Triangularization of real matrices

Question

Let $A = [a_{ij}] \in \mathbb{K}^{n \times n}$, where $\mathbb{K} = \mathbb{R}$ or $\mathbb{K} = \mathbb{C}$.

In the case $\mathbb{K} = \mathbb{C}$, for example, via the Jordan decomposition theorem, there exists an invertible matrix $B$ with coefficients from $\mathbb{K} = \mathbb{C}$ such that $C := B^{-1}AB$ is an upper triangular matrix, i.e., $C=[c_{ij}]$ with $c_{ij}=0$ for $i>j$.

Is it true that for every matrix $A$ of type $n \times n$ with real coefficients, which characteristic polynomial has $n$ real roots not necessarily different, there exists an invertible matrix $B$ of the same type with real coefficients such that $B^{-1}AB$ is upper triangular?

Thanks.

Answer 1

The fact that the coefficients are different are different or not in $\chi_A(x)$ makes no change. The only thing that matters is whether all the roots are real or not. It is true that every real matrix with real eigenvalues (roots of $\chi_A(x)$) is triangulable. To prove this, you need the following two facts:

1. If $\lambda$ is an eigenvalue of $A$ then there exists $0\neq v\in\mathbb{R}^n$ such that $A\cdot v=\lambda v$.

2. If $W\lneqq \mathbb{R}^n$ and all roots of $\chi_A(x)$ are real, then there exists $v\in\mathbb{R}^n$ and $\lambda\in\mathbb{R}$ such that $v\notin W$ and $Av-\lambda v\in W$

Using those two claims, you can construct a basis of $\mathbb{R}^n$, $b=(v_1,...,v_n)$ such that for all $1\leq i\leq n$ $Av_i=a_{i,1}v_1+...+a_{i,i}v_i$. Now let $B$ be a matrix whose columns are $(v_1,...,v_n)$. It's easily checked that $B^{-1}AB$ is upper-triangular.

Answer 2

In the Schaum's series book on matrices by Frank Ayres jr., "Matrizen, Theorie und Anwendungen" (German) it is explicitely stated that

"We can prove: VI. each n-dimensional square matrix A is 'similar' to a triangular matrix, whose elements on the diagonal are the characteristic roots of A"
(Ed. 1987, Chap 20, my translation)

Similarity means here the multiplication $\small A=B^{-1}\cdot C \cdot B $ Also here is no restriction that the general field of complex numbers were required.
Moreover I think to remember, that B can be a rotation (unitary) matrix, but don't have a reference at hand.
(Perhaps that fact should also be included in wikipedia/similarity)