Nonsymmetric matrix ($A^T\ne A$)

A nonsymmetric matrix is a square matrix unequal to its transpose. Its right eigenvectors need not be orthogonal, and it need not be a diagonalizable matrix. For instance, $A=\left(\begin{array}{cc}1& 1\\ 0& 2\end{array}\right)$ has eigenvectors $(1,0{)}^T$ and $(1,1{)}^T$ for its distinct eigenvalues. A starting vector $(1,0{)}^T$ has nonzero inner product with $(1,1{)}^T$ but zero coefficient in that direction in the eigenbasis. This is why power iteration and inverse iteration require a nonzero spectral component rather than an ordinary nonorthogonality test in the general case.