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.

Filter by
Sorted by
Tagged with
1 vote
1 answer
34 views

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 ...
user avatar
  • 11
1 vote
1 answer
21 views

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, ...
user avatar
-1 votes
0 answers
10 views

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 $(λ−1)^3 (λ+1)$ ...
user avatar
  • 1
0 votes
1 answer
35 views

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−...
user avatar
  • 1
0 votes
2 answers
37 views

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 ...
user avatar
2 votes
1 answer
57 views

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 $A$-...
user avatar
0 votes
0 answers
36 views

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 $V$. ...
user avatar
0 votes
1 answer
37 views

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 ...
user avatar
0 votes
1 answer
52 views

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 ...
user avatar
  • 11.5k
0 votes
2 answers
30 views

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 ...
user avatar
  • 488
0 votes
0 answers
23 views

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 ...
user avatar
  • 1,187
1 vote
0 answers
23 views

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....
user avatar
  • 488
0 votes
1 answer
51 views

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 $0$ ...
user avatar
  • 403
0 votes
0 answers
61 views

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 ...
user avatar
  • 6,071
2 votes
0 answers
32 views

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 ...
user avatar
2 votes
1 answer
63 views

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 ...
user avatar
  • 1,187
1 vote
0 answers
32 views

$5 \times 5$ 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' ...
user avatar
0 votes
0 answers
25 views

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 ...
user avatar
3 votes
2 answers
54 views

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 ...
user avatar
1 vote
2 answers
52 views

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'(\...
user avatar
0 votes
0 answers
38 views

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 ...
user avatar
1 vote
0 answers
26 views

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{...
user avatar
1 vote
0 answers
58 views

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 $a=1$...
user avatar
  • 2,947
1 vote
0 answers
24 views

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}(\...
user avatar
1 vote
0 answers
16 views

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 ...
user avatar
1 vote
1 answer
48 views

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 & \...
user avatar
  • 427
1 vote
1 answer
35 views

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 \...
user avatar
  • 1,585
2 votes
1 answer
37 views

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 ...
user avatar
  • 193
1 vote
0 answers
33 views

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 & ...
user avatar
0 votes
0 answers
9 views

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 $w_{i}$ ...
user avatar
0 votes
0 answers
31 views

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 ...
user avatar
  • 754
0 votes
1 answer
61 views

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\...
user avatar
1 vote
1 answer
49 views

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=...
user avatar
  • 782
0 votes
0 answers
23 views

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 $...
user avatar
  • 782
2 votes
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 ...
user avatar
  • 1,563
1 vote
1 answer
65 views

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 ...
user avatar
  • 782
1 vote
0 answers
75 views

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 ...
user avatar
  • 1,495
3 votes
2 answers
67 views

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 $...
user avatar
  • 3,288
2 votes
1 answer
89 views

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 $B$ ...
user avatar
  • 3,288
0 votes
0 answers
19 views

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 ...
user avatar
0 votes
2 answers
64 views

$T: \mathbb{C}_{2020}[x]\to \mathbb{C}_{2020}[x]$, $\sum_{i=0}^{2020}a_ix^i \mapsto \sum_{i=0}^{2020}a_i(x-1)^i$

$T: \mathbb{C}_{2020}[x]\to \mathbb{C}_{2020}[x]$ $\sum_{i=0}^{2020}a_ix^i \mapsto \sum_{i=0}^{2020}a_i(x-1)^i$ I have to find the presentation matrix of $T$ in order to find Jordan form. I tried to ...
user avatar
  • 1,549
0 votes
2 answers
35 views

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 $(5.32)$.
user avatar
  • 1,811
0 votes
2 answers
49 views

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 ...
user avatar
  • 123
3 votes
0 answers
54 views

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 ...
user avatar
  • 1,495
1 vote
2 answers
96 views

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&\...
user avatar
0 votes
0 answers
25 views

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, ...
user avatar
1 vote
0 answers
61 views

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 ...
user avatar
  • 6,071
2 votes
0 answers
59 views

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). ...
user avatar
0 votes
0 answers
27 views

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 ...
user avatar
  • 403
11 votes
1 answer
321 views

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 ...
user avatar

1
2 3 4 5
25