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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1answer
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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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0answers
20 views
$J={J_r}({\lambda})$ 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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1answer
9 views
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 ...
4
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1answer
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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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1answer
45 views
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 ...
1
vote
1answer
29 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 ...
0
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1answer
25 views
Power of a Jordan Normal Form
In my notes I have that the Jordan normal form of $B^2$ is $$\begin{bmatrix}
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 &1 \\
0 & 0 & 0 &...
0
votes
1answer
32 views
Complement space of a finite dimensional space over a finite field
Let $V$ be a finite dimensional space over the field $\mathbb{F}_q$ of $q$ elements and let $U\subset V$ a subspace of $V$. How many subspaces $W\subset V$ are there such that $W\cap U = 0 $ and $V=W+...
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2answers
47 views
Finding the Jordan Form of a matrix…
I know that this type of question has been asked on here before but I am still having a hard understanding what is going on. The text that I am learning from is "Linear Algebra Done Right by Sheldon ...
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0answers
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Invariant factors and Jordan reduction : how to find the adapted basis?
I have a question about the Jordan reduction using the module theory and especially the invariant factors. If we have a vector space $E$ of dimension $n$ over a field $k$, and $f \in End(E)$ then it's ...
1
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1answer
33 views
Calculation of Jordan Normal form to determine similarity
I wanted to find a way to tell whether or not two matrices are similar.
Of course, first you check that similarity isn’t ruled out by matrices having different determinants, eigenvalues, trace.
Is ...
2
votes
3answers
51 views
find the Jordan basis of a matrix
I'm trying to find the Jordan basis of the matrix $$A =\begin{bmatrix} 8 & 1 & 2 \\ -3 & 4 & -2\\ -3 & -1 & 3\end{bmatrix}$$ I've got the characteristic equation to be $CA(x) = ...
2
votes
0answers
32 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 ...
2
votes
1answer
41 views
Can a matrix be similar to a block matrix with Jordan Block or companion matrix of the non-linear irreducible factors in its diagonal block?
Let $A$ be $3 \times 3$ real matrix with minimal polynomial $f(X)=(X-1)(X^2 +1)=X^3-X^2+X-1.$ Then By Rational Canonical Form we know that $A$ is similar to the Companion matrix of $f(X)$ which is $\...
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2answers
43 views
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))...
2
votes
1answer
60 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 ...
5
votes
1answer
45 views
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 ...
2
votes
0answers
39 views
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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0answers
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exponential family distribution are main source to generate distribution characteristics
my question is that why we say this distribution is from exponential family and by comparing it with exponential form we find cummulant function ,sufficient statistics and more things, distributions ...
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2answers
43 views
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)$$
and want to find a Jordan Form for it and a basis for the ...
0
votes
0answers
67 views
Finding Jordan basis of $5 \times 5$ nilpotent matrix
I have $5 \times 5$ real matrix, which is nilpotent:
$$
A = \begin{bmatrix}
-2 & 2 & 1 & 3 & -1 \\
3 & -8 & -2 & -9 & 3 \\
-2 &-8&0 & -6 & 2 \\
-4 &...
1
vote
2answers
49 views
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 &...
1
vote
2answers
56 views
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 ...
0
votes
1answer
31 views
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 ...
2
votes
0answers
22 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 ...
5
votes
3answers
94 views
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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0answers
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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 ...
3
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1answer
109 views
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 ...
0
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0answers
51 views
How to show that eigenvalues of AB and BA are same?
Let $A$ be an $m\times n$ matrix and $B$ be an $n\times m$ matrix. How to use Jordan canonical form to prove the matrix shown below are similar.
Given matrices are $$\begin{bmatrix} AB&0\\ B&...
0
votes
1answer
22 views
Geometric multiplicity of an eigenvalue, solely from the characteristic polynomial
I've been asked to enumerate all of the possible Jordan Canonical Forms for a complex matrix A. The issue is that the only information I have about A is that it's characteristic polynomial is:
$$p_A(...
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1answer
33 views
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).
...
1
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1answer
25 views
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
$$
M = \left(
\begin{matrix} 1 & 0 & 2 & 0 \\
0 & 0 & 0 & 0 \\
1 & 0 ...
2
votes
1answer
27 views
$B^2 = A$ for non diagonalizable $B \in \mathbb{C}^{4x4}$
I have to prove that exist a non diagonalizable matrix $B \in \mathbb{C}^{4x4}$ such that $A = B^{2}$ with
$
A=
\begin{bmatrix}
1 & 2 & 6 & 0 \\
-1 & -2 & -6 & 0 \\
...
-2
votes
1answer
33 views
Find the Jordan canonical form [closed]
$N$ is a nilpotent $15\times15$ matrix over $\mathbb{R}$ such that $$\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$$ ...
0
votes
1answer
38 views
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 & * ...
0
votes
0answers
16 views
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 ...
1
vote
2answers
25 views
A relation satisfied by the real Jordan form of a $2 \times 2$ matrix with complex eigenvalues
Let $A$ be a $2 \times 2$ matrix with complex eigenvalues, $\lambda_1 = \alpha + i \beta$ and $\lambda_2 = \alpha - i \beta$ ($\alpha, \beta \in \mathbb{R}$). We know that $u + iv$ is an eigenvector ...
0
votes
1answer
59 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 ...
0
votes
0answers
17 views
Basis of Jordan block = cyclic subspace formed by generalised eigenvectors
I think jordan matrix which are formed by jordan blocks of a particular eigenvalue , have a form of direct sum of companion matrix of a cyclic subspaces. Then basis of such cyclic subspace containg ...
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0answers
20 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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votes
2answers
73 views
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) ...
0
votes
0answers
59 views
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 ...
0
votes
1answer
20 views
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, ...,...
1
vote
1answer
49 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 &...
1
vote
1answer
45 views
Finding a Jordan Basis after finding the Jordan Canonical Form
The question asked to find the Jordan Canonical Form and Jordan Basis of $\begin{bmatrix}1 & 1 & 0 & -1\\0 & 1 & 0 & 1\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\...
1
vote
1answer
33 views
Maximum index of a generalized eigenvector of $A$ associated to $\lambda$
My textbook says the following:
Let $\lambda$ be an eigenvalue of a matrix $A$. Then max-ind($λ$), the maximum
index of a generalized eigenvector of $A$ associated to $\lambda$, is the
largest ...
1
vote
1answer
41 views
Dimension of centraliser of an $n \times n$ matrix is atleast n.
Let $A$ be an $n \times n$ matrix over a field $\mathbb{F}.$ Then consider the subspace of $M_n(\mathbb{F})$ given by $W=\{ B \in M_n(\mathbb{F}): AB =BA\}.$ Then I want to show that $\text{dim}(W) \...
2
votes
1answer
45 views
Proving there are as many generalized eigenvectors as algebraic multiplicity eigenvalue without the Jordan canonical form
I'm reading some treatment of generalized eigenvectors in a differential equations book. They want to derive that there are as many generalized eigenvectors to a certain eigenvalue $\lambda_i$, as the ...
0
votes
0answers
43 views
Connection between regular elements of a Lie algebra and the Jordan normal form
I need a source fornthe following statement:
If $\mathfrak{g}$ is a subalgebra of $\mathrm{Mat}_n$, an element $X$ is regular if and only if its Jordan normal form contains a single Jordan block for ...
1
vote
2answers
27 views
Does the position of the $1$ in a Jordan block matters? [duplicate]
I am studying the Jordan Canonical Form of a matrix and I noticed that most of the books put the $1's$ of the Jordan blocks on the superdiagonal like this for example :
\begin{pmatrix}
\lambda &...
