that \(A^{2} = 0\).
Show that \(A\) is similar to \(A^{T}\) for all \(2 \times 2\) matrices \(A\). [Hint: Let \(A = \left[ \begin{array}{rr} a & b \\ c & d \end{array} \right]\). If \(c = 0\) treat the cases \(b = 0\) and \(b \neq 0\) separately. If \(c \neq 0\), reduce to the case \(c = 1\) using Exercise [ex:5_5_12](d).]
Refer to Section [sec:3_4] on linear recurrences. Assume that the sequence \(x_{0}, x_{1}, x_{2}, \dots\) satisfies
\[x_{n+k} = r_0x_n + r_1x_{n+1} + \dotsb + r_{k-1}x_{n+k-1} \nonumber \]
for all \(n \geq 0\). Define
\[A = \scriptsize \left[ \begin{array}{ccccc} 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & & \vdots \\ 0 & 0 & 0 & \cdots & 1 \\ r_0 & r_1 & r_2 & \cdots & r_{k-1} \end{array} \right], V_n = \left[ \begin{array}{cccc} x_n \\ x_{n+1} \\ \vdots \\ x_{n+k-1} \end{array} \right]. \nonumber \]
Then show that:
- \(V_{n} = A^{n}V_{0}\) for all \(n\).
- \(c_{A}(x) = x^{k} - r_{k-1}x^{k-1} - \dots - r_{1}x - r_{0}\)
- If \(\lambda\) is an eigenvalue of \(A\), the eigenspace \(E_{\lambda}\) has dimension 1, and \(\mathbf{x} = (1, \lambda, \lambda^{2}, \dots, \lambda^{k-1})^{T}\) is an eigenvector. [Hint: Use \(c_{A}(\lambda) = 0\) to show that \(E_{\lambda} = \mathbb{R}\mathbf{x}\).]
- \(A\) is diagonalizable if and only if the eigenvalues of \(A\) are distinct. [Hint: See part (c) and Theorem [thm:016090].]
- If \(\lambda_{1}, \lambda_{2}, \dots, \lambda_{k}\) are distinct real eigenvalues, there exist constants \(t_{1}, t_{2}, \dots, t_{k}\) such that \(x_n = t_1\lambda_1^n + \dots + t_k\lambda_k^n\) holds for all \(n\). [Hint: If \(D\) is diagonal with \(\lambda_{1}, \lambda_{2}, \dots, \lambda_{k}\) as the main diagonal entries, show that \(A^{n}\) = \(PD^{n}P^{-1}\) has entries that are linear combinations of \(\lambda_1^n, \lambda_2^n, \dots, \lambda_k^n\).]
Suppose \(A\) is \(2 \times 2\) and \(A^2=0\). If \(\func{tr} A \neq 0\) show that \(A=0\).