Find an invertible matrix \(A\) and a diagonal matrix \(B\) such that \[ \begin{pmatrix}4&-6\\3&-5\end{pmatrix}=ABA^{-1}. \]
Solution
The eigenvalues are \(1\) and \(-2\). Corresponding eigenvectors are \((2,1)^t\) and \((1,1)^t\). Thus one can take \[ A=\begin{pmatrix}2&1\\1&1\end{pmatrix}, \qquad B=\begin{pmatrix}1&0\\0&-2\end{pmatrix}. \] The columns of \(A\) are linearly independent, and the eigenvector equations give \(MA=AB\) for the given matrix \(M\). Hence \(M=ABA^{-1}\).