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Mathematics Stack Exchange

Questions tagged [triangularization]

A square matrix $A$ is called triangularizable if it is similar to a triangular matrix.

54 questions
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Is there an abstract definition of a matrix being "upper triangular"?

Another question brought this up. The only definition I have ever seen for a matrix being upper triangular is, written in component forms, "all the components below the main diagonal are zero." But of course that property is basis dependent. It…
RobertTheTutor
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Commuting matrices are simultaneously triangularizable

Let $A$, $B$ be two $n\times n$ matrices over $\mathbb{C}$. If $AB=BA$ then there exists a basis $\mathcal B$ such that under this basis, the matrices of $A$, $B$ are both upper triangular. How to prove this? I know how to prove the following:…
Shiquan
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Proof of a theorem on simultaneous diagonalization from Hoffman and Kunze.

Now I am reading Linear Algebra from the book of Hoffman and Kunze second edition. I am trying to understand theorem $8$ on pg number $207$ which is based on Simultaneous diagonalization. I have seen plenty of proofs on this simultaneous…
user371231
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1 answer

Triangularization of matrix over PID

Let $R$ be a PID and let $A \in Mat_n(R)$ with all of its eigenvalues in R. Is it true that I can always find $P \in GL_n(R)$ such that $P^{-1}AP$ is uppertriangular? If so can I have a reference for this. Otherwise what additional hypotheses on…
Larry
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Prove that commuting matrices over an algebraically closed field are simultaneously triangularizable.

Given an algebraically closed field $\mathbb K$ and matrices $A, B \in \mathbb K^{n \times n}$ such that $A B = B A$, show that $A$ and $B$ are simultaneously triangularizable, i.e., show that there exists a matrix $T$ such that $T^{-1} A T$ and…
spoonispooni
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Triangularizing a Matrix

In Hoffman & Kunze, it is mentioned several times that any matrix with a minimal polynomial that splits over your field into linear factors is similar to an upper triangular matrix. A proof is given, but it does not give an algorithm for finding…
squiggles
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Triangularization of matrices over algebraically closed field

A friend of mine is studying physics in first semester and for his next assignment, he has to prove the following theorem: Let $V$ be a finite dimensional vector space over an algebraically closed field $K$. Further, let $f: V \to V$ be an…
SometimesBlind
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2 answers

Show that the matrix $A = \begin{bmatrix} 2 & 0 \\ -1 & 2 \end{bmatrix}$ is not diagonalizable.

I have this question: Show that the matrix $$A = \begin{bmatrix} 2 & 0 \\ -1 & 2 \end{bmatrix}$$ is not diagonalizable. So is the general strategy is To Find the eigenvectors and then Show that the matrix of eigenvectors is not invertible? If…
Kitty Capital
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Simplifying a compact proof of triangularization

I am going through this interesting paper A simultaneous triangularization result by Dan Shemesh. The Theorem 2 read If $\textbf{A}[\textbf{A},\textbf{B}]=\textbf{0}$ and $\textbf{B}[\textbf{A},\textbf{B}]=\textbf{0}$, then matrices $\textbf{A}$…
Mike
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3 answers

Short way for upper triangularization

We are given a matrix $$A = \begin{bmatrix} 3 & 0 & 1 \\ -1 & 4 & -3 \\ -1 & 0 & 5 \\ \end{bmatrix}$$ and we are asked to find a matrix $P$ such that $P^{-1}AP$ is upper triangular. Here, we first find one eigenvalue as $\lambda= 4$. Then the…
ArsenBerk
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3
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2 answers

Determinant of a block skew-symmetric matrix

If I have a matrix of the form $$\begin{pmatrix} A & -B \\ B & A \end{pmatrix}$$ where all blocks are square and have the same dimensions, how do I turn it into something like $$\begin{pmatrix} X & Y \\ 0 & Z \end{pmatrix}$$ so that the determinant…
izikgo
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Conditions for Schur decomposition and its generalization

Let $M$ be a $n$ by $n$ matrix over a field $F$. When $F$ is $\mathbb{C}$, $M$ always has a Schur decomposition, i.e. it is always unitarily similar to a triangular matrix, i.e. $M = U T U^H$ where $U$ is some unitary matrix and $T$ is a triangular…
Tim
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Simultaneous triangularization of matrices

Let $\mathcal{F}=\{A_1,A_2,\ldots,A_r\}$ be a triangulable commuting family of $n\times n$ matrices (that is, each $A_i$ is triangulable and $A_iA_j=A_jA_i$ for every $i,j$). I know that $\mathcal{F}$ can be simultaneous triangularization, but what…
boaz
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Triangulation of matrices

Suppose that $A$ is some triangularizable matrix in $M_n(\mathbb R)$. The usual approach I know of to find a triangular matrix similar to it is to find bases for all the eigenspaces, then find their union. If the union does not form a basis of…
user230734
2
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1 answer

$AB-BA=c(A-B)$, then $A,B$ are simultaneously triangularizable

The problem goes like this: Suppose $A, B\in M_n(\mathbb{C})$ and $AB-BA=c(A-B)$ with nonzero $c\in \mathbb C$. Then $A, B$ are simultaneously triangularizable and those upper triangular matrices coincide on the main diagonal. I've proved that the…
SuperSupao
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