Chapter 7

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. $$A=\left[\begin{array}{rrr} 3 & 1 & 0 \\ -1 & 5 & 0 \\ 0 & 0 & 4 \end{array}\right]$$

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 \(S^{-1} A S=D\). $$A=\left[\begin{array}{rrr} -2 & 1 & 1 \\ 1 & 3 & 6 \\ 1 & 6 & -1 \end{array}\right].$$

Let \(A\) be a \(5 \times 5\) matrix with eigenvalues \(\lambda_{1}, \lambda_{1}\) \(\lambda_{1}, \lambda_{2}, \lambda_{2},\) where \(\lambda_{1} \neq \lambda_{2}\) (a) Determine the complete list of possible Jordan canonical forms of \(A\) (b) Assume further that \(\left(A-\lambda_{1} I\right)^{2}=0_{5} .\) Among the matrices listed in part (a), which of them are the possible Jordan canonical form of \(A\) in light of this new information?

If \(A=\left[\begin{array}{rl}-3 & 0 \\ 0 & 5\end{array}\right],\) determine \(e^{A t}\) and \(e^{-A t}\)

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. $$\left[\begin{array}{rrr} 1 & 1 & -1 \\ 1 & 1 & 1 \\ -1 & 1 & 1 \end{array}\right]$$

The effect of the linear transformation \(T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) with matrix \(A=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]\) is to reflect each vector across the line \(y=x .\) 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 $$S^{-1} A S=\operatorname{diag}\left(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}\right).$$ $$A=\left[\begin{array}{lll}-2 & 1 & 4 \\\\-2 & 1 & 4 \\\\-2 & 1 & 4\end{array}\right]$$

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. $$A=\left[\begin{array}{rrr} 3 & 0 & 0 \\ 2 & 0 & -4 \\ 1 & 4 & 0 \end{array}\right]$$

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 \(S^{-1} A S=D\). $$A=\left[\begin{array}{lll} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 1 & 1 \end{array}\right].$$

Suppose \(A\) is a \(6 \times 6\) matrix with eigenvalue \(\lambda\) (of multiplicity 6 ). If it is known that \((A-\lambda I)^{3}=0\) but \((A-\lambda I)^{2} \neq 0,\) write down all possible Jordan canonical forms of \(A\).

Prove that for all values of the constant \(\lambda\) $$ e^{\lambda I_{n} t}=e^{\lambda t} I_{n} $$

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. $$\left[\begin{array}{rrr} 1 & 0 & -1 \\ 0 & 1 & 1 \\ -1 & 1 & 0 \end{array}\right]$$

The linear transformation \(T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) with matrix \(A=\left[\begin{array}{lr}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right]\) rotates vectors in the \(x y\) -plane counterclockwise through an angle \(\theta,\) where \(0 \leq \theta<\) \(2 \pi .\) By arguing geometrically, determine all values of \(\theta\) for which \(A\) has real eigenvalues. Find the real eigenvalues and the corresponding eigenvectors.

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

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. $$A=\left[\begin{array}{rrr} 4 & 1 & 6 \\ -4 & 0 & -7 \\ 0 & 0 & -3 \end{array}\right]$$

Find the Jordan canonical form of each matrix. $$A=\left[\begin{array}{rrr} 5 & 8 & 16 \\ 4 & 1 & 8 \\ -4 & -4 & -11 \end{array}\right].$$ [Hint: The eigenvalues of \(A\) are \(\lambda=1\) and \(\lambda=-3.1]\)

The characteristic polynomial \(p(\lambda)\) for a square matrix \(A\) is given. Write down a set \(S\) of matrices such that every square matrix with characteristic polynomial \(p(\lambda)\) is guaranteed to be similar to exactly one of the matrices in the set \(S\). \(p(\lambda)=(4-\lambda)^{2}(-6-\lambda)\).

Consider the matrix \(A=\left[\begin{array}{ll}a & b \\ 0 & a\end{array}\right] .\) We can write \(A=B+C,\) where \(B=\left[\begin{array}{ll}a & 0 \\ 0 & a\end{array}\right]\) and \(C=\left[\begin{array}{ll}0 & b \\ 0 & 0\end{array}\right]\)(a) Verify that \(B C=C B\) (b) Verify that \(C^{2}=0_{2},\) and determine \(e^{C t}\) (c) Use property (1) of the matrix exponential function to find \(e^{A t}\)

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. $$\begin{aligned} &\left[\begin{array}{lll} 3 & 3 & 4 \\ 3 & 3 & 0 \\ 4 & 0 & 3 \end{array}\right]\\\ &\text { You may assume that } p(\lambda)=(\lambda+2)(\lambda-3)(8-\lambda) \end{aligned}$$

Determine whether the given matrix \(A\) is diagonalizable. Where possible, find a matrix \(S\) such that $$S^{-1} A S=\operatorname{diag}\left(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}\right).$$ $$A=\left[\begin{array}{rrr}4 & 0 & 0 \\\3 & -1 & -1 \\\0 & 2 & 1\end{array}\right]$$

The linear transformation \(T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}\) with matrix \(A=\left[\begin{array}{lll}0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0\end{array}\right]\) takes vectors \((x, y, z)\) in \(\mathbb{R}^{3}\) and moves them to the corresponding point \((0, y, 0)\) on the \(y\) -axis. By arguing geometrically, determine all eigenvalues and eigenvectors of \(A\).

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. $$A=\left[\begin{array}{lll} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{array}\right]$$

Find the Jordan canonical form of each matrix. $$A=\left[\begin{array}{rrr} 2 & 1 & 1 \\ 2 & 1 & -2 \\ -1 & 0 & -2 \end{array}\right].$$ [Hint: The eigenvalues of \(A \text { are } \lambda=-1 \text { and } \lambda=3 .]\)

The characteristic polynomial \(p(\lambda)\) for a square matrix \(A\) is given. Write down a set \(S\) of matrices such that every square matrix with characteristic polynomial \(p(\lambda)\) is guaranteed to be similar to exactly one of the matrices in the set \(S\). \(p(\lambda)=(4-\lambda)^{3}(-1-\lambda)^{2}\).

If \(A=\left[\begin{array}{rl}a & b \\ -b & a\end{array}\right],\) use property (1) of the matrix exponential function and Definition 7.4 .1 to show that \(e^{A t}=e^{a t}\left[\begin{array}{rc}\cos b t & \sin b t \\ -\sin b t & \cos b t\end{array}\right]\)

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. $$\begin{aligned} &\left[\begin{array}{rrr} -3 & 2 & 2 \\ 2 & -3 & 2 \\ 2 & 2 & -3 \end{array}\right]\\\ &\text { You may assume that } p(\lambda)=(1-\lambda)(\lambda+5)^{2} \end{aligned}$$

Determine all eigenvalues and corresponding eigenvectors of the given matrix. $$\left[\begin{array}{ll}5 & -4 \\\8 & -7\end{array}\right]$$.

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

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. $$A=\left[\begin{array}{rrr} 7 & -8 & 6 \\ 8 & -9 & 6 \\ 0 & 0 & -1 \end{array}\right]$$

Write down all of the possible Jordan canonical form structures, up to a rearrangement of the blocks, for matrices of the specified type. For each Jordan canonical form structure, list the number of linearly independent eigenvectors of a matrix with that Jordan canonical form, and list the maximum length of a cycle of generalized eigenvectors of the matrix. \(4 \times 4\) matrices with eigenvalues \(\lambda=-1,-1,-1,2.\)

The characteristic polynomial \(p(\lambda)\) for a square matrix \(A\) is given. Write down a set \(S\) of matrices such that every square matrix with characteristic polynomial \(p(\lambda)\) is guaranteed to be similar to exactly one of the matrices in the set \(S\). \(p(\lambda)=(3-\lambda)^{2}(-2-\lambda)^{3} \lambda^{2}\).

An \(n \times n\) matrix \(A\) that satisfies \(A^{k}=0\) for some \(k\) is called nilpotent. Show that the given matrix is nilpotent, and use Definition 7.4 .1 to determine \(e^{A t}\). $$A=\left[\begin{array}{rr} 1 & 1 \\ -1 & -1 \end{array}\right]$$

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. $$\begin{aligned} &\left[\begin{array}{lll} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{array}\right]\\\ &\text { You may assume that } p(\lambda)=(\lambda+1)^{2}(2-\lambda) \end{aligned}$$

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

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

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. $$A=\left[\begin{array}{lll} 2 & 2 & -1 \\ 2 & 1 & -1 \\ 2 & 3 & -1 \end{array}\right]$$

Write down all of the possible Jordan canonical form structures, up to a rearrangement of the blocks, for matrices of the specified type. For each Jordan canonical form structure, list the number of linearly independent eigenvectors of a matrix with that Jordan canonical form, and list the maximum length of a cycle of generalized eigenvectors of the matrix. \(5 \times 5\) matrices with eigenvalues \(\lambda=4,4,4,4,4.\)

The characteristic polynomial \(p(\lambda)\) for a square matrix \(A\) is given. Write down a set \(S\) of matrices such that every square matrix with characteristic polynomial \(p(\lambda)\) is guaranteed to be similar to exactly one of the matrices in the set \(S\). \(p(\lambda)=(-2-\lambda)^{2}(6-\lambda)^{5}\).

An \(n \times n\) matrix \(A\) that satisfies \(A^{k}=0\) for some \(k\) is called nilpotent. Show that the given matrix is nilpotent, and use Definition 7.4 .1 to determine \(e^{A t}\). $$A=\left[\begin{array}{ll} -3 & 9 \\ -1 & 3 \end{array}\right]$$

Determine a set of principal axes for the given quadratic form, and reduce the quadratic form to a sum of squares. $$\mathbf{x}^{T} A \mathbf{x}, \quad A=\left[\begin{array}{ll} 1 & 3 \\ 3 & 1 \end{array}\right]$$

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

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

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. $$A=\left[\begin{array}{lll} 1 & -1 & 2 \\ 1 & -1 & 2 \\ 1 & -1 & 2 \end{array}\right]$$

Write down all of the possible Jordan canonical form structures, up to a rearrangement of the blocks, for matrices of the specified type. For each Jordan canonical form structure, list the number of linearly independent eigenvectors of a matrix with that Jordan canonical form, and list the maximum length of a cycle of generalized eigenvectors of the matrix. \(6 \times 6\) matrices with eigenvalues \(\lambda=6,6,6,6,-3,-3.\)

An \(n \times n\) matrix \(A\) that satisfies \(A^{k}=0\) for some \(k\) is called nilpotent. Show that the given matrix is nilpotent, and use Definition 7.4 .1 to determine \(e^{A t}\). $$A=\left[\begin{array}{rrr} -1 & -6 & -5 \\ 0 & -2 & -1 \\ 1 & 2 & 3 \end{array}\right]$$

Determine a set of principal axes for the given quadratic form, and reduce the quadratic form to a sum of squares. $$\mathbf{x}^{T} A \mathbf{x}, \quad A=\left[\begin{array}{ll} 5 & 2 \\ 2 & 5 \end{array}\right]$$

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

Determine whether the given matrix \(A\) is diagonalizable. Where possible, find a matrix \(S\) such that $$S^{-1} A S=\operatorname{diag}\left(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}\right).$$ $$A=\left[\begin{array}{rrrr}1 & 2 & 3 & 4 \\\\-1 & -2 & -3 & -4 \\\2 & 4 & 6 & 8 \\\\-2 & -4 & -6 & -8 \end{array}\right]$$

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. $$A=\left[\begin{array}{rrr} 2 & 3 & 0 \\ -1 & 0 & 1 \\ -2 & -1 & 4 \end{array}\right]$$

Write down all of the possible Jordan canonical form structures, up to a rearrangement of the blocks, for matrices of the specified type. For each Jordan canonical form structure, list the number of linearly independent eigenvectors of a matrix with that Jordan canonical form, and list the maximum length of a cycle of generalized eigenvectors of the matrix. \(7 \times 7\) matrices with eigenvalues \(\lambda=2,2,2,2,-4,-4,-4.\)