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.

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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: $...
Arcticmonkey's user avatar
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804 views

Jordan Block of Kronecker Product

Let $A$ be a $(p\times p$)-Jordan block of generalized eigenvalue $\lambda$. Let $B$ be a $(q\times q$)-Jordan block of generalized eigenvalue $\mu$. Then what is the Jordan canonical form for $A\...
Y.H. Chan's user avatar
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7 votes
2 answers
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Can basis of kernel be extended to a Jordan basis?

Let $A\in\mathbb C^{n\times n}$ be nilpotent. A Jordan basis of $A$ is a basis of $\mathbb C^n$ with respect to which $A$ has Jordan normal form. Assume that we do not know the Jordan structure of $A$....
Friedrich Philipp's user avatar
6 votes
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An inverse of Jordan matrix - basis

Let $A\in M_{n\times n}$ be and invertible matrix over complex field and we assume it's already at Jordan form where $B=\{v_1,…,v_n \}$ is Jordan basis for A. Find Jordan form and Jordan basis for $...
Jessy's user avatar
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Conjecture: Any two matrices of size $n×n$ with same characteristic and minimal polynomial are similar implies $n\le 3$.

Notations: $\mathcal{M}_n(\Bbb{R}) $: the set of all $n×n$ matrices over $\Bbb{R}$ $\chi_A(x)$: Characteristic polynomial of $A$ $m_A(x)$ : Minimal polynomial of $A$ $A\sim B$ : $\exists P\in Gl_n(\...
Sourav Ghosh's user avatar
5 votes
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Different approaches to Jordan Canonical Form

I know two different proofs of the existence of JCF. Let $V$ be a finite-dimensional vector space over base field $\mathbb{C}$ and $\alpha \in \mathsf{End}_{\mathbb{C}}(V)$. Given transformation $\...
Jackson Harris's user avatar
5 votes
2 answers
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Have I found the Jordan form correctly?

I am given that the minimal polynomial and characteristic polynomial of a matrix are both $(x-1)^2(x+1)^2$. I have found the Jordan form to be $$\begin{bmatrix}1&1&0&0\\0&1&0&0\...
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Lost on rational and Jordan forms

I'm having a lot of trouble trying to understand rational canonical form, primary rational canonical form, and Jordan form. I've looked at the posts about this, but I haven't been able to understand ...
Who knows's user avatar
  • 522
4 votes
2 answers
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Calculate the Jordan normal form

I have the matrix $A=\begin{bmatrix} -2 & -3 & 6 \\ 1 & 2 & -2\\ -1 & -1 &3 \end{bmatrix}$ and I have to find the transformation matrix and its Jordan normal form. This is ...
Dada's user avatar
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Why block-diagonal form for nilpotent matrices?

I am currently reading Jim Hefferon's Linear Algebra. In chapter 5, nilpotence, strings, he goes through the process of finding a string basis of a map, and proves that there exists a string basis ...
Buster Bie's user avatar
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Prove the direct sum of generalized eigenspaces is the whole vector space

Given a $n\times n$ matrix $A$ over an algebraically closed field, let $\lambda_1,...,\lambda_k$ be its eigenvalues, and let $V_{\lambda_i}$ be the generalized eigenspace of $\lambda_i$. The question ...
Ralph B.'s user avatar
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Differential Equations: Jordan Form of a Matrix

I am using Lawrence Perko's book Differential Equations and Dynamical Systems, for my Differential Equations course. At the moment we are going over Jordan Forms of a linear system $x^{'}(t) = Ax$, ...
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Jordan Canonical Form and Minimal Polynomial

I was wondering what the relationship between the minimal polynomial and the Jordan Canonical Form is. Given a matrix, all one needs to do is to compute the characteristic polynomial to determine the ...
mathnoob's user avatar
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Is there some geometric understanding of a Jordan normal matrix?

We know that an eigenvalue decomposition of a matrix is to find those eigenvectors that are just scale for some coefficients. But what about Jordan matrix decomposition? I just learn how it is ...
Yuki's user avatar
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Number of similarity classes of matrices $A$ in $M_6(\mathbb{R})$ satisfying $(A-2I)^3=0.$

Find the number of similarity classes of matrices $A$ in $M_6(\mathbb{R})$ satisfying $(A-2I)^3=0.$ If $(A-2I)^3$ then the characteristic polynomial is $p_A(x)=(x-2)^6$ and we have 3 posibilities for $...
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Number of Jordan Blocks

Determine the Jordan Normal Form over $\Bbb Q$ of the following matrix. $$A=\begin{bmatrix}0&1&1&1&0\\0&2&0&0&0\\-1&3&0&0&0\\0&-1&1&0&1\\...
Jerry Holmes's user avatar
3 votes
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72 views

Prove that the rational form of a matrix remains the same over subfield

Let F be a subfield of the complex number. How to show that the rational form of a matrix over complex is the same over F? I think we need to use the cyclic decomposition theorem. Finding the rational ...
Guria Sona's user avatar
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What is the fastest way to find the characteristic polynomial of $4\times4$ matrix and its change of basis matrix?

Let: $$A=\begin{pmatrix} 3 & 0 & -2 & -3\\ 4 & -8 & 14 & -15\\ 2 & -4 & 7 & -7\\ 0 & 2 & -4 & 3 \end{pmatrix}$$ Find a change of basis matrix $P$ such ...
HMGB1's user avatar
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Jordan form with -1 in block

I'm having troubles finding the right base, I keep getting a weird Jordan form. I need to find the Jordan form and basis of $$ \left[\begin{array}{cc} 1 & 1\\ -1 & 3 \end{array}\right] $$ this ...
Danny Blozrov's user avatar
3 votes
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137 views

Confusion about the Jordan-Chevalley/Dunford decomposition in $\mathbb{R}$, example of a rotation (solved!!!)

I'm writing some notes on Jordan-Chevalley decompositions in which I want to treat both the real and complex case in one statement. One could of course write as in the french wikipedia article: an ...
Noix07's user avatar
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Upper bound of the remainder term of $\int_0^t e^{A \tau} d\tau$ using Lagrange Remainder of the Taylor series

$\newcommand{\mat}[1]{\begin{bmatrix}#1\end{bmatrix}}$ $\newcommand{\norm}[1]{\left\lVert#1\right\rVert}$Let $A$ be an $n\times n$ real matrix which has real block diagonal form and each block has a ...
obareey's user avatar
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minimal polynomial of a matrix B given minimal polynomial of $B^2$

If we are given a minimal polynomial for a matrix $B^2$ can we deduce the minimal polynomial for $B$ $?$ Example: if the minimal polynomial for $B^2$ is $m(\lambda) = \lambda^4$ then can we deduce ...
Scosh_lr's user avatar
3 votes
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63 views

Find the Jordan Canonical Form of the given transformation

The transformation here is $T(f(x)) = f(x + 1) + f(x − 1)$ which is a linear endomorphism on V, where $V={f(x) ∈ R[x] : deg f(x) ≤ 2017}$ So I have to find the jcf J of T. Along with a basis of B. $...
Bugcatcher123's user avatar
3 votes
1 answer
612 views

Minimal polynomial and possible Jordan forms

Let $A$ be an $8\times 8$ complex matrix with characteristic polynomial $$p_A(x)=(x-1)^4(x+2)^2(x^2+1)$$ and minimal polynomial $$m_A(x)=(x-1)^2(x+2)^2(x^2+1).$$ Determine all possible Jordan ...
Julie's user avatar
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Classification of bilinear forms: operator $A^{-1} A^T$ for bilinear form $A$

I would like to understand a classification of non-degenerate (not necessary symmetric or skew-symmetric) bilinear forms over an algebraically closed field via an operator $\kappa=A^{-1} A^T$ for a ...
evgeny's user avatar
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Jordan normal form theorem proof question

Theorem: Assume that the characteristic polynomial $x_f$ splits into linear factors. Then there exists a Jordan normal form for f. The Jordan normal form is unique up to the order of the Jordan blocks....
user10024395's user avatar
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Jordan basis of $\mathcal{M}_{\mathcal{T}}(A)$

Let $A\in M_{n\times n}(\mathbb{R})$ be a matrix. Let $\mathcal{B}$ be a basis of $\mathbb{R}^n$ and $X:=\mathcal{M}_{\mathcal{B}}(A)$. If $\mathcal{S}$ is the basis for which $\mathcal{M}_{\mathcal{S}...
user72870's user avatar
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3 votes
1 answer
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relation between minimal polynomial and jordan normal form

I just solved some exercises on minimal polynomials and i remember that there is a relation between the minimal polynomial and the jordan normal form. But my question is the following : knowing the ...
sigmatau's user avatar
  • 2,622
2 votes
1 answer
49 views

Finding all possible Jordan forms from the Characteristic polynomial

Let A be a 7 x 7 matrix with characteristic polynomial $(t − 2)^4(3 − t)^3$. It is known that in the Jordan form of A, the largest blocks for both the eigenvalues are of order 2. Show that there are ...
Ddh Hhd's user avatar
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Finding all invariant subspaces of $T_A:\mathbb{C}^n \to \mathbb{C}^n$ where $A=J_{n_1}(\lambda_1)\oplus ... \oplus J_{n_k}(\lambda_k)$

Let $A=J_{n_1}(\lambda_1)\oplus ... \oplus J_{n_k}(\lambda_k)\in \text{Mat}_n(\mathbb{C})$ when $\lambda_i\neq\lambda_j$ for every $i\neq j$. I need to describe all the $T$-invariant subspaces of $T=...
Staltus's user avatar
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0 answers
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Properties of the Jordan Normal Form

So I have a question about the additive Jordan decomposition in Springer's book on linear algebraic groups. If we have a morphism $f:V\Rightarrow W$ with $V,W$ vector spaces and $a\in End(V), b \in ...
Adronic's user avatar
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2 votes
0 answers
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Switch from 1 to random real in Jordan Decomposition

Context : Let's suppose $L$ is a linear map from $\mathbb{R^k}\rightarrow \mathbb{R}^k$ , $k$ strictly positive integer. Let's suppose $\epsilon$ is a strictly positive real. In an exercice , i have ...
Francis Benjamin's user avatar
2 votes
0 answers
66 views

Finding generalized eigenvectors of a matrix

I would like to know how to find the generalized eigenvectors to the following matrix $A$, so that I can express $A$ as $PJP^{-1}$. $$ A = \begin{bmatrix} 1 & -3 & 1\\ 1 & 5 & -1\\2 &...
Dev's user avatar
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Connection between uniqueness of Jordan normal form and Jordan–Chevalley decomposition

Let $f$ be an endomorphism of a finite-dimensional vector space $V$. Definition. A Jordan–Chevalley decomposition of $f$ is a decompositon $f = d + n$ where $d$ and $n$ are endomorphisms of $V$ such ...
Jendrik Stelzner's user avatar
2 votes
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175 views

Which of the following options are correct about $X = \left \{C \in GL_3 (\Bbb R)\ |\ CAC^{-1}\ \text {is triangular} \right \}\ $?

Let $A \in M_3 (\Bbb R)$ and let $X = \left \{C \in GL_3 (\Bbb R)\ |\ CAC^{-1}\ \text {is triangular} \right \}.$ Then $(1)$ $X \neq \varnothing$ $(2)$ If $X = \varnothing,$ then $A$ is not ...
Anil Bagchi.'s user avatar
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2 votes
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Looking for an intuition of the definition of Generalized Eigenspaces

The eigenspace of (a square matrix) $A$ corresponding to $\lambda$ is the collection of all vectors $\mathbf{x}$ that satisfy $A\mathbf{x}=\lambda\mathbf{x}$, or equivalently, $(A-\lambda I)\mathbf{x}=...
Taleofwoe's user avatar
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92 views

Jordan normal form in a reductive group

Let $G$ be a connected complex reductive group. Given an element $g \in G$, there is a canonical decomposition $g = s u$ such that $s$ is semisimple, $u$ is unipotent and $s$ and $u$ commute. ...
unknownymous's user avatar
2 votes
1 answer
103 views

Is it true that for jordan block with zero eigenvalue we can choose basis where all diagonal elements are non zero?

Is it true that for jordan block with zero eigenvalue we can choose basis where all diagonal elements are non zero? if there is a proper number 0, then you can try to find a matrix in the form of J^(-...
Alex_Lesley's user avatar
2 votes
0 answers
280 views

Bounding 2-norm of powers of a matrix

Suppose that $A$ is a $n \times n$ matrix with $\rho(A) \leq 1$ and $\|A\|_2 \leq R$, where $R>1$. How can I show an upper bound on $\|A^k\|_2$ that is polynomial in $k$? A trivial upper bound is ...
KRL's user avatar
  • 1,170
2 votes
0 answers
174 views

Jordan normal form of upper bidiagonal matrix

I am trying to find the Jordan normal form of a general matrix form, which is an $N \times N$ real upper bidiagonal matrix with non-zero diagonal entries and rows that sum to one: $$\mathbf{M} = \...
Ben's user avatar
  • 4,079
2 votes
0 answers
149 views

Jordan Canonical Form of Nilpotent matrix

Let $A\in \mathbb{C}^{n\times n}$ be a nilpotent matrix. I want to find a nonsingular matrix $P$, s.t. $P^{-1}AP$ is in Jordan canonical form. Denote $F_i=N(A^i).$ (where $N(A)$ is the null space of $...
Lee's user avatar
  • 1,910
2 votes
0 answers
81 views

Properties of blocks in blockmatrix A if the matrix pair (E,A) is regular

I am working on a problem from the book "Differential-Algebraic Equations: Analysis and Numerical Solution" by Kunkel, Mehrmann. It is the Exercise 3 from Page 53 concerning matrix pairs $(E,A)$ and ...
bambuk's user avatar
  • 33
2 votes
0 answers
98 views

How many similarity classes are zeroes of a given polynomial.

I am looking for an easy way of calculating the number of similarity classes of complex matrices that satisfy some polynomial $p(t)$. As an example consider $p(t)=(t-1)^3(t+1)^4$ and $5\times 5$ ...
Sorfosh's user avatar
  • 3,266
2 votes
0 answers
75 views

Analogy of Jordon Normal Form for Antilinear Maps

Given complex vector spaces $V$, and antilinear $T:V \rightarrow V$, then if we fix a basis of $V$, we can represent $T$ by the matrix of the linear $T \circ J$, where $J$ is complex conjugation. I ...
James's user avatar
  • 1,585
2 votes
0 answers
135 views

How can I find the Jordan form of this upper triangular Toeplitz matrix?

Given an $n \times n$ matrix $A$ whose $(i,j)$ entry is $$a_{ij} = \begin{cases} n-j+i & \text{if } j \geq i\\ 0 & \text{otherwise}\end{cases}$$ find its Jordan form. I know that all the ...
Omer's user avatar
  • 2,490
2 votes
0 answers
38 views

For which values ​the matrix is ​diagonalizable

For which values ​​of $a$ matrix $A$ is ​​diagonalizable? $$A = \pmatrix{0&i\\i&a}$$ in the case that it is not diagonalizable determine a base of Jordan Attempt: The minimal polynomial ...
Ilovemath's user avatar
  • 2,921
2 votes
0 answers
339 views

When is possible to use an orthogonal matrix to put in Jordan form a matrix?

I know that if I have a symmetrical matrix defined on $R$, it is always diagonalisable and I can always find beetwen the matrix of its eigenspaces an orthogonal matrix. While if I have a non ...
Landau's user avatar
  • 219
2 votes
0 answers
235 views

Uniqueness of Jordan-Chevalley-Decomposition - why do the nilpotent matrices commute?

The Jordan-Chevalley-Theorem states that for a given endomorphism $f\in \mathrm{End}_K(V)$ of a $K$-vector space $V$, such that the characteristic polynomial $\chi_f$ splits into factors, there exist ...
Aaron Wild's user avatar
2 votes
0 answers
173 views

Square root of a matrix by looking at the eigenspaces

A problem that is well known in linear algebra is the existence of square root of a matrix, where square root of matrix $A \in M_n(\mathbb F)$ is defined to be $K \in M_n(\mathbb F)$ such that $K^2=A$ ...
Rab's user avatar
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2 votes
0 answers
81 views

Trouble in proof of Jordan Canonical Theorem

Jordan Canonical Theorem stated that: Let $K$ be an algebra closed field. Let $V$ be a nonzero, finite dimensional vector space over $K$, and let $\psi \in \operatorname{End}_K(V)$. Then there ...
Trần Linh's user avatar

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