Miklos Schweitzer 1965_3
Source:
September 25, 2008
linear algebramatrixcomplex numberslinear algebra unsolved
Problem Statement
Let a,b_0,b_1,b_2,...,b_{n\minus{}1} be complex numbers, a complex square matrix of order , and the unit matrix of order . Assuming that the eigenvalues of are given, determine the eigenvalues of the matrix
B\equal{}\begin{pmatrix} b_0E&b_1A&b_2A^2&\cdots&b_{n\minus{}1}A^{n\minus{}1} \\
ab_{n\minus{}1}A^{n\minus{}1}&b_0E&b_1A&\cdots&b_{n\minus{}2}A^{n\minus{}2}\\
ab_{n\minus{}2}A^{n\minus{}2}&ab_{n\minus{}1}A^{n\minus{}1}&b_0E&\cdots&b_{n\minus{}3}A^{n\minus{}3}\\
\vdots&\vdots&\vdots&\ddots&\vdots&\\
ab_1A&ab_2A^2&ab_3A^3&\cdots&b_0E
\end{pmatrix}