On tridiagonalization of matrices (Q1111647)

A square matrix \(B=(b_{ij})\) is called tridiagonal if \(b_{ij}=0\) for \(| i-j| >1\). A complex \(n\times n\) square matrix is called tridiagonalizable if it is unitarily similar to a tridiagonal matrix. The author presents a proof due to J. L. Noakes of the result due to \textit{B. Sturmfels} [ibid. 109, 165-166 (1988; reviewed below)] which, for \(n\geq 6\), gives a negative answer to the author's question of whether every complex matrix is tridiagonalizable. On the other hand, the author gives an affirmative answer to the question for \(n=3\) and exhibits some special classes of tridiagonalizable matrices.