91Ó°ÊÓ

Find the eigenvalues of the symmetric matrix. For each eigenvalue, find the dimension of the corresponding eigenspace. $$\left[\begin{array}{lll} 0 & 2 & 2 \\ 2 & 0 & 2 \\ 2 & 2 & 0 \end{array}\right]$$

Show that the matrix is not diagonalizable. $$\left[\begin{array}{rr} 1 & 0 \\ -2 & 1 \end{array}\right]$$

Show that the matrix is not diagonalizable. $$\left[\begin{array}{rrr} 2 & 1 & -1 \\ 0 & -1 & 2 \\ 0 & 0 & -1 \end{array}\right]$$

Show that the matrix is not diagonalizable. $$\left[\begin{array}{rrrr} 1 & -3 & 3 & 3 \\ -1 & 4 & -3 & -3 \\ -2 & 0 & 1 & 1 \\ 1 & 0 & 0 & 0 \end{array}\right]$$ (See Exercise 38 Section \(7.1 .)\)

Find (a) the characteristic equation and (b) the eigenvalues (and corresponding eigenvectors) of the matrix. $$\left[\begin{array}{lll} 3 & 2 & 1 \\ 0 & 0 & 2 \\ 0 & 2 & 0 \end{array}\right]$$

Find an orthogonal matrix \(P\) such that \(P^{T} A P\) diagonalizes \(A .\) Verify that \(P^{T} A P\) gives the proper diagonal form. $$A=\left[\begin{array}{rr} 2 & \sqrt{2} \\ \sqrt{2} & 1 \end{array}\right]$$

Write out the system of first-order linear differential equations represented by the matrix equation \(\mathbf{y}^{\prime}=A \mathbf{y} .\) Then verify the indicated general solution. $$A=\left[\begin{array}{ll} 1 & 1 \\ 0 & 1 \end{array}\right], \begin{array}{l} y_{1}=C_{1} e^{t}+C_{2} t e^{t} \\ y_{2}=C_{2} e^{t} \end{array}$$

Find an orthogonal matrix \(P\) such that \(P^{T} A P\) diagonalizes \(A .\) Verify that \(P^{T} A P\) gives the proper diagonal form. $$A=\left[\begin{array}{rrr} 0 & 10 & 10 \\ 10 & 5 & 0 \\ 10 & 0 & -5 \end{array}\right]$$

Use a graphing utility with matrix capabilities or a computer software program to find the eigenvalues of the matrix. $$\left[\begin{array}{rrr} \frac{1}{2} & 0 & 5 \\ -2 & \frac{1}{5} & \frac{1}{4} \\ 0 & 0 & 3 \end{array}\right]$$

Find a basis \(B\) for the domain of \(T\) such that the matrix of \(T\) relative to \(B\) is diagonal. $$\begin{aligned} T: P_{2} \rightarrow P_{2}: T\left(a_{0}+a_{1} x+a_{2} x^{2}\right)=&\left(2 a_{0}+a_{2}\right)+ \left(3 a_{1}+4 a_{2}\right) x+a_{2} x^{2} \end{aligned}$$