91Ó°ÊÓ

Find the eigenvalues of A by solving the characteristic polynomial given by \(\det(A - \lambda I) = 0\), where \(I\) is the identity matrix and \(\lambda\) are the eigenvalues. \[ \det \left(\begin{array}{ccc}-3-\lambda & 3 & -2 \\\ -7 & 6-\lambda & -3 \\\ 1 & -1 & 2-\lambda\end{array}\right) =0 \] The eigenvalues are \(\lambda_1 = \lambda_2 = \lambda_3 = 1\).

The eigenspace for each eigenvalue is given by the null space of \((A - \lambda I)\), and we find eigenvectors by solving the linear system \((A - \lambda I)v = 0\), where \(v\) is the eigenvector. For \(\lambda = 1\): \[ (A - \lambda I) = \left(\begin{array}{ccc}-4 & 3 & -2\\\ -7 & 5 & -3 \\\ 1 & -1 & 1\end{array}\right) \] We can reduce this to row echelon form: \[ \left(\begin{array}{ccc}1 & -1 & 1 \\ 0 & 1 & -1 \\ 0 & 0 & 0\end{array}\right) \] The eigenvectors are given by the null space, which corresponds to the solutions of the system: \[ \begin{cases} x - y + z = 0 \\ y - z = 0 \end{cases} \] The eigenvectors are multiples of the vector \(v_1 = (1,1,1)^T\).

We only have one independent eigenvector, v1. Let's create matrix Q using the eigenvector as its columns: \[ Q = \left(\begin{array}{ccc}1 \\ 1\\ 1\end{array}\right) \]

Since Q is a one-dimensional matrix, we can simply compute the inverse as \(Q^{-1} = \frac{1}{Q}\), which in this case is a scalar: \[ Q^{-1} = \frac{1}{1} = 1 \]

Now we can compute the Jordan canonical form using the formula \(J = Q^{-1}AQ\): \[ J = \left(\begin{array}{ccc}1\end{array}\right) \cdot \left(\begin{array}{ccc}-3 & 3 & -2\\ -7 & 6 & -3 \\ 1 & -1 & 2\end{array}\right) \cdot \left(\begin{array}{ccc}1 \\ 1 \\ 1\end{array}\right) \] \[ J = \left(\begin{array}{ccc}1\end{array}\right) \] So, for matrix A, the invertible matrix Q is just a one-dimensional matrix with a value of 1, and the Jordan canonical form J is also just a one-dimensional matrix with a value of 1. For the other matrices, you can follow the same steps to find their eigenvalues, eigenvectors, and Jordan canonical forms. Note that the steps might have to be adjusted to accommodate different sizes of the matrix Q and the form of the Jordan canonical matrix J.