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.