Question

show that matrix A and transpose of A have same eigenvalues. Give an example 2by2 matrix A, where A and A^T have different eigenvectors

Answer 1

\[A= \left[\begin{matrix}a & b \\ c & d\end{matrix}\right]\]
\[A^T=\left[\begin{matrix}a & c \\ b & d\end{matrix}\right]\]

these two matrices have same eigenvalues but different eigenvectors