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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90 views

$n \ge 2, (p,n) \ne (2,2).$ Prove that for any $v \in \mathbb{F}_p^n,$ the sequence $v, (I+M)v, (I+M+M^2)v, \dots$ has period $<p^n.$

Let $p$ be prime and $n \ge 2, (p,n) \ne (2,2), M \in \mathbb{M}_{n \times n}(\mathbb{F}_p).$ Prove that for any $v \in \mathbb{F}_p^n,$ the sequence $v, (I+M)v, (I+M+M^2)v, \dots$ has period $<p^n....
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3answers
107 views

Find a Jordan Canonical Form

Let $A$ and $B$ be matrices over the real numbers, such that $A$ is $3 \times 5$ and $B$ is $5 \times 3$, and the product $AB$ is $$ \left( \begin{array} \\ 1& 1 & 0 \\ 0 & 1 & 0 \\ 0 &...
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2answers
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Jordan normal form of sum of two commuting nilpotent matrices over a finite field (variant on a linear matrix pencil problem)

This question comes up with trying to construct Lie subalgebras of (large) Lie algebras that are invariant under a finite group $H$. I have two isomorphic $H$-invariant nilpotent subalgebras and am ...
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1answer
27 views

What does subspace A-matrix invariance tells me in terms of A Jordan canonical form.

I am asked to show that the semi group $(e^{tA})_{t\geq0}$ for $A \in M_n(\mathbb{C})$ is hyperbolic i.e. there exists direct decomposition $\mathbb(C)^n=X_s \oplus X_u$ in to A-invariant subspaces $...
2
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1answer
99 views

How to find the Jordan form of an anti-diagonal matrix?

How to find the Jordan form of an anti-diagonal matrix? $$\begin{bmatrix} &&&{}a_{1}\\ &&\ddots &\\ &a_{\text{} n-1}&&\\ a_{n}&&& \end{bmatrix}$$ It ...
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2answers
78 views

Matrix whose square is in Jordan normal form

Let $$A = \begin{bmatrix}J_0^2 \\ & J_0^2 & \\ && J_{1/4}^3\end{bmatrix}\in M_7(\mathbb{Q})$$ Find, with proof, a matrix $B$ so that $B^2 = A$. I'm not sure how to find this matrix. ...
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1answer
71 views

Jordan normal form of $\;\begin{pmatrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & a & b \end{pmatrix},\; a,b\in\mathbb{R}$

If possible, compute the Jordan normal form of $\begin{pmatrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & a & b \end{pmatrix}\in\mathbb{R}^{3\times 3}$ with $a,b\in\mathbb{R}$. In the ...
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1answer
156 views

Jordan normal form of powers of Jordan normal form

Previous related question: Jordan normal form powers Let $A$ be a $n\times n$ Matrix such that $A=PBP^{-1}$ where $B$ is in Jordan normal form with $\lambda_i(k)_j$ Where $i$ is the size, $k$ is the ...
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1answer
94 views

If $T:\mathbb{C}^n \to \mathbb{C}^n$ is diagonalizable when restricted to any two-dimensional invariant subspace, then $T$ is diagonalizable

Suppose $n\geq 2$. Let $T:\mathbb{C}^n\to \mathbb{C}^n$ be linear. Prove that the following are equivalent: (i) $T$ is diagonalizable. (ii) For every two-dimensional subspace $W\subseteq \mathbb{C}^n$ ...
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1answer
80 views

Jordan normal form powers

Let $A$ be a $n\times n$ such that $A=PBP^{-1}$ where $B$ is in Jordan normal form with $\lambda_i(k)_j$ Where $i$ is the size, $k$ is the eigenvalue and $j$ the order. If $A$ was diagonal($i=1$) ...
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0answers
66 views

Function (Taylor series) of Jordan canonical form about arbitrary point

On the Wikipedia page for Jordan matrices, under the section on functions $f$ of matrices $A = PJP^{-1}$ (that being the Jordan Canonical Form), the following can be found (I've paraphrased it a tiny ...
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1answer
24 views

Chain of kernels, generalised eigenvector

Given $A\in M_{n,n}(\mathbb{R})$ and $\lambda$ an eigenvalue, a generalized eigenvector of rank $i$ is defined as $v \in ker(A-\lambda E)^i\setminus ker(A-\lambda E)^{i-1}$. Why does such vector ...
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1answer
80 views

$W$ is a $T$-invariant subspace of $V$, prove a Jordan form of $T|_W$ contained the Jordan form of $T$.

The meaning of the title is to show that each block of the Jordan form of $T|_W$ corresponds to a block in the Jordan form of $T$ of equal or greater size. I know that each Jordan block in the form of ...
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1answer
30 views

From generalized eigenvector to Jordan form

I can't figure out the following part of Chen's Linear Systems book. How does he "readily obtain" $Av_2=v_1+\lambda v_2$?
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Jordan Normal Form eigenvalues

I need to find the Jordan Normal Form of a square matrix $\Phi$ such that the entries in the diagonal (or near diagonal) matrix has its elements ordered by absolute value. In particular, I need to ...
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0answers
49 views

A linear operator on a $5$ dimensional complex vector space

A linear operator $T$ on a complex vector space $V$ has the characteristics polynomial $x^3(x-5)^2$ and the minimal polynomial $x^2(x-5)$ .Choose all correct options. $(a)$ The Jordan Form of $T$ is ...
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1answer
97 views

Multiplicity of each eigenvalue in a minimal polynomial of a matrix

It is well known that for a $n \times n$ matrix $A$ , the charicteristic polynomial $p(x)$ satisfies $p(x)=\prod_{\lambda : eigenvector} (x-\lambda)^{a(\lambda)}$ where $a(\lambda )$ is the ...
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1answer
245 views

Proving Jordan chain for nilpotent matrix is linearly independent

I am looking for a proof for the Jordan chain $\{L^ix, L^{i-1}x, \dots, x\}$ being independent, where $L$ is a nilpotent matrix of index $k$, $i<k$ and $x \neq 0$. I have tried the following. ...
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50 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. ...
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Proof about diagonalizable Matrix [duplicate]

The proof said the following, let $A \in M_{n \times n}(\mathbb{C})$ such that $A^r=I_{n \times n}$ for some $r \in \mathbb{N},r>0$ then $A$ is diagonalizable. Mi attempt is the next, since $A$ is ...
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1answer
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Stuck on finding the canonical form an endomorphism.

When asked to find the Jordan form of an endomorphism I'm usually given a matrix associated with the endomorphism from which I can compute the Jordan, yet this isn't the case with this excercise; ...
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2answers
352 views

Is the Jordan normal form uniquely determined by the characteristic and minimal polynomial?

I was looking into this answer to a question about obtaining the Jordan normal form given the characteristic and minimal polynomials of a matrix. In this answer, it is stated that "The ...
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0answers
48 views

Minimum annihilating polynomial of matrix $A$ of order $n$ is equal to $(λ + 1)^2(λ - 2)$.

Minimum annihilating polynomial of matrix $A$ of order $n$ is equal to $(λ + 1)^2(λ - 2)$. What can be said about the minimal annihilating polynomial of matrix $A^2$? Justify the answer. While working ...
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1answer
137 views

Minimal polynomial = chratacteristic polynomial $\iff$ distinct eigenvalues associated with distinct Jordan blocks?

"Let M be the given matrix of order n and its Jordan Canonical Form be J . Prove that the minimal and characteristic polynomial of M are same, if and only if, distinct eigenvalues of M ...
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1answer
46 views

Prove that the Jordan Canonical Form of $T$ contains the Jordan Canonical Form of $T|_W$ for any $T$-invariant $W.$

Let $V$ be a complex vector space with a linear operator $T : V \to V$ and a $T$-invariant subspace $W \subseteq V.$ Prove that the Jordan Canonical Form of $T$ contains the Jordan Canonical Form of $...
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1answer
182 views

Calculating matrix exponential for $n \times n$ Jordan block [duplicate]

I want to calculate exponential of the matrix which on diagonal has some $a \in \mathbb{R}$ and ones above. The $n\times n$ matrix looks like following $$ A = \left( \begin{matrix} a & 1 & 0 ...
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29 views

Find nondiagonalizable matrix A

Find nondiagonalizable matrix A such that matrix $A^2-6A$ is diagonizible. Simpliest nondiagonizable matrix is a Jordan block 2 by 2 $\begin{pmatrix} a & 1\\ 0 & a \end{pmatrix}$. $\begin{...
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1answer
50 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^(-...
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1answer
63 views

Relation between Jordan Normal Form and cyclic modules

I've just started reading about the relation between cyclic modules and Jordan Normal Form and, being honest, I've quite a doubt. The text I am using says that "clearly", the following assumption is ...
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1answer
41 views

Jordan matrix form and polynomial proof.

let $ f\in F[x] $ be a polynomial. and prove that the matrix $ f\left(J_{n}\left(\lambda\right)\right) $ satisfies $ [f\left(J_{n}\left(\lambda\right)\right)]_{ij}=\begin{cases} \frac{1}{\left(j-i\...
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1answer
39 views

Question involving eigenspaces and generalized eigenspaces

Question image Let $T$ be a linear operator on a finite-dimensional vector space $V$, and let $\lambda$ be an eigenvalue of $T$ with corresponding eigenspace and generalized eigenspace $E_{\lambda}$ ...
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1answer
79 views

Prove that $U(E_{\lambda})=E_{\lambda}$ and $U(K_{\lambda})=K_{\lambda}$.

Let T be a linear map on a finite-dimensional vector space V , and let $\lambda$ be an eigenvalue of T with corresponding eigenspace and generalized eigenspace $E_{\lambda}$ and $K_{\lambda}$. Let U ...
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1answer
39 views

Alll the matrices $A\in M_{7x7}\left(\mathbb{C}\right)$, with characteristic polynomial is: $\left(x-1\right)^3\left(x-2\right)^4$, ...

I need to find all the matrices $A\in M_{7x7}\left(\mathbb{C}\right)$, all I know is the characteristic polynomial is: $$\left(x-1\right)^3\left(x-2\right)^4$$ $$\dim\:\ker\:\left(A-2I\right)=3$$ $$\...
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1answer
30 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 ...
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1answer
23 views

Show that there exist $a_1,\ldots, a_{2n-1}$ such that $ a_{2n-1}J^{2n-1}+\cdots+a_1 J=I_n,$ where $J$ is a Jordan matrix

Let $J\in\mathbb{C}^{n\times n}$ be a Jordan normal form and assume that ${\rm tr~}J<2n$. Prove or disprove that there exist $a_1,\ldots, a_{2n-1}\in\mathbb{R}$ such that \begin{equation} a_{2n-1}...
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3answers
203 views

Calculate power of a matrix using jordan form

I need to calculate: $$ \begin{bmatrix} 1&1\\ -1&3 \end{bmatrix}^{50} $$ The solution i have uses jordan form and get to: There are some points that i dont understand: $1.$ In the right ...
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1answer
37 views

Find a matrix B such that B^3 = ((9,1),(-1,7)). B can have complex values.

I have found the Jordan Normal form of the matrix ((9,1),(-1,7)) which is J = ((8,0)(1,8)). We have learnt that the 1's in the Normalform come under the main diagonal so that its a lower triangular ...
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0answers
24 views

Spectral norm of Jordan basis

Suppose that $A \in \mathbb{R}^{n \times n}$ with $\|A\|_2 \leq R$. and let $A = P J P^{-1}$ be a Jordan canonical form. Are there any upper bounds on the norms $\|P\|_2$ and $\|P^{-1}\|_2$ in terms ...
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0answers
69 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 ...
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26 views

If $A$ has a Jordan form in $\mathbb{R}$, could it be that $A$ diagonalizes in $\mathbb{C}$?

I was finding the Jordan form for a matrix $A \in M_{4}(\mathbb{R})$ given by: $$ A= \begin{pmatrix} 0 & -1 & -1 & -1 \\ 1 & 2 & 1 & 1 \\ 0 & 0 & 0 & -1 \\ 0 & ...
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1answer
71 views

Are the subspaces corresponding to Jordan blocks unique?

Let $T$ be a linear operator on a complex vector space $V$, where $n<\infty$, and let $A_1,\dots,A_m$ be the Jordan blocks of the matrix of $T$ with respect to some Jordan basis. For each $A_i$ (of ...
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0answers
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Do Jordan chains single out special directions of an eigenspace?

Let $T$ be a linear operator on $\mathbb C^n$ where $n<\infty$ and let $U$ be the eigenspace associated with eigenvalue $\lambda$. For simplicity, assume $\lambda$ is the only eigenvalue of $T$. ...
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1answer
197 views

Relation between Jordan block sizes and multiplicity in characteristic/minimal polynomial

Given an unknown matrix $A \in \mathbb{R}^{n \times n}$ and assuming that for some eigenvalue $\lambda$ of $A$ I know the multiplitiy of the root corresponding to $\lambda$ in both the characteristic ...
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1answer
26 views

prove a bound for 2-norm of Jordan block [closed]

Let J be a Jordan block.$$J=\begin{bmatrix}x&\epsilon&\\&x&\epsilon&\\&&\ddots&\ddots\\&&&x\end{bmatrix}$$ I recently find out that many J satisfy $||J||_2\...
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0answers
48 views

Is it necessary to check for linear independence when finding Jordan basis?

There a several ways to finding a jordan basis of a matrix $A$. For this question I have in mind two methods. Let $N=A-I\lambda$. Say we have already discovered several strings, say these two $v_1,...
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1answer
50 views

Generator for a Jordan basis associated with a Jordan block

Let $M$ be an $n\times n$ matrix whose characteristic polynomial splits into $n$ linear factors, so that there exists a Jordan canonical form $\mathcal{J}_M$ of $M$. Now suppose $\lambda$ is an ...
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1answer
52 views

Subspaces of generalised eigenspace

If $A$ is an endomorphism of a vector space $V$ and $\lambda$ an eigenvalue then $Ker (A-\lambda I) $ is the eigenspace corresponding to $\lambda$ and $V_\lambda=Ker (A-\lambda I)^{dim \ V} $ is the ...
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1answer
63 views

Find all possible Jordan canonical forms of $4\times 4$ complex matrices which satisfy the following three conditions simultaneously?

Find all possible Jordan canonical forms of $4\times 4$ complex matrices which satisfy the following three conditions simultaneously: $A$ is not diagonalizable; $A$ has characteristic polynomial $(x-...
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2answers
555 views

How do I find the Jordan canonical form of this 4x4 matrix?

A = $\begin{bmatrix} 1 & -1 & 0 & -1\\ 0 & 2 & 0 & 1\\ -2 & 1 & -1 & 1\\ 2 & -1 & 2 & 0\\ \end{bmatrix}$ I've been given this matrix - I'm supposed ...
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2answers
73 views

How does the size of Jordan blocks in the Jordan form of matrix A relate to exponents in the factorization of the minimal polynomial?

How does the size of Jordan blocks in the Jordan canonical form of matrix A relate to exponents $d_i$ in the factorization of the minimal polynomial $m_A(x) = (x-\lambda_1)^{d_1}(x-\lambda_2)^{d_2}\...

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