Characteristic Equation

The characteristic equation is the equation which is solved to find a matrix's eigenvalues, also called the characteristic polynomial. For a general matrix , the characteristic equation in variable is defined by

(1)

where is the identity matrix and is the determinant of the matrix . Writing out explicitly gives

(2)

so the characteristic equation is given by

(3)

The solutions of the characteristic equation are called eigenvalues, and are extremely important in the analysis of many problems in mathematics and physics. The polynomial left-hand side of the characteristic equation is known as the characteristic polynomial.