Questions tagged [jordan-normal-form]

The Jordan normal form of a matrix is a similar block matrix having diagonal blocks when the matrix is diagonalizable and diagonal + nilpotent blocks more generally.

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62 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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807 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 $...
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121 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\...
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967 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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176 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 ...
3
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0answers
42 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 ...
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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. $...
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143 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 ...
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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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317 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....
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39 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}...
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44 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 ...
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24 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 ...
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26 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 ...
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64 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 ...
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42 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. ...
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31 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 ...
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69 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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118 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 ...
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77 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 ...
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22 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....
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96 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 ...
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347 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 ...
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38 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
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306 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 ...
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166 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 ...
2
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508 views

Complex eigenvalues Jordan real matrix

As I posted here and here I'm studying Jordan forms and similar concepts. I've got a problem with complex eigenvalues in jordan real matrices. I know (at least I think so) how to compute the Jordan ...
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77 views

What does power of a factor in a minimal polynomial mean in rational form?

Let $V$ be a finite dimensional vector space and $T$ be some linear operator. Suppose the minimal polynomial has a factor $(x-c)^2$. If $T$ has Jordan form, then we can assert that the biggest jordan ...
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45 views

Lipschitz continuity of invariant subspaces for parametrized matrices

Let $A(t)$ be a one-dimensional parametrized family of linear operators on $\mathbb{R}^m$ that has smooth dependence on $t$. Let $V_0\subset \mathbb{R}^n$ be an $n$-dimensional invariant subspace for ...
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62 views

How do I interpret “K2 mod K1”?

I've been doing a bit of work these past couple of nights on computing cyclic subspace decompositions, finding cyclic bases, and then computing the Jordan canonical form of matrices. My question is: ...
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237 views

Relationship Jordan Form and Rational Canonical Form

If $A$ is a matrix over a field whose characteristic polynomial splits, then how is the Jordan form related to the rational Canonical form and can we recover one from the other in a computationally ...
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0answers
136 views

Jordan decomposition algorithm

I'm trying to calculate the value of a matrix function. As far as I understood, this is done by first decomposing my matrix $A$ into $PJP^{-1}$. Where $J$ is in Jordan normal form. However, this ...
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653 views

About Jordan-Chevalley decomposition

I have this problem: Let $K$ be a field. Let $J\in M_n(K)$ a Jordan matrix. Prove that there exists a diagonal matrix $D$ and a nilpotent matrix $N$ such that $J=D+N$ and $DN=ND$. I saw that this ...
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917 views

Decompose $A=D+N$ with $DN=ND$, $N$ nilpotent, $D$ diagonalizable

Can anyone help me out with the following question: For the matrix $A$ give a diagonalizable matrix $D$ and a nilpotent matrix $N$ so that $A=D+N$ and $ND=DN$. $\begin{bmatrix} 1 & 4 \\ -1 & ...
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28 views

Galois descent for a semisimple automorphism

Let $K$ be a perfect field and $\overline{K}$ be the algebraic/separable closure. Let $V$ be a finite dimensional $K$-vector space, and let $V_{\overline{K}} = V \otimes_K \overline{K}$. Given an ...
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0answers
14 views

I want to find an explicit direct sum decomposition of the space into T-cyclic subspaces

I construct a linear operator $T \in \mathcal{L(\mathbf{C}^7)}$ , where the minimal polynomial is $m_T (x) = x^2(x-1)^2$ and the caraterisitic polynomial is $p_T (x) = x^3(x - 1)^4$. The linear ...
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0answers
40 views

Canonical Jordan form contradiction

I am faced with the following problem: Given endomorphism $f$ whose characteristic polynomial is $$P_c(x) = (x+1)^{10} (x-1)^{10} x^{10}$$ and whose minimal polynomial is $$P_m (x) = (x+1)^5 (x-1)^...
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30 views

Finding possible Jordan forms of a matrix

Find the Jordan forms of a matrix $A$ subject to the following conditions: the characteristic polynomial is $(x-1)^4(x+3)^5$. matrix $A-I$ has nullity $4$ and matrix $A+3I$ has nullity $1$. ...
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51 views

Given a Jordan canonical basis, how to find out to which generalized eigenspace picked generalized eigenvector belongs

Suppose we have finite-dimensional linear operator $A:V\to V$ , that has eigenvalues $\lambda_1 ,\lambda_2, ... \lambda_n$ . It is known that we can decompose $V$ into direct sum of generalized ...
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0answers
31 views

Proving that a matrix is nonnegative if its powers are nonnegative

I am working on a problem involving doubly stochastic matrices where I must prove that $P$ is doubly stochastic if and only if $P^k$ is doubly stochastic for $k = 1, 2, ...$ It is easy to show that if ...
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20 views

Jordan form of the matrices of a group

Let's consider a set of $m$ generic square matricies $(N;N) $ defined on $R$ which forms a group. Chosen one of these $ m $ matrices, I know that, by changing the base on my vectorial space, I can ...
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29 views

What is the difference between ones and zeroes on the superdiagonal in a Jordan normal form matrix?

I've observed that in Jordan normal form the eigenvalues are obviously on the diagonal, but on the superdiagonal there can sometimes be only ones, sometimes be only zeroes and sometimes both. What is ...
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64 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 ...
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0answers
32 views

Question regarding possibilities for Jordan normal form

In this problem I am given a matrix $$B = \begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}$$ ...
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37 views

A Step in The Proof of Jordan Form: Trivial Space

Let $\mathcal{R}$ denote the range space and $\mathcal{N}$ the null space of a linear transformation. We write $$ \mathcal{R}(t^n) = \mathcal{R}(t^{n+1}) = \dots = \mathcal{R}_{\infty}(t)$$ and ...
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0answers
75 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 $K^2=A$ ...
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62 views

Jordan Normal Form from characteristic and minimal polynomials

I'm trying to find Jordan Normal Form of a linear transformation $F: V \to V$ with characteristic polynomial $$P_{F}(t) =(t+1)^3 (t-1)^3$$ and minimal polynomial $$M_{F}(t) =(t+1)^2 (t-1)^2$$ It has ...
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0answers
162 views

How to prove that every real skew-symmetric matrix is congruent to a block diagonal matrix by using bilinear forms?

For any real $n \times n$ skew-symmetric matrix $A,$ show there exists an orthogonal matrix $P$ such that: $PAP^T = \begin{pmatrix}{} \Lambda_1 & \\ & \Lambda_2 & \\ & & \ddots &...
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0answers
9 views

Proving existence of basis that can be constructed using Jordan Blocks.

Let A be an n x n complex Jordan Block. I am trying to show that there exists a vector $\vec{v}$ such that $\vec{v} , A \vec{v}, \dots , A^{n-1} \vec{v}$ constitute a basis for $\mathbb{C}^n$. As I ...
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0answers
71 views

Prove that, over a Euclidean domain $R, I + cE_{ij}$ generate $SL_n(R)$.

Let $R$ be a euclidean domain. Let $E_n(R)$ be the subgroup of $GL_n(R)$ generated by all matrices of the form $I + λ$ where $λ$ is a matrix with precisely one non zero entry and this entry does not ...