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)
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where is the identity matrix and is the determinant of the matrix . Writing out explicitly gives
(2)
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so the characteristic equation is given by
(3)
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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.