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.
1,208
questions
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Finding $2$-dimensional invariant subspace
I just want to check that my understanding is correct of invariant subspaces. I was given a matrix A in which I have found that it is invertible, so I know that a $2$-dimensional invariant subspace ...
- 11
1
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1
answer
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What do you call the converse of an invariant subspace of an operator?
Question. I am looking for the concept converse to invariance: what do we call a set $W$, such that $$T(w) \in W \implies w \in W ?\tag{1}$$
I feel there was a word for this, but I can't recall it, ...
- 226
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10
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Finding Jordan canonical basis from Jordan canonical form for 4x4 matrix [duplicate]
The matrix in standard basis:
$$A =\begin{bmatrix}-3&1&3&3\\-10&2&9&9\\-4&0&5&4\\2&1&-3&-2\end{bmatrix}$$
characteristic polynomial is $(λ−1)^3 (λ+1)$ ...
- 1
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1
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35
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Jordan canonical form and basis for 4 by 4 matrix with two eigenvalues
So, I am given the matrix in standard basis
$$A =\begin{bmatrix}-3&1&3&3\\-10&2&9&9\\-4&0&5&4\\2&1&-3&-2\end{bmatrix}$$
characteristic polynomial is $(λ−...
- 1
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votes
2
answers
37
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Matrices similarity using the jordan form
Given $A \in M_{n}(\mathbb{C}),$ a matrix of size $n\times n$ with characteristic polynomial $\chi_{A}(X) = (X-1)^n$, I need to show that $A$ is similar to $A^k$ for every $k \in \mathbb{N}$. I've ...
- 21
2
votes
1
answer
57
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How to find all the invariant subspaces in relation to Matrix
Let
$$A=\begin{bmatrix}1 & 1 & 0 & 0\\
-1 & 1 & 2 & 1\\
0 & 0 & 3 & 1\\
0 & 0 & -1 & 1
\end{bmatrix}\in M_4(\mathbb C)$$
I need to find all the $A$-...
- 21
0
votes
0
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36
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Invariant and cyclic subspaces
Iv'e been given this question and I don't really have a clue how to do it.
Let $V$ be a finite dimensional vector space of dimension $n$ over a field $\mathbb{F}$, and let $f$ be an operator over $V$.
...
0
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1
answer
37
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Matrices with given conditions on minimal and characteristic polynomial are similar
Assuming we have two matrices, $A_{1}$ and $A_{2}$. Determine whether they are similar or not given the fact that both matrices satisfies the following conditions (there is no connection between the ...
0
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1
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52
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Missing statement in a proof Jordan normal form
I am reading the proof for the existence of the Jordan normal form.
I know the Fitting lemma.
I am missing one thing though, I mean the proof in my book seems to have this particular thing missing.
I ...
- 11.5k
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2
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30
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An example for an ambiguous Jordan-Normalform
Give an example where the knowledge of the characteristic and minimal polynomial, as well as the geometric multiplicity, is not sufficient to fully determine the Jordan Canonical Form.
I'm struggling ...
- 488
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23
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What is the index of nilpotency of $A$?
Here is the question I am trying to answer:
Find the Jordan form of
$$A = \begin{pmatrix}
1 & 0 & 1 \\
0 & 0 & 0\\
0 & 0 & -1
\end{pmatrix}$$
I know that the first step is to ...
- 1,187
1
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0
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23
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Jordan Normalform and geometric multiplicity.
Let $K$ be an algebraically closed Field, $V$ a $K$-vector space with $\dim V < ∞ $ and
$f ∈ End(V)$. For an eigenvalue $λ$, let $r(f, λ)$ denote k the multiplicity of $λ$ in the minimal polynomial....
- 488
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51
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Calculating Jordan Form of matrix - what to do with $rank(A^2)$?
I am told $m(x)$ divides $x^2(x-1)$, $rank(A) = n-1$ & $rank(A^2) = n-2$. So I have the base form: $$J_2(0)^a \oplus J_1(0)^b\oplus J_1(1)^c$$So I know the geometric multiplicity of the $0$ ...
- 403
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61
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Any nilpotent matrix $N$ has the unique normal form. ("Linear Algebra" by Ichiro Satake) Did the author really prove the uniqueness?
I am reading "Linear Algebra" by Ichiro Satake.
Example 3 says that any nilpotent matrix $N$ has the unique normal form.
I understood any nilpotent matrix $N$ has a normal form.
But I cannot ...
- 6,071
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0
answers
32
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Properties of the Jordan Normal Form
So I have a question about the additive Jordan decomposition in Springer's book on linear algebraic groups. If we have a morphism $f:V\Rightarrow W$ with $V,W$ vector spaces and $a\in End(V), b \in ...
2
votes
1
answer
63
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A basis that makes a matrix triangular.
Find a basis for $\mathbb C^3$ so that the following matrix is in triangular form:
\begin{pmatrix}
0 & 1 & 0 \\
0 & 0 & 1 \\
1 & 0 & 0
\end{pmatrix}
What are the eigenvalues?
I ...
- 1,187
1
vote
0
answers
32
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$5 \times 5$ nilpotent matrices with the same minimal polynomial and nullity must be similar.
Problem. Suppose $A, B$ are both $5 \times 5$ nilpotent complex matrices with the same nullity and the same minimal polynomial. Prove that $A$ and $B$ are similar.
My Question. Is there a 'clever' ...
- 2,077
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25
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Random nilpotent and upper triangular matrix question
Suppose a $4$ by $4$ matrix $A$ is nilpotent and upper triangular, and all $(i, j)$ entries for $i < j$ are chosen randomly and uniformly in the interval $[−1, 1]$. What are the probabilities that ...
3
votes
2
answers
54
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How can I guarantee that all the number of generalized eigenvectors are equal to the algebraic multiplicity?
I have found many questions in the Stack but, I couldn't find what I want...
Since my major is physics, I'm not good at the terms in mathematics.
So it was hard to reach understanding Jordan normal ...
1
vote
2
answers
52
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Jordan canonical form of $p(\alpha)$ in terms of that of $\alpha$
Let $\alpha$ be a linear transformation defined in a finite-dimensional vector space $V$ over a field $F$. If polynomials $p(x)\in F[x]$ are such that for all eigenvalues $\lambda$ of $\alpha$, $p'(\...
- 27
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nth root of non diagonalizable matrix
Good day,
I am interest in math, and while doing the exercise. I can find the nth root of the matrix if it is diagonalizable, using the $A^n = P*Q*P^-$, where Q is the diagonal, and P are the ...
1
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0
answers
26
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Help with Jordan form of matrix
I was studying Jordan descomposition of a Matrix, and I am really stuck in the algorithm to do this.
I start with the matrix $$A=\begin{pmatrix}1 & -3 & 1\\
1&5&-1\\
2&6&0
\end{...
- 11
1
vote
0
answers
58
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Why is Jordan normal form possible?
We know that we are able to put a 2x2 matrix $A$ into the following Jordan normal form: $A=PJP^{-1}$ with
$J = \begin{pmatrix}
\lambda_1 & a \\
0 & \lambda_2
\end{pmatrix}$
Where $a=0$ or $a=1$...
- 2,947
1
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0
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Prove $\lambda^np_{AB}(\lambda) = \lambda^mp_{BA}(\lambda)$ for non-square $A$ and $B$.
Problem. Suppose $A \in \mathcal{M}_{m,n}(\mathbb{C})$ and $B \in \mathcal{M}_{n,m}(\mathbb{C})$. Prove that the characteristic polynomials for the matrices $AB$ and $BA$ satisfy $\lambda^np_{AB}(\...
- 2,077
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0
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Determining the dimension of the Eigenspace without knowing the eigenvalues (Segre Classification)
This has been a burning problem in my head for a while now. Any help/suggestions are greatly appreciated. I'll use a concrete 3x3 matrix as an example, but I'd like to know thoughts especially for 4x4 ...
- 33
1
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Jordan Normal Form and its conjugates
Is it true that a matrix of the form
$$
\begin{bmatrix}
0 & x_1 & \ldots & \ldots & 0 \\
0 & 0 & x_2 & \ldots & 0 \\
\vdots & \vdots & \ldots & \...
- 427
1
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1
answer
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Given $\Delta_{T}(x)=m_{T}(x)=(x-\alpha)^{n}$ and $ TS = ST $ where $T,S$ are linear maps. Prove there is a polynomial $f(x)$ such that $S=f(T)$.
Problem: Let $V$ denote a vector space over $\mathbb{C}$. Let $T: V \rightarrow V$ denote a linear map such that $\Delta_{T}(x)=m_{T}(x)=(x-\alpha)^{n}$ for some $\alpha \in \mathbb{C} .$ Let $S: V \...
- 1,585
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votes
1
answer
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Computing the Jordan form of matrix - Getting the null matrix?
I am trying to diagonalise the following matrix
$$
A = \begin{pmatrix}
\sqrt{8} & 0 & 0 \\
4 & \sqrt{8} & -4 \\
0 & 0 & \sqrt{8}
\end{pmatrix}
$$
finding its eigenvalues.
The ...
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How to determine the basis of an endomorphism Jordan normal form?
$T$ is an endomorphism on $ \mathbb{R}^4 $ represented by the matrix $ A $:
$$
A = \begin{pmatrix}
-1 & 1 & 0 & 0 \\
-1 & -3 & 0 & 0 \\
0 & 1 &-1 & 1 \\
1 & ...
- 1,091
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0
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Adding vectors to extend basis of Jordan chains?
The outline of a proof of existence of a basis of Jordan chains:
While extending $B_{u}$ to $B_{v}$ is to include what is in $Ker(S^{k+1})$ but not yet in $Ker(S^{k})$, what are those $w_{i}$ ...
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Show that the minimal polynomial is $M_{f}(t)=(t-\lambda_{1})^{d_{1}}...(t-\lambda_{k})^{d_{k}}$, $d_{i}$ is$(f|_{V_{i}}-\lambda_{i}I_{n})^{d_{i}}=0$.
I would like to show that the minimal polynomial is given by $M_{f}(t)=(t-\lambda_{1})^{d_{1}}...(t-\lambda_{k})^{d_{k}}$.
I use the following definition of the minimal polynomial:
Def.(Minimal ...
- 754
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1
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Is there a 4x4 real matrix that has the following Jordan Canonical Form
I'd like to know if there is a 4x4 matrix with real entries that can be transformed to the following matrix A. $x,y\in\mathbb{R}, i=\sqrt{-1}$,$$ \ $$ $A=\begin{bmatrix} (x+iy) & 1 & 0 & 0\...
- 33
1
vote
1
answer
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Real Eigenvalues and Similarity
Let $A$ be a complex $n \times n $ matrix.
a) Prove that if all the eigenvalues of $A$ are real, then $A$ is similar to a real matrix.
b) Classify up to similarity all the matrices $A$ such that $A^n=...
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Find a Jordan canonical form and a basis
Let $V=span_{\mathbb{R}} \{e^t, te^t, t^2e^t, e^{2t} \}$ and $T(f)=f'$.
Find a Jordan canonical form $J $ and a basis $\beta$.
For the characteristic polynomial, I got $(\lambda-1)^3(\lambda-2)$.
So $...
- 782
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1
answer
59
views
Let $M$ and $N$ be two $3\times 3$ matrices such that $MN=NM$, Further if $M\neq N^2$ and $M^2=N^4$ then
Let $M$ and $N$ be two $3\times 3$ matrices such that $MN=NM$, Further if $M\neq N^2$ and $M^2=N^4$ then
(A) Determinant of $(M^2+MN^2)$ is $0$
(B) There is a non-zero $3\times 3$ matrix $U$ such ...
- 1,563
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vote
1
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Find the Jordan canonical form and an invertible $Q$ such that $A=QJQ^{-1}$
$$ A = \begin{bmatrix}
-3 & 3 & -2 \\
-7 & 6 & -3 \\
1 & -1 & 2
\end{bmatrix} $$
The characteristic polynomial can be found to be $p(t)= -(t-1)(t-2)^2$. For
$t=1$, I have that ...
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0
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75
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How to show that the matrices $A$ and $B$ are similar?
Let $A, B$ be two $n \times n$ matrices over a field $F$ and $A,B$ have the same characteristic and minimal polynomials. If no eigenvalue has algebraic multiplicity greater than $6$ and the solution ...
- 1,495
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votes
2
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Is the inverse of a block matrix also a block matrix?
If $N$ is a nilpotent matrix then $N^t$ and $N$ are similar. Use the jordan form and this to prove that a complex matrix is similar to transpose.
Let $N$ be a $k \times k$ nilpotent matrix such that $...
- 3,288
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votes
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Let $A$ and $B$ be $n \times n$ matrices over a field $F$ which have the same characteristic and minimal polynomial
Let $A, B$ be two $n \times n$ matrices over a field $F$ and $A,B$ have the same characteristic and minimal polynomials. If no eigenvalue has algebraic multiplicity greater than $3,$ then $A$ and $B$ ...
- 3,288
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0
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How to calculate Jordan basis
I got a question regarding how to find a basis to Jordan matrix with this way I tried here.
I saw a lot of different ways here about that but I need in this way if its possible.
A represent T in the ...
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2
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$T: \mathbb{C}_{2020}[x]\to \mathbb{C}_{2020}[x]$, $\sum_{i=0}^{2020}a_ix^i \mapsto \sum_{i=0}^{2020}a_i(x-1)^i$
$T: \mathbb{C}_{2020}[x]\to \mathbb{C}_{2020}[x]$
$\sum_{i=0}^{2020}a_ix^i \mapsto \sum_{i=0}^{2020}a_i(x-1)^i$
I have to find the presentation matrix of $T$ in order to find Jordan form.
I tried to ...
- 1,549
0
votes
2
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What form is the complex Jordan form allowed to take?
Could you please explain to me why is the last matrix not included in $(5.32)$?
How is the last matrix different form the three matrices in $(5.32)$.
- 1,811
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49
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How to order the basis vectors that put a matrix in jordan canonical form
I have the following matrix A \begin{pmatrix} 1 & -3 & -1\\ 1 & 5 & 1 \\ -2 & -6 & 0 \end{pmatrix}
I want to find a basis that puts the matrix in Jordan canonical form. The ...
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3
votes
0
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54
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Prove that the rational form of a matrix remains the same over subfield
Let F be a subfield of the complex number.
How to show that the rational form of a matrix over complex is the same over F?
I think we need to use the cyclic decomposition theorem. Finding the rational ...
- 1,495
1
vote
2
answers
96
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Square roots of the basic Jordan block of order $n$ associated with the eigenvalue $1$
Let $F$ be a field. The basic Jordan block of order $n$ associated with the eigenvalue $1$ will be denoted by $J_n$ where
$$J_n=\begin{pmatrix}1&1&&\\&1&1\\ &&\ddots&\...
- 401
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votes
0
answers
25
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How can I "recreate" a nilpotent operator given a Jordan matrix with just zeroes on diagonal?
I have been given a problem where for an algebraically closed field $\mathbb{F}$, a vector space $\dim{V}=n\in\mathbb{N}^+$ ,
$$ 0 <n_1<n_2<\ldots<n_{k-1},n_k=n$$
A sequence of integers, ...
1
vote
0
answers
61
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Please tell me good linear algebra books which use the elementary divisors to prove the Jordan normal form theorem.
I want to read linear algebra books which use the elementary divisors to prove the Jordan normal form theorem.
Please tell me good linear algebra books which use the elementary divisors to prove the ...
- 6,071
2
votes
0
answers
59
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Finding Geometric Multiplicity and Size of Jordan block of Eigen Value
Let A be square matrix such that
$$|A-xI|=x^4×(x-1)^2×(x-2)^3$$
If $rank(A^3)=rank(A^4)<rank(A^2)$
then geometric multiplicity of Eigen value 0 is?
I found that rank of A = 9 - dim eigenspace(0).
...
0
votes
0
answers
27
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Find all possible Jordan forms with a given minimal polynomial
Let $f(x) = (x-5)^2(x^2-1)$. Find all possible Jordan forms for $7 \times 7$ matrices over $\mathbb{C}$ consiting of 5 elementary Jordan blocks whose minimal polynomial is $f(x)$.
I found three ...
- 403
11
votes
1
answer
321
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On the complex matrix equation $AX-XA=B$
I want to show that there exists solution to the matrix equation $AX-XA=B$ if and only if
$$
\begin{pmatrix}
A&0\\
0&A
\end{pmatrix},
\begin{pmatrix}
A&B\\
0&A
\end{pmatrix}
$$
are ...
- 164