Triangularization

Let $$A$$ represent an $$n\times n$$ matrix that have $$n$$ (distinct or non-distinct) eigenvalues, and let $$d_1, d_2, \dots, d_n$$ represent the (not necessarily distinct) eigenvalues of $$A$$ in arbitrary order. Then, there exists an orthogonal matrix $$Q$$ such that

$$Q'AQ = T$$

where $$T$$ is an $$n\times n$$ upper triangular matrix whose diagonal elements are $$d_1, d_2, \dots, d_n$$ (see T21.5.11 ).

What's the poitn of triangularization?
Among other possible advantages, triangularization facilitates the computation of determinant and trace

$$\det(A) = \prod_{i=1}^n d_i \\

\text{tr}(A) = \sum_{i=1}^n d_i$$