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Let $A=[a_{ij}]$ be a matrix of type $n\times n$ with coefficients with the field $K$, where $K=\mathbb{R}$ or $K=\mathbb{C}$.

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

Is it true that for every matrice $A$ 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.

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Do you mean "$n$ not necessarily different real roots"? –  fpqc Feb 8 '12 at 9:31
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2 Answers

up vote 2 down vote accepted

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 claim, 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.

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How to find $v$ and $\lambda$ from the fact 2) ? –  Richard Feb 8 '12 at 9:56
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@Richard: Use the Cayley–Hamilton theorem. Take any $v\notin W$. Since $\chi_A(A)=0$, we have $\chi_A(A)v=0$. Since all the roots of $\chi_A(x)$ are real, we can write $\chi_A(A)=(A-\lambda_1I)\cdot\ldots\cdot(A-\lambda_nI)$. Denote $w=(A-\lambda_iI)\cdot\ldots\cdot(A-\lambda_nI)v$ such that $w\notin W$ and $(A-\lambda_{i-1}I)w\in W$ (exists since $v\notin W$ and $\chi_A(A)v=0\in W$) –  Dennis Gulko Feb 8 '12 at 16:45
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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)

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Thanks for answer and references. –  Richard Feb 9 '12 at 12:33
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