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 ).