The matrix \(\mathbf A\) is given by

\(\mathbf A=\begin{pmatrix}2&2&-3\\2&2&3\\-3&3&3\end{pmatrix}\).

The matrix \(\mathbf A\) has an eigenvector \(\begin{pmatrix}1\\-1\\1\end{pmatrix}\). Find the corresponding eigenvalue.

The matrix \(\mathbf A\) also has eigenvalues \(4\) and \(6\). Find corresponding eigenvectors.

Hence find a matrix \(\mathbf P\) and a diagonal matrix \(\mathbf D\) such that \(\mathbf A=\mathbf P\mathbf D\mathbf P^{-1}\).

The matrix \(\mathbf B\) is such that \(\mathbf B=\mathbf Q\mathbf A\mathbf Q^{-1}\), where \(\mathbf Q=\begin{pmatrix}4&11&5\\1&4&2\\1&2&1\end{pmatrix}\).

By using the expression \(\mathbf P\mathbf D\mathbf P^{-1}\) for \(\mathbf A\), find the set of eigenvalues and a corresponding set of eigenvectors for \(\mathbf B\).