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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Computing transformation matrix between similar matrices

If I have the matrix $A = \begin{pmatrix} 2 & 0 & 1 & -3 \\ 0 & 2 & 10 & 4 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 3 \end{pmatrix}$ and I've calculated its ...
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In the Jordan-Chevalley decomposition $M=D+N$, how obtaining $D$ and $N$ as polynomials in $M$?

The Jordan-Chevalley expresses a linear operator $M$ as $$ M = D + N, $$ where $D$ is semisimple (diagonalizable), $N$ is nilpotent and $DN=ND$. Although it is stated in many sources that $D$ and $N$ ...
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$A$ and $B$ be $n \times n$ matrices over the field $\mathbb F$ which have the same characteristic polynomial

Lemma: Let $N_1$ and $N_2$ be $3 \times 3$ nilpotent matrices over field $\mathbb F$. Then, $N_1$ and $N_2$ are similar if and only if they have the same minimal polynomial. Use the result above ...
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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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Finding characteristic and minimal polynomials and the Jordan normal form of $f$, knowing some relations for $f$.

Given a vector space $V$ of dimension $4$ and a base $\{v_1,v_2,v_3,v_4\}$, let $f$ be an endomorphism of $V$ such that $f^3=0$ and moreover $f(v_1)=f(v_2)=v_3$, $f(v_3)=kv_4$, and $f(v_4)\in\left<...
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How many realizations are there then in a structured set of matrices yielding characteristic polynomial $(t+1)^4$?

Let us consider a subset $S$ of $M_4(\mathbb R)$ which has following form \begin{align*} \begin{pmatrix} 0 & * & 0 & * \\ 1 & * & 0 & * \\ 0 & * & 0 & * \\ 0 & *...
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Finding ch. polynomial and Jordan normal form of $f$ knowing $\dim\ker f=2$ and there are $a,b$ not in $\ker f$ such that $f^2(a)=0, f(b)=b$

Given a vector space V of dimension $4$, let $f$ be an endomorphism such that $\dim(\ker f)=2$. Assuming there exist $a,b\in V\setminus \ker f$ such that $f^2(a)=0, f(b)=b$ I should find $\chi_f$ and ...
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Let $\;A\;$ be a $\;2\times 2-$matrix with only one eigenvalue $\;x=5.\;$ Show that $\;(5I −A)^2 = 0.$

I know that every matrix is conjugate to an upper triangle form matrix and conjugate matrices have the same characteristic polynomial. I then try to get the characteristic polynomial of the upper ...
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Yet Another Question Regarding Jordan Form [duplicate]

The Problem: Let $A$ be a $5 \times 5$ matrix with characteristic polynomial $(x-2)^3(x+1)^2$ and minimal polynomial $(x-2)^2(x+1)^2$. What are the possible Jordan forms for $A$. My Approach: There ...
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$J={J_r}({\lambda})$ is a Jordan Block matrix for $\lambda$, $s{\leq}r$ is an integer. Find formula for $J^s$.

So The title is part (i) and part (ii) is "use the formula to show that if A is a square matrix with identity $A^l$ for some $l$, then A is diagonalizable. I'm totally new to Jordan block matrices ...
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Is the transformation matrix of an upper triangular matrix to its Jordan normal form always triangular?

Assume that $A$ is an upper triangular matrix. In the case where $A$ is 2x2, I've checked that a transformation matrix $P$ such that $J = P^{-1}AP$, with $J$ Jordan normal form of $A$, is always upper ...
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A linear map $T: \mathbb{R^3 \to \mathbb{R^3}}$ has a two dimensional invariant subspace.

Let $T: \mathbb{R^3 \to \mathbb{R^3}}$ be an $\mathbb{R}$-linear map. Then I want to show that $T$ has a $2$ dimensional invariant subspace of $\mathbb{R^3}.$ I considered all possible minimal ...
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Non-integral powers of a matrix

Question Given a square complex matrix $A$, what ways are there to define and compute $A^p$ for non-integral scalar exponents $p\in\mathbb R$, and for what matrices do they work? My thoughts ...
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62 views

Matrix Exponential Jordan Form Linear System

Given the Linear System $\dot{x}(t)=A x(t)$ with $x_0=(x_{01},x_{02})$ as initial state and $A=\begin{pmatrix} 0 & 1 \\ -k/M & -h/M \end{pmatrix}$, when $h^2=4Mk$ the matrix A has a single ...
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Jordan Form of a 3x3 Matrix with an eigenvalue of multiplicity 3…

Let $$A= \begin{bmatrix} 2&2&3\\ 1&3&3\\ -1&-2&-2 \end{bmatrix} . $$ Find the Jordan Form, $J$, of this matrix, and an invertible matrix $Q$ such that $A = QJQ^{-1}$. I have ...
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Power of a Jordan Normal Form

In my notes I have that the Jordan normal form of $B^2$ is $$\begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 &1 \\ 0 & 0 & 0 &...
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Complement space of a finite dimensional space over a finite field

Let $V$ be a finite dimensional space over the field $\mathbb{F}_q$ of $q$ elements and let $U\subset V$ a subspace of $V$. How many subspaces $W\subset V$ are there such that $W\cap U = 0 $ and $V=W+...
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Finding the Jordan Form of a matrix…

I know that this type of question has been asked on here before but I am still having a hard understanding what is going on. The text that I am learning from is "Linear Algebra Done Right by Sheldon ...
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57 views

Calculation of Jordan Normal form to determine similarity

I wanted to find a way to tell whether or not two matrices are similar. Of course, first you check that similarity isn’t ruled out by matrices having different determinants, eigenvalues, trace. Is ...
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Invariant factors and Jordan reduction : how to find the adapted basis?

I have a question about the Jordan reduction using the module theory and especially the invariant factors. If we have a vector space $E$ of dimension $n$ over a field $k$, and $f \in End(E)$ then it's ...
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find the Jordan basis of a matrix

I'm trying to find the Jordan basis of the matrix $$A =\begin{bmatrix} 8 & 1 & 2 \\ -3 & 4 & -2\\ -3 & -1 & 3\end{bmatrix}$$ I've got the characteristic equation to be $CA(x) = ...
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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 ...
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Jordan form of an $n \times n$ Jordan block with eigenvalue $0$

Suppose $J$ is an $n \times n$ Jordan block with eigenvalue $0$, what is the Jordan form of $J^2$? My solution: I squared the matrix, it follows that the eigenvalues are $0$ again, and $\dim(N(J^2))...
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131 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 ...
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What is the simplified form of the generalized eigen space when the characteristic polynomial does not split in the given field.

Let $V$ be a finite dimensional vector space over a field $\mathbb{F}.$ Let $T$ be a inear operator on $V$ and $\lambda \in \mathbb{F}$ be an eigenvalue of $T$ of algebraic multiplicity $m.$ Now ...
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Reference for the size of Jordan blocks.

Can someone give any reference or pdf to determine the size of Jordan blocks ? There are some exercises related to the size of Jordan blocks, but I couldn't solve. Any reference will be appreciated.
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Jordan Normal Form: Two times the same basis vector?!

I have 3 dimensional matrix $$A = \left(\begin{array}{c} 2 & 1 & 0 \\ -1 & 0 & 1 \\ 1 & 3 & 1\end{array}\right)$$ and want to find a Jordan Form for it and a basis for the ...
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Finding Jordan basis of $5 \times 5$ nilpotent matrix

I have $5 \times 5$ real matrix, which is nilpotent: $$ A = \begin{bmatrix} -2 & 2 & 1 & 3 & -1 \\ 3 & -8 & -2 & -9 & 3 \\ -2 &-8&0 & -6 & 2 \\ -4 &...
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Put a matrix $A$ in Jordan Normal Form and find a $P$ such that $P^{-1}AP=J$

I have a linear algebra exam tomorrow and this is a frequent question. $A= \begin{pmatrix} 4 & 0 & 1 & 0 \\ 2 & 2 & 3 & 0 \\ -1 & 0 & 2 & 0 \\ 4 & 0 & 1 &...
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Minimal polynomial for any power of Jordan block is same as the minimal polynomial of the Jordan block.

Let $J$ be the $n \times n$ Jordan block corresponding to the eigen value $1$. For any natural number $r$ is it true that the minimal polynomial for $J^r$ is $(X-1)^n$ ? Another way to think about it ...
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Find the Jordan forms of a matrix from just the ranks of its eigenspaces

Let $C$ be a $10\times 10$ matrix whose characteristic polynomial is $(t+2)^5(t-3)^5$. Suppose that $rank((C+2I)^2)=6$ and $rank((C-3I))=8$. What are the possible Jordan forms of $C$? This is the ...
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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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Prove that there exists number $k\in \mathbb{N}$ such that $ V = \operatorname{Ker}A^{k} \dot{+} \operatorname{Im}A^{k}$

Problem: Let A be linear operator A $\in L(V)$. Prove that there exists number $k\in \mathbb{N}$ such that $ V = \operatorname{Ker}A^{k} \dot{+} \operatorname{Im}A^{k}$. Then prove that operator $\...
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An intuitive approach to the Jordan Normal form

I want to understand the meaning behind the Jordan Normal form, as I think this is crucial for a mathematician. As far as I understand this, the idea is to get the closest representation of an ...
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Commutant of Jordan Block

Matrices $B \in \mathbf{C}^{n\times n}$ commuting with a given Jordan block $A$ are known to be upper triangular Toeplitz matrices. I have seen convincing proofs, but I wanted to derive this fact by ...
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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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61 views

Geometric multiplicity of an eigenvalue, solely from the characteristic polynomial

I've been asked to enumerate all of the possible Jordan Canonical Forms for a complex matrix A. The issue is that the only information I have about A is that it's characteristic polynomial is: $$p_A(...
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Eigenspace, Jordan Canonical form

Let $A\in \mathbb{C}^{n\times n}$ with all eigenvalues equal to $\lambda$, i.e. the characteristic function $c_A(z)=(z-\lambda)^n$. Denote $F_i=N[(A-\lambda I)^i]$ ($N$ represents the null space). ...
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How to find Jordan Basis and Jordan Form

I need to find the Jordan Normal form $J$ and a matrix $S$ such that $J = S^{-1} AS$. The matrix is $$ M = \left( \begin{matrix} 1 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 ...
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$B^2 = A$ for non diagonalizable $B \in \mathbb{C}^{4x4}$

I have to prove that exist a non diagonalizable matrix $B \in \mathbb{C}^{4x4}$ such that $A = B^{2}$ with $ A= \begin{bmatrix} 1 & 2 & 6 & 0 \\ -1 & -2 & -6 & 0 \\ ...
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Find the Jordan canonical form [closed]

$N$ is a nilpotent $15\times15$ matrix over $\mathbb{R}$ such that $$\dim(\ker N) = 5, \quad \dim (\ker{N^2}) =8, \quad \dim(\ker{N^3})= 11,$$ $$\dim (\ker{N^4}) = 13, \quad \dim(\ker{N^5}) =15$$ ...
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Does this structured matrix yielding a specified characteristic polynomial admit uniquely defined Jordan blocks?

Let $\mathcal E \subset M_5(\mathbb R)$ be a subset of matrices defined by: for every $A \in \mathcal E$, $A$ admits the structure \begin{align*} A = \begin{pmatrix} 0 & * & 0 & 0 & * ...
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Every eigenvector generating a cyclic subspaces, which contains some generalised eigenvectors .

Is every generalised eigenvector belongs to some cyclic subspace generated by an eigenvector ? I think 'yes' and for JNF , this cyclic subspace be a jordan block corresponding to that generating ...
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61 views

Distinguishing between two possibilities for the Jordan normal form of a matrix

Let $A=\begin{pmatrix} 2&2&0&0 \\ -2&-2&0&0 \\ 0&0&-1&1\\0&0&-1&1 \end{pmatrix}$. I noticed that $A^2=0_4$, and I deduced that the characteristic ...
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Center of a non-abelian subgroup of $GL(2, \mathbb{C})$

I'm trying to do the following exercise: Let be $G$ a non-abelian subgroup of $GL(2, \mathbb{C})$. Prove that the center of $G$ is contained in the center of $GL(2, \mathbb{C})$ My (very partial) ...
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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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Find nth power of symmetric matrix of size m*m where m<=1000 and n can be any positive integer?

i have to find nth power of matrix which is symmetric about main diagonal and all elements of main diagonals are zero. For example sample matrix $\left( \begin{array}{cc} 0 & 1 & 0 & 0\\ 1 ...
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Verifying understanding of uniqueness of the Jordan form?

I would just like to make sure I understand the uniqueness part correctly. Suppose I know that the Jordan blocks $J_1, ..., J_n$ make up the Jordan form of $A$. Then I can arrange the blocks $J_1, ...,...
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297 views

Minimal Polynomial VS Jordan Normal Form.

So I just felt the need to refresh my concepts a bit, so I started reading about minimal polynomials. The minimal polynomial $p_A$ for a matrix ${\bf A}$, can be defined as the smallest degree monic ...
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1answer
78 views

Find Jordan Decomposition without calculating $\ker(A - \lambda E)$

Without calculating the null space of $(A - \lambda E)$, find the Jordan decomposition of $$ A = \begin{pmatrix} 0 & 1 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 & 1 \\ 0 & 0 &...