# Characteristic polynomial

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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.

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.