Chapter 7

Determine all eigenvalues and corresponding eigenvectors of the given matrix. $$\left[\begin{array}{rrr}7 & -8 & 6 \\\8 & -9 & 6 \\\0 & 0 & -1\end{array}\right]$$.

Deal with the eigenvalue/eigenvector problem for \(n \times n\) real skew- symmetric matrices. It follows from the previous problem that the only real eigenvalue that a real skew-symmetric matrix can possess is \(\lambda=0 .\) Use this to prove that if \(A\) is an \(n \times n\) real skew-symmetric matrix, with \(n\) odd, then \(A\) necessarily has zero as one of its eigenvalues.

Determine a basis for each eigenspace of \(A\) and sketch the eigenspaces. $$A=\left[\begin{array}{ll} 2 & 1 \\ 3 & 4 \end{array}\right]$$

Find the Jordan canonical form \(J\) for the matrix \(A_{1}\) and determine an invertible matrix \(S\) such that \(S^{-1} A S=J\). \(A=\left[\begin{array}{rrr}2 & -2 & 14 \\ 0 & 3 & -7 \\ 0 & 0 & 2\end{array}\right]\).

Let \(A\) be a nondefective matrix. Then $$S^{-1} A S=D,$$ where \(D\) is a diagonal matrix. This can be written as $$A=S D S^{-1},$$ Use this result to show that $$A^{2}=S D^{2} S^{-1},$$ and that for every positive integer \(k\) $$A^{k}=S D^{k} S^{-1}.$$

Deal with the eigenvalue/eigenvector problem for \(n \times n\) real skew- symmetric matrices. Determine all eigenvalues and corresponding eigenvectors of the matrix $$ A=\left[\begin{array}{rrr} 0 & 4 & -4 \\ -4 & 0 & -2 \\ 4 & 2 & 0 \end{array}\right] $$

Determine all eigenvalues and corresponding eigenvectors of the given matrix. $$\left[\begin{array}{rrr}0 & 1 & -1 \\\0 & 2 & 0 \\\2 & -1 & 3\end{array}\right]$$.

Determine a basis for each eigenspace of \(A\) and sketch the eigenspaces. $$A=\left[\begin{array}{ll} 2 & 3 \\ 0 & 2 \end{array}\right]$$

Find the Jordan canonical form \(J\) for the matrix \(A_{1}\) and determine an invertible matrix \(S\) such that \(S^{-1} A S=J\). \(A=\left[\begin{array}{rrr}7 & -2 & 2 \\ 0 & 4 & -1 \\ -1 & 1 & 4\end{array}\right]\).

If \(D=\operatorname{diag}\left(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}\right),\) show that for every positive integer \(k\) $$D^{k}=\operatorname{diag}\left(\lambda_{1}^{k}, \lambda_{2}^{k}, \ldots, \lambda_{n}^{k}\right).$$

Determine all eigenvalues and corresponding eigenvectors of the given matrix. $$\left[\begin{array}{rrr}1 & 0 & 0 \\\0 & 0 & 1 \\\0 & -1 & 0\end{array}\right]$$.

Find the Jordan canonical form \(J\) for the matrix \(A_{1}\) and determine an invertible matrix \(S\) such that \(S^{-1} A S=J\). \(A=\left[\begin{array}{rrr}-1 & -1 & 0 \\ 0 & -1 & -2 \\ 0 & 0 & -1\end{array}\right]\).

Determine all eigenvalues and corresponding eigenvectors of the given matrix. $$\left[\begin{array}{rrr}-2 & 1 & 0 \\\1 & -1 & -1 \\\1 & 3 & -3\end{array}\right].$$

Determine a basis for each eigenspace of \(A\) and sketch the eigenspaces. $$\begin{aligned} &A=\left[\begin{array}{rrr} 3 & 1 & -1 \\ 1 & 3 & -1 \\ -1 & -1 & 3 \end{array}\right]\\\ &\text { characteristic polynomial } p(\lambda)=(5-\lambda)(\lambda-2)^{2} \end{aligned}$$

We call a matrix \(B\) a square root of \(A\) if \(B^{2}=A\) (a) Show that if \(D=\operatorname{diag}\left(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}\right),\) then thematrix $$\sqrt{D}=\operatorname{diag}(\sqrt{\lambda_{1}}, \sqrt{\lambda_{2}}, \ldots, \sqrt{\lambda_{n}})$$ is a square root of \(D .\) (b) Show that if \(A\) is a nondefective matrix with \(S^{-1} A S=D\) for some invertible matrix \(S\) and diagonal matrix \(D,\) then \(S \sqrt{D} S^{-1}\) is a square root of \(A\). (c) Find a square root for the matrix $$A=\left[\begin{array}{rr}6 & -2 \\\\-3 & 7\end{array}\right]$$

Find the Jordan canonical form \(J\) for the matrix \(A_{1}\) and determine an invertible matrix \(S\) such that \(S^{-1} A S=J\). \(A=\left[\begin{array}{rrrr}2 & -1 & 0 & 1 \\ 0 & 3 & -1 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & -1 & 0 & 3\end{array}\right]\).

Determine all eigenvalues and corresponding eigenvectors of the given matrix. $$\left[\begin{array}{rrr}2 & -1 & 3 \\\3 & 1 & 0 \\\2 & -1 & 3\end{array}\right]$$.

Determine a basis for each eigenspace of \(A\) and sketch the eigenspaces. $$A=\left[\begin{array}{rrr} -3 & 1 & 0 \\ -1 & -1 & 2 \\ 0 & 0 & -2 \end{array}\right]$$

Find the Jordan canonical form \(J\) for the matrix \(A_{1}\) and determine an invertible matrix \(S\) such that \(S^{-1} A S=J\). \(A=\left[\begin{array}{rrrr}2 & -4 & 2 & 2 \\ -2 & 0 & 1 & 3 \\ -2 & -2 & 3 & 3 \\ -2 & -6 & 3 & 7\end{array}\right] .\) [The characteristic polynomial is \(\left.p(\lambda)=(2-\lambda)^{2}(4-\lambda)^{2} .\right]\)

Prove the following properties for similar matrices: (a) A matrix \(A\) is always similar to itself. (b) If \(A\) is similar to \(B,\) then \(B\) is similar to \(A\). (c) If \(A\) is similar to \(B\) and \(B\) is similar to \(C,\) then \(A\) is similar to \(C .\)

Determine all eigenvalues and corresponding eigenvectors of the given matrix. $$\left[\begin{array}{lll}5 & 0 & 0 \\\0 & 5 & 0 \\\0 & 0 & 5\end{array}\right]$$.

The matrix $$ A=\left[\begin{array}{lll} 2 & -2 & 3 \\ 1 & -1 & 3 \\ 1 & -2 & 4 \end{array}\right] $$ has eigenvalues \(\lambda_{1}=1\) and \(\lambda_{2}=3\) (a) Determine a basis for the eigenspace \(E_{1}\) corresponding to \(\lambda_{1}=1\) and then use the GramSchmidt procedure to obtain an orthogonal basis for \(E_{1}\) (b) Are the vectors in \(E_{1}\) orthogonal to the vectors in \(E_{2},\) the eigenspace corresponding to \(\lambda_{2}=3 ?\)

If \(A\) is similar to \(B,\) prove that \(A^{T}\) is similar to \(B^{T}\).

Find the Jordan canonical form \(J\) for the matrix \(A\). You need not determine an invertible matrix \(S\) such that \(S^{-1} A S=J\). \(A=\left[\begin{array}{lllll}2 & 1 & 1 & 1 & 1 \\ 0 & 2 & 0 & 0 & 1 \\ 0 & 0 & 2 & 0 & 1 \\ 0 & 0 & 0 & 2 & 1 \\ 0 & 0 & 0 & 0 & 2\end{array}\right]\).

Determine all eigenvalues and corresponding eigenvectors of the given matrix. $$\left[\begin{array}{lll}0 & 2 & 2 \\\2 & 0 & 2 \\\2 & 2 & 0\end{array}\right]$$.

Find the Jordan canonical form \(J\) for the matrix \(A\). You need not determine an invertible matrix \(S\) such that \(S^{-1} A S=J\). \(A=\left[\begin{array}{lllll}0 & 0 & 0 & 1 & 4 \\ 0 & 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 6 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0\end{array}\right]\).

In Theorem \(7.3 .3,\) we proved that similar matrices have the same eigenvalues. This problem investigates the relationship between their eigenvectors. Let \(\mathbf{v}\) be an eigenvector of \(A\) corresponding to the eigenvalue \(\lambda\) Prove that if \(B=S^{-1} A S,\) then \(S^{-1} \mathbf{v}\) is an eigenvector of \(B\) corresponding to the eigenvalue \(\lambda\).

Determine all eigenvalues and corresponding eigenvectors of the given matrix. $$\left[\begin{array}{llll}1 & 2 & 3 & 4 \\\4 & 3 & 2 & 1 \\\4 & 5 & 6 & 7 \\ 7 & 6 & 5 & 4\end{array}\right]$$.

The matrix $$ A=\left[\begin{array}{lll} a & b & c \\ a & b & c \\ a & b & c \end{array}\right] $$ has cigenvalues \(0,0,\) and \(a+b+c .\) Determine all values of the constants \(a, b,\) and \(c\) for which \(A\) is nondefective.

Let \(A\) be a nondefective matrix and let \(S\) be a matrix such that \(S^{-1} A S=\operatorname{diag}\left(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}\right),\) where all \(\lambda_{i}\) are nonzero. (a) Prove that \(A\) is invertible. (b) Prove that $$S^{-1} A^{-1} S=\operatorname{diag}\left(\frac{1}{\lambda_{1}}, \frac{1}{\lambda_{2}}, \ldots, \frac{1}{\lambda_{n}}\right)$$

Find the Jordan canonical form \(J\) for the matrix \(A\). You need not determine an invertible matrix \(S\) such that \(S^{-1} A S=J\). \(A=\left[\begin{array}{llllllll}1 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right]\).

Consider the characteristic polynomial of an \(n \times n\) matrix \(A\); namely, $$ p(\lambda)=\operatorname{det}(A-\lambda I)=\left|\begin{array}{cccc} a_{11}-\lambda & a_{12} & \dots & a_{1 n} \\ a_{21} & a_{22}-\lambda & \dots & a_{2 n} \\ \vdots & \vdots & & \vdots \\ a_{n 1} & a_{n 2} & \dots & a_{n n}-\lambda \end{array}\right| $$ which can be written in either of the following equivalent forms: $$ \begin{array}{l} p(\lambda)=(-1)^{n} \lambda^{n}+b_{1} \lambda^{n-1}+\cdots+b_{n},(7.2 .5) \\ p(\lambda)=\left(\lambda_{1}-\lambda\right)\left(\lambda_{2}-\lambda\right) \ldots\left(\lambda_{n}-\lambda\right),(7.2 .6) \end{array} $$ where \(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}\) (not necessarily distinct) are the eigenvalues of \(A\) (a) Use Equations (7.2.4) and (7.2.5) to show that $$ \begin{array}{l} b_{1}=(-1)^{n-1}\left(a_{11}+a_{22}+\cdots+a_{n n}\right) \\ b_{n}=\operatorname{det}(A) \end{array} $$ Recall that the quantity \(a_{11}+a_{22}+\cdots+a_{n n}\) is called the trace of the matrix \(A\), denoted \(\operatorname{tr}(A)\) (b) Use Equations (7.2.5) and (7.2.6) to show that $$ \begin{array}{l} b_{1}=(-1)^{n-1}\left(\lambda_{1}+\lambda_{2}+\cdots+\lambda_{n}\right) \\ b_{n}=\lambda_{1} \lambda_{2} \ldots \lambda_{n} \end{array} $$ (c) Use your results from (a) and (b) to show that $$ \operatorname{det}(A)=\text { product of the eigenvalues of } A $$ \(\operatorname{tr}(A)=\) sum of the eigenvalues of \(A\)

Determine all eigenvalues and corresponding eigenvectors of the given matrix. $$\left[\begin{array}{rrrr}0 & 1 & 0 & 0 \\\\-1 & 0 & 0 & 0 \\\0 & 0 & 0 & -1 \\\ 0 & 0 & 1 & 0\end{array}\right]$$.

Use the result of Problem 32 to determine the sum and the product of the eigenvalues of the given matrix \(A\). $$A=\left[\begin{array}{rrr} -1 & -2 & 0 \\ 6 & -3 & -8 \\ -2 & 2 & 1 \end{array}\right]$$

Let \(A\) be a nondefective matrix and let \(S\) be a matrix such that \(S^{-1} A S=\operatorname{diag}\left(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}\right)\) (a) Prove that if \(Q=\left(S^{T}\right)^{-1},\) then $$Q^{-1} A^{T} Q=\operatorname{diag}\left(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}\right)$$ (b) If \(M_{C}\) denotes the matrix of cofactors of \(S,\) prove that the column vectors of \(M_{C}\) are linearly independent eigenvectors of \(A^{T} .\) [Hint: Use the adjoint method to determine \(S^{-1} .]\)

Use Jordan canonical forms to determine whether the given pair of matrices are similar. \(A=\left[\begin{array}{rrr}7 & 1 & 0 \\ -1 & 5 & 0 \\ 1 & 0 & 6\end{array}\right] ; B=\left[\begin{array}{rrr}6 & -1 & 1 \\ 0 & 6 & 0 \\ 0 & 0 & 6\end{array}\right]\).

Find all eigenvalues and corresponding eigenvectors of $$A=\left[\begin{array}{ccc} 1+i & 0 & 0 \\\2-2 i & 1-3 i & 0 \\\2 i & 0 & 1\end{array}\right]$$. Note that the eigenvectors do not occur in complex conjugate pairs. Does this contradict Theorem \(7.1 .8 ?\) Explain.

Use the result of Problem 32 to determine the sum and the product of the eigenvalues of the given matrix \(A\). $$A=\left[\begin{array}{rrr} 2 & 0 & 5 \\ 0 & -1 & 1 \\ 3 & -4 & 2 \end{array}\right]$$

If \(A=\left[\begin{array}{rr}-2 & 4 \\ 1 & 1\end{array}\right],\) determine \(S\) such that \(S^{-1} A S=\) \(\operatorname{diag}(-3,2),\) and use the result from the previous problem to determine all eigenvectors of \(A^{T}\).

Use Jordan canonical forms to determine whether the given pair of matrices are similar. \(A=\left[\begin{array}{rrr}7 & -2 & 2 \\ 0 & 4 & -1 \\ -1 & 1 & 4\end{array}\right] ; B=\left[\begin{array}{rrr}3 & -1 & -2 \\ 1 & 6 & 1 \\\ 1 & 0 & 6\end{array}\right]\).

Consider the matrix \(A=\left[\begin{array}{rr}1 & -1 \\ 2 & 4\end{array}\right].\) (a) Show that the characteristic polynomial of \(A\) is \(p(\lambda)=\lambda^{2}-5 \lambda+6\). (b) Show that \(A\) satisfies its characteristic equation. That is, \(A^{2}-5 A+6 I_{2}=0_{2} .\) (This result is known as the Cayley-Hamilton Theorem and is true for general \(n \times n\) matrices.) (c) Use the result from (b) to find \(A^{-1}\). [Hint: Multiply the equation in (b) by \(\left.A^{-1} .\right]\)

Deal with the generalization of the diagonalization problem to defective matrices. A complete discussion of this topic can be found in Section 7.6. Prove that \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\) must satisfy $$\begin{array}{c}(A-\lambda I) \mathbf{v}_{1}=\mathbf{0} \\\\(A-\lambda I) \mathbf{v}_{2}=\mathbf{v}_{1}\end{array}$$ Equation \((7.3 .11)\) is the statement that \(\mathbf{v}_{1}\) must be an eigenvector of \(A\) corresponding to the eigenvalue \(\lambda\) Any vectors that satisfy \((7.3 .12)\) are called generalized eigenvectors of \(A\). The subject of generalized eigenvectors and Jordan canonical form matrices will be taken up in detail in Section 7.6.

Use Jordan canonical forms to determine whether the given pair of matrices are similar. \(A=\left[\begin{array}{rrr}3 & 0 & 4 \\ 0 & 2 & 0 \\ -4 & 0 & -5\end{array}\right] ; B=\left[\begin{array}{rrr}-1 & -1 & 3 \\ 0 & -1 & 1 \\\ 0 & 0 & 2\end{array}\right]\).

Use the result of Problem 32 to determine the sum and the product of the eigenvalues of the given matrix \(A\). $$A=\left[\begin{array}{rrrr} 0 & -3 & 1 & 1 \\ 0 & 2 & -1 & 3 \\ -1 & 1 & 1 & 1 \\ 1 & 0 & 5 & -2 \end{array}\right]$$

Let \(A=\left[\begin{array}{rr}1 & 2 \\ 2 & -2\end{array}\right].\) (a) Determine all eigenvalues of \(A\) (b) Reduce \(A\) to row-echelon form, and determine the eigenvalues of the resulting matrix. Are these the same as the eigenvalues of \(A ?\)

Use the result of Problem 32 to determine the sum and the product of the eigenvalues of the given matrix \(A\). $$A=\left[\begin{array}{rrrr} 12 & 11 & 9 & -7 \\ 2 & 3 & -5 & 6 \\ 10 & 8 & 5 & 4 \\ 1 & 0 & 3 & 4 \end{array}\right]$$

Deal with the generalization of the diagonalization problem to defective matrices. A complete discussion of this topic can be found in Section 7.6. Show that \(A=\left[\begin{array}{rr}2 & 1 \\ -1 & 4\end{array}\right]\) is defective and use the previous problem to determine a matrix \(S\) such that $$S^{-1} A S=\left[\begin{array}{ll}3 & 1 \\\0 & 3\end{array}\right].$$

Determine the general solution to the system \(\mathbf{x}^{\prime}=A \mathbf{x}\) for the given matrix \(A\). \(A=\left[\begin{array}{rr}-4 & 1 \\ -1 & -6\end{array}\right]\).

If \(\mathbf{v}_{1}=(1,-1)\) and \(\mathbf{v}_{2}=(2,1)\) are eigenvectors of the matrix \(A\) corresponding to the eigenvalues \(\lambda_{1}=2, \lambda_{2}=-3,\) respectively, find \(A\left(3 \mathbf{v}_{1}-\mathbf{v}_{2}\right)\).

Let \(\mathbf{v}_{1}=(1,-1,1), \mathbf{v}_{2}=(2,1,3),\) and \(\mathbf{v}_{3}=\) (-1,-1,2) be eigenvectors of the matrix \(A\) corresponding to the eigenvalues \(\lambda_{1}=2, \lambda_{2}=-2,\) and \(\lambda_{3}=3,\) respectively, and let \(\mathbf{v}=(5,0,3)\). (a) Express \(\mathbf{v}\) as a linear combination of \(\mathbf{v}_{1}, \mathbf{v}_{2}\) and \(\mathbf{v}_{3}\). (b) Find \(A \mathbf{v}\).