Complex Vector Spaces

Abstract

A real matrix has real coefficients in its characteristic polynomial, but the eigenvalues may fail to be real. For instance, the matrix $A = \left[ {\begin{array}{*{20}{l}} 1&{ - 1} \\ 1&1 \end{array}} \right]$ has no real eigenvalues, but it has the complex eigenvalues λ= 1 ± i. Thus, it is indispensable to work with complex numbers to find the full set of eigenvalues and eigenvectors. Therefore, it is natural to extend the concept of real vector spaces to that of complex vector spaces, and then develop the basic properties of complex vector spaces. With this extension, all the square matrices of order n will have n eigenvalues.