Originally Posted by

**dwsmith** A is matrix n x n over field F, similar to an upper triangular matrix iff. the characteristic polynomial can be factored into an expression of the form $\displaystyle \displaystyle (\lambda_1-\lambda)(\lambda_2-\lambda)...(\lambda_n-\lambda)$

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

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

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

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

$\displaystyle \displaystyle (-1)^{n-1}=tr(A)=\sum_{i=1}^{n}\lambda_i$

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

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

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