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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6
votes
0answers
98 views

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: $...
6
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0answers
965 views

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 $...
5
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2answers
202 views

Show The Jordan Normal Form Of $\varphi$.

Fix a nonnegative integer $n$, and consider the linear space $$\mathbb{R}_n\left [x,y \right ] := \left\{ \sum_{\substack{ i_1,i_2;\\ i_1+i_2\leq n}}a_{i_1i_2}x^{i_1}y^{i_2}\quad\Big|{}_{\quad}a_{...
5
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2answers
49 views

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\...
5
votes
1answer
77 views

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 $A$....
5
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0answers
209 views

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 ...
5
votes
2answers
752 views

Generalization of the Jordan form for infinite matrices

Under what conditions is it the case that for a matrix $M$ whose rows and columns are indexed by a countably infinite set $S$ one has a Hamel basis consisting of generalized eigenvectors (i.e. $v \in \...
5
votes
2answers
1k views

Jordan form exercise

What am I doing wrong? I've been learning how to put matrices into Jordan canonical form and it was going fine until I encountered this $4 \times 4$ matrix: $A=\begin{bmatrix} 2 & 2 & 0 &...
4
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0answers
269 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\...
4
votes
1answer
197 views

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 ...
4
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0answers
1k views

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 ...
4
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1answer
437 views

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 ...
3
votes
1answer
60 views

Possible Jordan Canonical Forms

Suppose I have a matrix $A \in M_{n \times n}(\mathbb{C})$ such that its minimal polynomial is either $x-1$ or $(x-1)^{2}$. What are its possible Jordan Canonical Forms? I was thinking that if its ...
3
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0answers
45 views

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 ...
3
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1answer
182 views

Using Jordan Normal Form to determine when characteristic and minimal polynomials are identical

Say I want to immediately write down a matrix with an identical minimal and characteristic polynomial. Say, $$ (t-1)^{3}(t-2). $$ My first instinct is to write down Jordan Blocks in a block ...
3
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0answers
59 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. $...
3
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1answer
317 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 ...
3
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0answers
172 views

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 ...
3
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0answers
1k views

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 $x^{'}(t) = Ax$, ...
3
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0answers
364 views

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....
3
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0answers
40 views

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}...
3
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0answers
1k views

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 ...
2
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0answers
41 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. ...
2
votes
1answer
26 views

find all the matrices (which are not similiar) which fulfill this formula

I need to find all the matrices $A\in M_{4x4}\left(\mathbb{C}\right)\:$ such that: $$A^4-2A^2+I\:=\:0$$ which means $\left(A^2-I\right)^2=0$ So I see that there is a few groups of which can give ...
2
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0answers
29 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 ...
2
votes
2answers
63 views

Difficulties with Jordan normal form

i'm studying in German and because of corona virus we had only a video lecture, so unfortunately I have not understood how to deal in cases when I do not have the matrix. If I had it, I think I got ...
2
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0answers
34 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 ...
2
votes
1answer
15 views

Basics of Jordan matrix, please clarify the following

Let $$J=\oplus_{i=1}^{k} J_{n_i}(\lambda)$$ where $J_{n_i}(\lambda)$ is a jordan block of size $n_i$ with $\lambda $ on its diagonal, and $\sum_{i=1}^{k}n_i = n $, so $J$ is $n\times n$ matrix, ...
2
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0answers
24 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 $5\times 5$ ...
2
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0answers
44 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 ...
2
votes
1answer
69 views

How to find a Jordan basis and a Jordan matrix for a nilpotent matrix?

I am trying to find a general step-by-step "easy" / "intuitive" solution to finding Jordan basis and Jordan matrix (based on the basis) for a nilpotent matrix. If you can add an intuition for the ...
2
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0answers
51 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 ...
2
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0answers
32 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 ...
2
votes
1answer
172 views

Linear Algebra : Jordan Canonical form (Jordan blocks and the Super-Diagonal)

In terms of Jordan Canonical Form, and more specifically about Jordan Blocks. When there is a definition about Jordan Blocks they say the eigenvalues go on the principle diagonal and the diagonal ...
2
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0answers
55 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 ...
2
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0answers
112 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 ...
2
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0answers
68 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 ...
2
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0answers
61 views

Classification of Matrices and normal forms

As is shown in couses of Linear Algebra, for every square matrix $A$ one can choose $S,T,P\in GL(n,K)$ so that $SAT^{-1}=\operatorname{diag}(1,...1,0,...,0)$ and $PAP^{-1}$ is in Jordan normal form. ...
2
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0answers
40 views

Polar Decomposition in Real Algebraic Groups

Every element $g \in GL(n,\mathbb{C})$ has a unique Jordan decomposition $$ g = g_u g_s $$ where $g_u$ is unipotent, $g_s$ is semisimple (i.e. diagonalizable over $\mathbb{C})$ and $g_ug_s=g_sg_u$. It ...
2
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0answers
70 views

Calculate $\det(p(A))$

Let $n \in \mathbb N$, $A \in \mathbb C^{n \times n}$ be nilpotent and $l\in \mathbb N$. Further, let $$p = \sum_{i=0}^n \alpha_i A^i \in \mathbb C[t]$$ be a polynomial. Show that zero is the only ...
2
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0answers
152 views

Problem on characteristic polynomial and minimal polynomial

I am given two matrices a and b such that characteristic polynomial and minimal polynomial of a and b are equal I have to check If they are similar JC form of a and b are same Rank(a) = rank(b) If ...
2
votes
0answers
82 views

Linear transformation with diagonalizable power

Let $f: \mathbb{C}^5 \to \mathbb{C}^5 $ a linear transformation such that $f^3$ is diagonalizable, but $f^2$ is it not. Is it true that $f$ necessarily has a $3 \times 3$ jordan block with a null ...
2
votes
0answers
23 views

Let $p$ be prime and $K$ a field with $char(K)=p$. Let $A \in M_n(K)$ such that $A^p=I$. Find the Jordan-Chevalley decomposition of A

Let $p$ be prime and $K$ a field with $char(K)=p$. Let $A \in M_n(K)$ such that $A^p=I$. Find the Jordan-Chevalley. I got a hint which says, that I should write $A^p-I$ as a power of some other matrix....
2
votes
0answers
120 views

If all eigenvalues are < 1, fixed point iterations converges to the only solution

Theorem states that for every initial value fixed point iteration x = Bx+b converges to the only solution of the system if all $|\lambda| $ < 1. Prove it using Jordans normal form. Initial form is ...
2
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0answers
434 views

Show the form of $J$ and $P$ for Leslie matrix $A$ when $A = PJP^{-1}$

I'm trying to solve this for a homework assignment. The Jordan Normal form theorem states that every complex $n \times n$ matrix $A$ van be written as $A=PJP^{-1}$, where $J$ is the diagonal matrix ...
2
votes
1answer
81 views

Describing the space of matrices which “jordanize” a given matrix

This is a naive linear algebra question. I apologize for the level but I could not find an answer in the literature. Let $A$ be a $n$ by $n$ matrix (say over $\mathbb C$). Suppose the Jordan form of $...
2
votes
0answers
43 views

Show that jordan matrix with k blocks in diagonal has exactly k independent eigenvectors?

How can I prove that a Jordan matrix with k blocks in the diagonal has exactly k independent eigenvectors. Can you help me to find a formal proof of this statement?
2
votes
1answer
43 views

Direct sums of invariant subspaces

Let $A$ be a complex $n\times n$ matrix, with its Jordan carnonical form as $J=diag(J_1,\cdots,J_s)$. Then there exists an invertible matrix $P$ such that $P^{-1}AP=J$. It is easy to verify that $\Bbb ...
2
votes
0answers
409 views

Jordan canonical form when field is not algebraically closed

Suppose we have a linear operator $T : V \to V$, where $V$ is a vector space $V$ over a field $F$. Now if $F$ is not algebraically closed, we don't necessarily know that $T$ has a Jordan canonical ...
2
votes
0answers
194 views

Jordan Normal Form of a self-adjoint Linear Transformation

Let $V$ a finite inner product space, $dim V = n \geq 3$. Let $w_1,w_2 \in V$ such that: $<w_1,w_2>=0$, $||w_1||=||w_2||=1$ where $||w||$ is the norm of a vector $w$. The inner product space is ...

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