Chapter 7

Decide whether or not the given matrix \(A\) is diagonalizable. If so, find an invertible matrix \(S\) and a diagonal matrix \(D\) such that \(S^{-1} A S=D\). $$A=\left[\begin{array}{rr} 3 & 0 \\ 16 & -1 \end{array}\right].$$

For Problems \(1-5,\) determine how many Jordan canonical forms are with the given eigenvalues (not counting rearrangements of the Jordan blocks) and list each of them.A \(3 \times 3\) matrix with eigenvalues \(\lambda=-4,0,9\).

Show that \(A\) is non defective and use Theorem 7.4 .3 to find \(e^{A t}\). $$A=\left[\begin{array}{ll} 1 & 2 \\ 0 & 3 \end{array}\right]$$

Determine an orthogonal matrix \(S\) such that \(S^{T} A S=\operatorname{diag}\left(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}\right),\) where \(A\) denotes the given matrix. $$\left[\begin{array}{rr} 1 & 4 \\ 4 & -5 \end{array}\right]$$

Determine whether the given matrix \(A\) is diagonalizable. Where possible, find a matrix \(S\) such that $$S^{-1} A S=\operatorname{diag}\left(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}\right).$$ $$A=\left[\begin{array}{rr}-9 & 0 \\\4 & -9\end{array}\right]$$

Use Equation \((7.1 .1)\) to verify that \(\lambda\) and v are an eigenvalue/eigenvector pair for the given matrix \(A\). $$\lambda=2, \quad v=(2,9), \quad A=\left[\begin{array}{cc}-7 & 2 \\\\-9 & 4 \end{array}\right].$$

Determine the multiplicity of each eigenvalue and a basis for each eigenspace of the given matrix \(A\). Hence, determine the dimension of each eigenspace and state whether the matrix is defective or nondefective. $$A=\left[\begin{array}{rr} -7 & 0 \\ -3 & -7 \end{array}\right]$$

Decide whether or not the given matrix \(A\) is diagonalizable. If so, find an invertible matrix \(S\) and a diagonal matrix \(D\) such that \(S^{-1} A S=D\). $$A=\left[\begin{array}{cc} 13 & -9 \\ 25 & -17 \end{array}\right].$$

Determine how many Jordan canonical forms are with the given eigenvalues (not counting rearrangements of the Jordan blocks) and list each of them. A \(3 \times 3\) matrix with eigenvalues \(\lambda=1,1,1\).

Show that \(A\) is non defective and use Theorem 7.4 .3 to find \(e^{A t}\). $$A=\left[\begin{array}{ll} 3 & 1 \\ 1 & 3 \end{array}\right]$$

Determine an orthogonal matrix \(S\) such that \(S^{T} A S=\operatorname{diag}\left(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}\right),\) where \(A\) denotes the given matrix. $$\left[\begin{array}{rr} 2 & 2 \\ 2 & -1 \end{array}\right]$$

Determine whether the given matrix \(A\) is diagonalizable. Where possible, find a matrix \(S\) such that $$S^{-1} A S=\operatorname{diag}\left(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}\right).$$ $$A=\left[\begin{array}{rr}-1 & -2 \\\\-2 & 2\end{array}\right]$$

Use Equation \((7.1 .1)\) to verify that \(\lambda\) and v are an eigenvalue/eigenvector pair for the given matrix \(A\). $$\lambda=4, \quad \mathbf{v}=(1,1), \quad A=\left[\begin{array}{ll}1 & 3 \\\2 & 2 \end{array}\right].$$

Determine the multiplicity of each eigenvalue and a basis for each eigenspace of the given matrix \(A\). Hence, determine the dimension of each eigenspace and state whether the matrix is defective or nondefective. $$A=\left[\begin{array}{ll} 1 & 4 \\ 2 & 3 \end{array}\right]$$

Decide whether or not the given matrix \(A\) is diagonalizable. If so, find an invertible matrix \(S\) and a diagonal matrix \(D\) such that \(S^{-1} A S=D\). $$A=\left[\begin{array}{rrr} -4 & 3 & 0 \\ -6 & 5 & 0 \\ 3 & -3 & -1 \end{array}\right].$$

Determine how many Jordan canonical forms are with the given eigenvalues (not counting rearrangements of the Jordan blocks) and list each of them. A \(4 \times 4\) matrix with eigenvalues \(\lambda=1,1,3,3\).

Show that \(A\) is non defective and use Theorem 7.4 .3 to find \(e^{A t}\). $$A=\left[\begin{array}{rl} 0 & 2 \\ -2 & 0 \end{array}\right]$$

Determine an orthogonal matrix \(S\) such that \(S^{T} A S=\operatorname{diag}\left(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}\right),\) where \(A\) denotes the given matrix. $$\left[\begin{array}{ll} 4 & 6 \\ 6 & 9 \end{array}\right]$$

Determine whether the given matrix \(A\) is diagonalizable. Where possible, find a matrix \(S\) such that $$S^{-1} A S=\operatorname{diag}\left(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}\right).$$ $$A=\left[\begin{array}{ll}-7 & 4 \\\\-4 & 1\end{array}\right]$$

Use Equation \((7.1 .1)\) to verify that \(\lambda\) and v are an eigenvalue/eigenvector pair for the given matrix \(A\). $$\lambda=3, \mathbf{v}=(2,1,-1), A=\left[\begin{array}{rrr}1 & -2 & -6 \\\\-2 & 2 & -5 \\\2 & 1 & 8\end{array}\right].$$

Determine the multiplicity of each eigenvalue and a basis for each eigenspace of the given matrix \(A\). Hence, determine the dimension of each eigenspace and state whether the matrix is defective or nondefective. $$A=\left[\begin{array}{ll} 3 & 0 \\ 0 & 3 \end{array}\right]$$

Decide whether or not the given matrix \(A\) is diagonalizable. If so, find an invertible matrix \(S\) and a diagonal matrix \(D\) such that \(S^{-1} A S=D\). $$A=\left[\begin{array}{rrr} 1 & 1 & 0 \\ -4 & 5 & 0 \\ 17 & -11 & -2 \end{array}\right].$$

Determine how many Jordan canonical forms are with the given eigenvalues (not counting rearrangements of the Jordan blocks) and list each of them. A \(5 \times 5\) matrix with eigenvalues \(\lambda=2,2,2,2,2\).

Show that \(A\) is non defective and use Theorem 7.4 .3 to find \(e^{A t}\). $$A=\left[\begin{array}{rr} -1 & 3 \\ -3 & -1 \end{array}\right]$$

Determine an orthogonal matrix \(S\) such that \(S^{T} A S=\operatorname{diag}\left(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}\right),\) where \(A\) denotes the given matrix. $$\left[\begin{array}{ll} 1 & 2 \\ 2 & 1 \end{array}\right]$$

Determine whether the given matrix \(A\) is diagonalizable. Where possible, find a matrix \(S\) such that $$S^{-1} A S=\operatorname{diag}\left(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}\right).$$ $$A=\left[\begin{array}{ll}1 & -8 \\\2 & -7\end{array}\right]$$

Determine the multiplicity of each eigenvalue and a basis for each eigenspace of the given matrix \(A\). Hence, determine the dimension of each eigenspace and state whether the matrix is defective or nondefective. $$A=\left[\begin{array}{rr} 1 & 2 \\ -2 & 5 \end{array}\right]$$

Decide whether or not the given matrix \(A\) is diagonalizable. If so, find an invertible matrix \(S\) and a diagonal matrix \(D\) such that \(S^{-1} A S=D\). $$A=\left[\begin{array}{rrr} -1 & -1 & 3 \\ 4 & 4 & -4 \\ -1 & 0 & 3 \end{array}\right].$$ [Hint: The only eigenvalue of \(A\) is \(\lambda=2.1\)]

Determine an orthogonal matrix \(S\) such that \(S^{T} A S=\operatorname{diag}\left(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}\right),\) where \(A\) denotes the given matrix. $$\left[\begin{array}{rrr} 0 & 0 & 3 \\ 0 & -2 & 0 \\ 3 & 0 & 0 \end{array}\right]$$

Use Equation \((7.1 .1)\) to verify that \(\lambda\) and v are an eigenvalue/eigenvector pair for the given matrix \(A\). $$\lambda=10, \quad \mathbf{v}=c_{1}(1,-4,0)+c_{2}(0,0,1),$$ \(A=\left[\begin{array}{rrr}6 & -1 & 0 \\ -16 & 6 & 0 \\ -4 & -1 & 10\end{array}\right],\) where \(c_{1}\) and \(c_{2}\) are constants.

Determine whether the given matrix \(A\) is diagonalizable. Where possible, find a matrix \(S\) such that $$S^{-1} A S=\operatorname{diag}\left(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}\right).$$ $$A=\left[\begin{array}{rr}0 & 4 \\\\-4 & 0\end{array}\right]$$

Determine the multiplicity of each eigenvalue and a basis for each eigenspace of the given matrix \(A\). Hence, determine the dimension of each eigenspace and state whether the matrix is defective or nondefective. $$A=\left[\begin{array}{rr} 5 & 5 \\ -2 & -1 \end{array}\right]$$

Decide whether or not the given matrix \(A\) is diagonalizable. If so, find an invertible matrix \(S\) and a diagonal matrix \(D\) such that \(S^{-1} A S=D\). $$A=\left[\begin{array}{rrr} 9 & 5 & -5 \\ 0 & -1 & 0 \\ 10 & 5 & -6 \end{array}\right].$$ [Hint: The eigenvalues of \(A \text { are } \lambda=4 \text { and } \lambda=-1 .]\)

Determine how many Jordan canonical forms are possible with the given eigenvalues (not counting rearrangements of the Jordan blocks). You do not need to list them. An \(11 \times 11\) matrix with eigenvalues \(\lambda=2,2,2\) 2,6,6,6,6,8,8,8.

Determine an orthogonal matrix \(S\) such that \(S^{T} A S=\operatorname{diag}\left(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}\right),\) where \(A\) denotes the given matrix. $$\left[\begin{array}{lll} 1 & 2 & 1 \\ 2 & 4 & 2 \\ 1 & 2 & 1 \end{array}\right]$$

Given that \(\mathbf{v}_{1}=(-2,1)\) and \(\mathbf{v}_{2}=(1,1)\) are eigenvectors of $$A=\left[\begin{array}{rr}-5 & 2 \\\1 & -4\end{array}\right]$$ determine the eigenvalues of \(A.\)

Determine whether the given matrix \(A\) is diagonalizable. Where possible, find a matrix \(S\) such that $$S^{-1} A S=\operatorname{diag}\left(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}\right).$$ $$A=\left[\begin{array}{rrr}1 & 0 & 0 \\\0 & 3 & 7 \\\1 & 1 & -3\end{array}\right]$$

Determine the multiplicity of each eigenvalue and a basis for each eigenspace of the given matrix \(A\). Hence, determine the dimension of each eigenspace and state whether the matrix is defective or nondefective. $$A=\left[\begin{array}{rrr} 3 & -4 & -1 \\ 0 & -1 & -1 \\ 0 & -4 & 2 \end{array}\right]$$

Use some form of technology to find a complete set of orthonormal eigenvectors for \(A\) and an orthogonal matrix \(S\) and a diagonal matrix \(D\) such that \(S^{-1} A S=D\). $$A=\left[\begin{array}{lll} 2 & 2 & 1 \\ 2 & 5 & 2 \\ 1 & 2 & 2 \end{array}\right].$$

Determine how many Jordan canonical forms are possible with the given eigenvalues (not counting rearrangements of the Jordan blocks). You do not need to list them. A \(10 \times 10\) matrix with eigenvalues \(\lambda=2,2,2\) 2,5,5,5,5,5,5.

Show that \(A\) is non defective and use Theorem 7.4 .3 to find \(e^{A t}\). \(A=\left[\begin{array}{rrr}6 & -2 & -1 \\ 8 & -2 & -2 \\ 4 & -2 & 1\end{array}\right]\) and you may assume that \(p(\lambda)=-(\lambda-2)^{2}(\lambda-1)\)

Determine an orthogonal matrix \(S\) such that \(S^{T} A S=\operatorname{diag}\left(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}\right),\) where \(A\) denotes the given matrix. $$\left[\begin{array}{lll} 2 & 0 & 0 \\ 0 & 3 & 1 \\ 0 & 1 & 3 \end{array}\right]$$

Given that \(\mathbf{v}_{1}=(1,-2)\) and \(\mathbf{v}_{2}=(1,1)\) are eigenvectors of \(A=\left[\begin{array}{ll}4 & 1 \\ 2 & 3\end{array}\right],\) determine the eigenvalues of \(A\).

Determine whether the given matrix \(A\) is diagonalizable. Where possible, find a matrix \(S\) such that $$S^{-1} A S=\operatorname{diag}\left(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}\right).$$ $$A=\left[\begin{array}{lll}1 & -2 & 0 \\\2 & -3 & 0 \\\2 & -2 & -1\end{array}\right]$$

Use some form of technology to find a complete set of orthonormal eigenvectors for \(A\) and an orthogonal matrix \(S\) and a diagonal matrix \(D\) such that \(S^{-1} A S=D\). $$A=\left[\begin{array}{rrr} 0 & -1 & 4 \\ -1 & 5 & 2 \\ 4 & 2 & 2 \end{array}\right].$$

If it is known that \((A-5 I)^{2}=0\) for the matrix in Problem \(7,\) how many Jordan canonical form structures are possible for the matrix \(A\) ?

If \(A=\operatorname{diag}\left(d_{1}, d_{2}, \ldots, d_{n}\right),\) prove that $$ e^{A t}=\operatorname{diag}\left(e^{d_{1} t}, e^{d_{2} t}, e^{d_{3} t}, \ldots, e^{d_{n} t}\right) $$

Determine an orthogonal matrix \(S\) such that \(S^{T} A S=\operatorname{diag}\left(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}\right),\) where \(A\) denotes the given matrix. $$\left[\begin{array}{lll} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array}\right]$$

The effect of the linear transformation \(T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) with matrix \(A=\left[\begin{array}{rr}1 & 0 \\ 0 & -1\end{array}\right]\) is to reflect each vector in the \(x\) -axis. By arguing geometrically, determine all eigenvalues and eigenvectors of \(A\).

Determine whether the given matrix \(A\) is diagonalizable. Where possible, find a matrix \(S\) such that $$S^{-1} A S=\operatorname{diag}\left(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}\right).$$ $$A=\left[\begin{array}{rrr}0 & -2 & -2 \\\\-2 & 0 & -2 \\\\-2 & -2 & 0\end{array}\right]$$