# Matrices & char. polynomials.

• Apr 6th 2010, 04:18 AM
Also sprach Zarathustra
Matrices & char. polynomials.
Where I could find a nice proof of following theorem:

A is matrix n x n over field F , similar to upper triangular matrix if and only if her characteristic polynomial can be factored into an expression with the form:
(x-t_1)(x-t_2)...(x-t_n)
• May 14th 2010, 10:51 PM
dwsmith
Quote:

Originally Posted by Also sprach Zarathustra
Where I could find a nice proof of following theorem:

A is matrix n x n over field F , similar to upper triangular matrix if and only if her characteristic polynomial can be factored into an expression with the form:
(x-t_1)(x-t_2)...(x-t_n)

$\displaystyle det(A)=(a_{11}-\lambda)A_{11}+\sum_{i=2}^{n}a_{i1}A_{i1}$

$\displaystyle (a_{11}-\lambda)A_{11}=(a_{11}-\lambda)(a_{22}-\lambda)...(a_{nn}-\lambda)$

$\displaystyle =(-1)^n\lambda^n+...+(-1)^{n-1}\lambda^{n-1}$

$\displaystyle p(0)=det(A)=\lambda_1\lambda_2...\lambda_n$
$\displaystyle (-1)^{n-1}=tr(A)=\sum_{i=1}^{n}\lambda_i$

$\displaystyle p(\lambda)=0$ has exactly n solutions $\displaystyle \lambda_1,...,\lambda_n$

$\displaystyle p(\lambda)=(\lambda_1-\lambda)(\lambda_2-\lambda)...(\lambda_n-\lambda)$

$\displaystyle p(0)=(\lambda_1)(\lambda_2)...(\lambda_n)=det(A)$