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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76 views

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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3answers
787 views

How do I transform the matrix with similarity transformation to diagonal form and use this result to calculate $A^{10}$?

I have matrix $A=\left (\begin{matrix} -2 & -8 & -12\\ 1 & 4 & 4\\ 0& 0 &1 \end{matrix} \right )$ and I need to transform with similarity transformation to diagonal form. ...
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3answers
53 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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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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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 ...
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1answer
32 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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1answer
98 views

Jordan form — determining how large $n \times n$ matrix is

Can you determine how large $n$ can be if you have an $n \times n$ matrix $A$ such that $A^3 = 0$ and $A$ has a Jordan form of exactly $4$ blocks? If so, how?
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1answer
486 views

Index of nilpotency Jordan block

If $T$ is an endomorphism, there exists a basis, according to which $T$ will be a block-diagonal matrix. Because if $V$ is the domain of $T$, $V$ will be the direct sum of the generalized eigenspaces, ...
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2answers
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Uniqueness of the Jordan decomposition

I have seen it said that a matrix $M$ (over $\mathbb{C}$, say) has a unique decomposition $M = D + N$ where $D$ is diagonal and $N$ is nilpotent. I'm having trouble seeing this, since the Jordan form ...
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1answer
524 views

Jordan basis of $A$ when $A$ is the companion matrix?

The actual question: when $A$ is the companion matrix, why the general form of $M_i$ (the group of columns of the Jordan matrix $M$ that belongs to the block associated to $\lambda_i$) is: $$ M_i^{h,...
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2answers
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Showing companion matrix is similar to Jordan block using Jordan-Chevalley decomposition

The Jordan-Chevalley decomposition says that given a linear operator $L$, you can decompose it as $L = S + N$, where $S$ is diagonalizable and $N$ is nilpotent. My textbook (Linear Algebra by ...
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2answers
592 views

Eigenvalues and Jordan form

I have a $5\times 5$ matrix and I need to find the Jordan form and its inverse. I know how to find the inverse. But for the Jordan form I am screwed. The matrix is $$\begin{bmatrix}3 & 0 & 0 ...
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1answer
732 views

Jordan blocks and the characteristic polynomial

For $A \in \mathbb{C}^{n,n}$ and $\{ \lambda_1, \dots , \lambda_r\}$ are the eigenvalues of $A$. Then my lecture notes say that the characteristic polynomial of $A$ is $$(-1)^n\prod_{i=1}^r(x-\...
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2answers
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Jordan block and Jordan chain question?

I don't understand why it's apparently 'clear' that the matrix of $T$ with respect to the basis $v_1, \dots, v_n$ is a Jordan block of degree $n$ if and only if $v_1, \dots ,v_n$ is a Jordan chain for ...
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1answer
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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 $...
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1answer
64 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
43 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
6k views

Finding Jordan Canonical form given the minimal and characteristic polynomial.

I have the following information: the characteristic polynomial of $A$ is $p_A(t)=(t-4)^3(t+6)^2$ and the minimal polynomial is $q_A(t)=(t-4)^2(t+6).$ I'm having problems seeing how one would work ...
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1answer
65 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
55 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
62 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
99 views

Prove that $A$ is similar to $B$

Let $A, B \in M_n(\mathbb{F})$ such that $m_A(x) = m_B(x)$ and $$f_A(x)=f_B(x)=(x-\lambda_1)^{d_1}\cdots (x-\lambda_k)^{d_k}$$ for different $\lambda_1, \ldots, \lambda_k$ such that $1 \le d_l \le 3$ ...
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1answer
29 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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1answer
88 views

Jordan form of operator $X \mapsto AXA$ [closed]

Matrices $n \times n$ on complex field. Compute Jordan form of operator $X \mapsto AXA$: $$ A = \begin{bmatrix} 0 & 1 & & \\ & 0 & \ddots & \\ & & \...
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2answers
3k views

Why does the largest Jordan block determine the degree for that factor in the minimal polynomial?

Let $A$ be a square matrix, so $A$ has some Jordan Normal form. Then $A$ has a minimal polynomial, say $m(X)=\prod_{i=1}^k (t-\lambda_i)^{m_i}$. Wikipedia says The factors of the minimal ...
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2answers
300 views

How to find a matrix $C$ such that $C^{-1}AC$ is in Jordan block form.

$A:=\begin{bmatrix} 6 & -1\\ 4 & 2 \end{bmatrix}$ Now, just to show I've done some working, at least to find $A$'s eigenvalues and deduced that it's not diagonalisable: Any hints/advice? ...
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1answer
225 views

Alternative proof for Jordan form statement

I am trying to understand why given a nilpotent matrix $L,$ rank ($L^{k-1}$) - rank ($L^{k}$) is the number of Jordan blocks sized $\geq k \times k$ in the Jordan representation of $L.$ There is a ...
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1answer
1k views

Jordan normal form for complex matrices

Suppose we are given the  characteristic polynomial and minimal polynomial of a matrix, say, $(x-a)^4(x-b)^2$ and $(x-a)^2(x-b)$. Then, I can tell what the largest Jordan blocks are, and hence work ...
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1answer
123 views

Polynomials and Jordan form

Problem Let A ($n\times n$ matrix) be a single Jordan block and let $C$ be an $n\times n$ matrix that commutes with $A$. Prove that $C = f(A)$ for some polynomial $f$.
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2answers
338 views

Jordan form and base for a $n \times n$ Matrix

Given this matrix in size $n \times n$ $\begin{pmatrix} 1 &1 &1&\cdots & 1 & 1 & \\ 0 &1 & & & \\ . & &.& & & \\ . &...
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1answer
2k views

Determining the Jordan form of a matrix given the invariant factors

I am trying to recover the Jordan normal form of a matrix given a list of invariant factors and was wondering if I am proceeding correctly in constructing the Jordan blocks. Let $F = \mathbb{C}$ and ...
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1answer
1k views

Jordan Block Eigenvalue Proof [closed]

Let A be a near-Jordan block, that is, a matrix obtained from a Jordan block by possibly changing the first column. Prove that no two Jordan blocks in any Jordan canonical form for A have the same ...
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1answer
18 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
31 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
20 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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20 views

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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45 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
36 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
28 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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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. ...
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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 &...
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0answers
16 views

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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2answers
50 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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2answers
2k views

Find the invariant factors and elementary divisors from the relations matrix.

Let $V$ be a finite dimensional vector space over $\Bbb C$, and $T$ be a linear operator on $V$. Consider $V$ as an $\Bbb C[x]$-module by defining $xv = T(v)$ for each $v \in V$. Let $$A = \begin{...
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1answer
17 views

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
1k views

Jordan form of a specific 2x2 matrix

I'm trying to follow an algorithm for finding the Jordan form for the matrix: $$ \begin{pmatrix} 0 & -1 \\ 4 & 4 \\ \end{pmatrix} $$ Its eigenvalue is 2 with ...
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0answers
43 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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0answers
10 views

possible Jordan normal forms of endomorphism

Let $f:\mathbb{C}^6\to\mathbb{C}^6$ be a linear map with characteristic polynomial $ch_f(X)=(X+2)^4(X-1)^2$ and $rank(f+2id)>rank(f+2id)^2=1$, as well as $rank(f-id)=5$. Assignment: Find the ...
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1answer
46 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
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 ...

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