Questions tagged [jordan-normal-form]

This tag is for questions relating to the Jordan normal form, also known as a Jordan canonical form or JCF of a matrix which is a similar block matrix having diagonal blocks when the matrix is diagonalizable and diagonal + nilpotent blocks more generally.

The Jordan normal form of a matrix is a canonical form given by a similarity transformation. The Jordan normal form consists of diagonal blocks corresponding to its generalized eigenvectors, with blocks that are themselves constant diagonal when an eigenvalue has a basis of eigenvectors and otherwise blocks that are constant diagonal + nilpotent. For more information, see this Wikipedia article.

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On the complex matrix equation $AX-XA=B$

I want to show that there exists solution to the matrix equation $AX-XA=B$ if and only if $$ \begin{pmatrix} A&0\\ 0&A \end{pmatrix}, \begin{pmatrix} A&B\\ 0&A \end{pmatrix} $$ are similar, where all matrices ($A,B,X$) are complex and $A,B$ are…
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Jordan form step by step general algorithm

So I am trying to compile a summary of the procedure one should follow to find the Jordan basis and the Jordan form of a matrix, and I am on the lookout for free resources online where the algorithm to be followed is clearly explained in an amenable…
Jsevillamol
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Finding the Jordan canonical form of this upper triangular $3\times3$ matrix

I am supposed to find the Jordan canonical form of a couple of matrices, but I was absent for a few lectures. \begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 2 \\ 0 & 0 & 3 \end{bmatrix} Since this is an upper triangular matrix, its eigenvalues are the…
N3buchadnezzar
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Finding Jordan Basis of a matrix

Having trouble finding the Jordan base (and hence $P$) for this matrix $A = \begin{pmatrix} 15&-4\\ 49&-13 \end{pmatrix}$ I know that the eigenvalue is $1$, this gives an eigenvector $\begin{pmatrix} 2\\ 7 \end{pmatrix} $ Now to create the Jordan…
user65972
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Where does the Jordan canonical form show up in more advanced mathematics?

My bounty for this question expires soon :) Edit: in regards to the bounty offered, what current research trends use the Jordan canonical form? If one takes a second course in Linear Algebra — or a graduate level Linear Algebra course — one…
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Order of invertible matrices

I recently came across an interesting problem in Artin which says: If $A \in GL_2(\mathbb{Z})$ is of finite order then it has order $1,2,3,4,6$. I was looking for a generalization of this problem. For example, if they were invertible finite order…
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Possible Jordan Canonical Forms Given Minimal Polynomial

I was supposed to find all possible Jordan canonical forms of a $5\times 5$ complex matrix with minimal polynomial $(x-2)^2(x-1)$ on a qualifying exam last semester. I took the polynomial to mean that there were at least two 2's and one 1 on the…
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Criterion for deciding whether matrix is diagonalizable

Let $B \in$ GL$_n(\mathbb{C})$. In a paper I'm reading someone probably claims the following: Lemma: For showing that $B$ is diagonalizable it suffices to show the following: Let $\lambda$ be an eigenvalue of $B$ with algebraic multiplicity $\geq 2$…
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Similar Matrices and their Jordan Canonical Forms

Let $A$ and $B$ be two matrices in $M_n$. Is the following ture: $A$ and $B$ are similar $\iff$ $A$ and $B$ have the same jordan canonical form. Could someone explain?
MATH
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Prove that $ND = DN$ where $D$ is a diagonalizable and $N$ is a nilpotent matrix.

Let $A$ be an $n \times n$ complex matrix. Prove that there exist a diagonalizable matrix $D$ and a nilpotent matrix $N$ such that a. A = D + N b. DN = ND and show that these matrices are uniquely determined. I think I've solved the part a but…
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Conjugacy classes in $SU_2$

I'm trying to find all conjugacy classes in $SU_2$. Matrices in $SU_2$ are of the form: $M = \begin{bmatrix} \alpha & \beta \\ - \bar{\beta} & \bar{\alpha} \end{bmatrix}$ and $|\alpha|^2+|\beta|^2 =1$ I thought I could use…
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Let $J$ be a $k \times k$ jordan block, prove that any matrix which commutes with $J$ is a polynomial in $J$

Let $J$ be a $k \times k$ jordan block, prove that any matrix which commutes with $J$ is a polynomial in $J$. I appreciate your hints, Thanks
the8thone
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Prove Why $B^2 = A$ exists?

Define $$A = \begin{pmatrix} 8 & −4 & 3/2 & 2 & −11/4 & −4 & −4 & 1 \\ 2 & 2 & 1 & 0 & 1 & 0 & 0 & 0 \\ −9 & 8 & 1/2 & −4 & 31/4 & 8 & 8 & −2 \\ 4 & −6 & 2 & 5 & −7 & −6 & −6 & 0 \\ −2 & 0 & −1 & 0 & 1/2 & 0 & 0 & 0 \\ −1 & 0 & −1/2 & 0 & −3/4 &…
AriNubar
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What's the Jordan canonical form of this matrix?

given is the $6 \times 6$-matrix $A$: $A = \begin{pmatrix} 0 & 1 & 0 & -1 & 0 & 0 \\ 0 &0&1&1&-1&0\\ -1&0&0&0&-1&-1 \\ 1 & 0&0&0&1&0 \\ 0&1&0&0&0&1 \\ 0&0&1&1&0&0 \end{pmatrix}$ With only the information that $A$ has exactly two different…
Vazrael
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Find a flag to transform a matrix to an upper triangular one

Consider $F: \mathbb{R^3} \to \mathbb{R^3}$ represented by: $ A= \begin{bmatrix} 1 & 1 & 2 \\ -2 & 5 & 6 \\ 1 & -2 & -2 \\ \end{bmatrix} $ , eigenvalues: $(\lambda - 1)^2(\lambda -2)$ Now, $A$ admits Jordan canonical form: $…
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