# Characteristic polynomial

*of a matrix $ A = \| a _ {ij} \| _ {1} ^ {n} $*
over a field $ K $

The polynomial

$$ p ( \lambda ) = \ \mathop{\rm det} ( A - \lambda E) = \ \left \| \begin{array}{llll} a _ {11} - \lambda &a _ {12} &\dots &a _ {1n} \\ a _ {21} &a _ {22} - \lambda &\dots &a _ {2n} \\ \dots &\dots &\dots &\dots \\ a _ {n1} &a _ {n2} &\dots &a _ {nn} - \lambda \\ \end{array} \ \right \| = $$

$$ = \ (- \lambda ) ^ {n} + b _ {1} (- \lambda ) ^ {n - 1 } + \dots + b _ {n} $$

over $ K $. The degree of the characteristic polynomial is equal to the order of the square matrix $ A $, the coefficient $ b _ {1} $ is the trace of $ A $( $ b _ {1} = \mathop{\rm Tr} A = a _ {11} + \dots + a _ {nn} $, cf. Trace of a square matrix), the coefficient $ b _ {m} $ is the sum of all principal minors of order $ m $, in particular, $ b _ {n} = \mathop{\rm det} A $( cf. Minor). The equation $ p ( \lambda ) = 0 $ is called the characteristic equation of $ A $ or the secular equation.

The roots of the characteristic polynomial lying in $ K $ are called the characteristic values or eigen values of $ A $. If $ K $ is a number field, then the term "characteristic number of a matrix" is also used. Sometimes the roots of the characteristic polynomial are considered in the algebraic closure of $ K $. They are usually called the characteristic roots of $ A $. A matrix $ A $ of order $ n $ regarded over an algebraically closed field (for example, over the field of complex numbers) has $ n $ eigen values $ \lambda _ {1} \dots \lambda _ {n} $, if every root is counted according to its multiplicity. See also Eigen value.

Similar matrices have the same characteristic polynomial. Every polynomial over $ K $ with leading coefficient $ (- 1) ^ {n} $ is the characteristic polynomial of some matrix over $ K $ of order $ n $, the so-called Frobenius matrix.

For references see Matrix.

#### Comments

The characteristic roots are often also called the eigen values or characteristic values, thereby not distinguishing the roots of the characteristic polynomial in $ K $ and those in its algebraic closure. Given a polynomial $ b ( \lambda ) = ( - \lambda ) ^ {n} + b _ {1} ( - \lambda ) ^ {n-1} + \dots + b _ {n} $. The matrix in companion form

$$ A = \left \| \begin{array}{lllll} 0 & 1 & 0 &\dots & 0 \\ 0 &\dots &\dots &\dots &\dots \\ \dots &\dots &\dots &\dots &\dots \\ \dots &\dots &\dots &\dots & 0 \\ 0 &\dots &\dots & 0 & 1 \\ b _ {n} ^ \prime &\dots &\dots &\dots &b _ {1} ^ \prime \\ \end{array} \right \| $$

with $ b _ {k} ^ \prime = ( - 1 ) ^ {k+1} b _ {k} $, has $ b ( \lambda ) $ as its characteristic polynomial.

**How to Cite This Entry:**

Characteristic polynomial.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Characteristic_polynomial&oldid=46322