» Required Reading » Table of Contents » Chapter 4 4.5 Eigenvalues and the Characteristic Equation of a Matrix
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A vector v which satisfies Mv = cv for some number c is called an eigenvector of M and c is called its eigenvalue.

This condition can be rewritten as (M-cI)v = 0.  This equation says that the matrix M-cI takes v into the 0 vector, which implies that it cannot have an inverse so that its determinant must be 0.

The equation det (M – cI) = 0 is called the characteristic equation of the matrix M and can be solved to find its eigenvalues c.

The eigenvector or vectors corresponding to each eigenvalue can be obtained straightforwardly.

The trace of a square matrix M, written as Tr(M) is the sum of its diagonal elements.

The characteristic equation of a 2 by 2 matrix M is  c² – cTr(M) + det M = 0.