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.
338
questions with no upvoted or accepted answers
7
votes
0
answers
191
views
Find a flag to transform a matrix to an upper triangular one
Consider $F: \mathbb{R^3} \to \mathbb{R^3}$ represented by:
$ A=
\begin{bmatrix}
1 & 1 & 2 \\
-2 & 5 & 6 \\
1 & -2 & -2 \\
\end{bmatrix}
$
, eigenvalues: $...
7
votes
0
answers
804
views
Jordan Block of Kronecker Product
Let $A$ be a $(p\times p$)-Jordan block of generalized eigenvalue $\lambda$. Let $B$ be a $(q\times q$)-Jordan block of generalized eigenvalue $\mu$. Then what is the Jordan canonical form for $A\...
7
votes
2
answers
182
views
Can basis of kernel be extended to a Jordan basis?
Let $A\in\mathbb C^{n\times n}$ be nilpotent. A Jordan basis of $A$ is a basis of $\mathbb C^n$ with respect to which $A$ has Jordan normal form. Assume that we do not know the Jordan structure of $A$....
6
votes
0
answers
2k
views
An inverse of Jordan matrix - basis
Let $A\in M_{n\times n}$ be and invertible matrix over complex field and we assume it's already at Jordan form where $B=\{v_1,…,v_n \}$ is Jordan basis for A.
Find Jordan form and Jordan basis for $...
5
votes
0
answers
277
views
Conjecture: Any two matrices of size $n×n$ with same characteristic and minimal polynomial are similar implies $n\le 3$.
Notations:
$\mathcal{M}_n(\Bbb{R}) $: the set of all $n×n$ matrices over $\Bbb{R}$
$\chi_A(x)$: Characteristic polynomial of $A$
$m_A(x)$ : Minimal polynomial of $A$
$A\sim B$ : $\exists P\in Gl_n(\...
5
votes
0
answers
69
views
Different approaches to Jordan Canonical Form
I know two different proofs of the existence of JCF. Let $V$ be a finite-dimensional vector space over base field $\mathbb{C}$ and $\alpha \in \mathsf{End}_{\mathbb{C}}(V)$.
Given transformation $\...
5
votes
2
answers
63
views
Have I found the Jordan form correctly?
I am given that the minimal polynomial and characteristic polynomial of a matrix are both $(x-1)^2(x+1)^2$. I have found the Jordan form to be $$\begin{bmatrix}1&1&0&0\\0&1&0&0\...
5
votes
0
answers
307
views
Lost on rational and Jordan forms
I'm having a lot of trouble trying to understand rational canonical form, primary rational canonical form, and Jordan form. I've looked at the posts about this, but I haven't been able to understand ...
4
votes
2
answers
1k
views
Calculate the Jordan normal form
I have the matrix $A=\begin{bmatrix} -2 & -3 & 6 \\ 1 & 2 & -2\\ -1 & -1 &3 \end{bmatrix}$ and I have to find the transformation matrix and its Jordan normal form.
This is ...
4
votes
1
answer
388
views
Why block-diagonal form for nilpotent matrices?
I am currently reading Jim Hefferon's Linear Algebra.
In chapter 5, nilpotence, strings, he goes through the process of finding a string basis of a map, and proves that there exists a string basis ...
4
votes
0
answers
3k
views
Prove the direct sum of generalized eigenspaces is the whole vector space
Given a $n\times n$ matrix $A$ over an algebraically closed field, let $\lambda_1,...,\lambda_k$ be its eigenvalues, and let $V_{\lambda_i}$ be the generalized eigenspace of $\lambda_i$. The question ...
4
votes
0
answers
2k
views
Differential Equations: Jordan Form of a Matrix
I am using Lawrence Perko's book Differential Equations and Dynamical Systems, for my Differential Equations course.
At the moment we are going over Jordan Forms of a linear system $x^{'}(t) = Ax$, ...
4
votes
0
answers
1k
views
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 ...
3
votes
0
answers
46
views
Is there some geometric understanding of a Jordan normal matrix?
We know that an eigenvalue decomposition of a matrix is to find those eigenvectors that are just scale for some coefficients. But what about Jordan matrix decomposition? I just learn how it is ...
3
votes
0
answers
112
views
Number of similarity classes of matrices $A$ in $M_6(\mathbb{R})$ satisfying $(A-2I)^3=0.$
Find the number of similarity classes of matrices $A$ in $M_6(\mathbb{R})$ satisfying $(A-2I)^3=0.$
If $(A-2I)^3$ then the characteristic polynomial is $p_A(x)=(x-2)^6$ and we have 3 posibilities for $...
3
votes
0
answers
154
views
Number of Jordan Blocks
Determine the Jordan Normal Form over $\Bbb Q$ of the following matrix.
$$A=\begin{bmatrix}0&1&1&1&0\\0&2&0&0&0\\-1&3&0&0&0\\0&-1&1&0&1\\...
3
votes
0
answers
72
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 ...
3
votes
0
answers
85
views
What is the fastest way to find the characteristic polynomial of $4\times4$ matrix and its change of basis matrix?
Let:
$$A=\begin{pmatrix}
3 & 0 & -2 & -3\\
4 & -8 & 14 & -15\\
2 & -4 & 7 & -7\\
0 & 2 & -4 & 3
\end{pmatrix}$$
Find a change of basis matrix $P$ such ...
3
votes
0
answers
41
views
Jordan form with -1 in block
I'm having troubles finding the right base, I keep getting a weird Jordan form.
I need to find the Jordan form and basis of $$ \left[\begin{array}{cc}
1 & 1\\
-1 & 3
\end{array}\right] $$
this ...
3
votes
0
answers
137
views
Confusion about the Jordan-Chevalley/Dunford decomposition in $\mathbb{R}$, example of a rotation (solved!!!)
I'm writing some notes on Jordan-Chevalley decompositions in which I want to treat both the real and complex case in one statement.
One could of course write as in the french wikipedia article: an ...
3
votes
0
answers
95
views
Upper bound of the remainder term of $\int_0^t e^{A \tau} d\tau$ using Lagrange Remainder of the Taylor series
$\newcommand{\mat}[1]{\begin{bmatrix}#1\end{bmatrix}}$
$\newcommand{\norm}[1]{\left\lVert#1\right\rVert}$Let $A$ be an $n\times n$ real matrix which has real block diagonal form and each block has a ...
3
votes
0
answers
70
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 ...
3
votes
0
answers
63
views
Find the Jordan Canonical Form of the given transformation
The transformation here is $T(f(x)) = f(x + 1) + f(x − 1)$ which is a linear endomorphism on V, where $V={f(x) ∈ R[x] : deg f(x) ≤ 2017}$
So I have to find the jcf J of T. Along with a basis of B.
$...
3
votes
1
answer
612
views
Minimal polynomial and possible Jordan forms
Let $A$ be an $8\times 8$ complex matrix with characteristic polynomial $$p_A(x)=(x-1)^4(x+2)^2(x^2+1)$$ and minimal polynomial $$m_A(x)=(x-1)^2(x+2)^2(x^2+1).$$ Determine all possible Jordan ...
3
votes
0
answers
268
views
Classification of bilinear forms: operator $A^{-1} A^T$ for bilinear form $A$
I would like to understand a classification of non-degenerate (not necessary symmetric or skew-symmetric) bilinear forms over an algebraically closed field via an operator $\kappa=A^{-1} A^T$ for a ...
3
votes
0
answers
410
views
Jordan normal form theorem proof question
Theorem:
Assume that the characteristic polynomial $x_f$ splits into linear factors. Then there exists a Jordan normal form for f. The Jordan normal form is unique up to the order of the Jordan blocks....
3
votes
0
answers
45
views
Jordan basis of $\mathcal{M}_{\mathcal{T}}(A)$
Let $A\in M_{n\times n}(\mathbb{R})$ be a matrix. Let $\mathcal{B}$ be a basis of $\mathbb{R}^n$ and $X:=\mathcal{M}_{\mathcal{B}}(A)$. If $\mathcal{S}$ is the basis for which $\mathcal{M}_{\mathcal{S}...
3
votes
1
answer
983
views
relation between minimal polynomial and jordan normal form
I just solved some exercises on minimal polynomials and i remember that there is a relation between the minimal polynomial and the jordan normal form.
But my question is the following : knowing the ...
2
votes
1
answer
49
views
Finding all possible Jordan forms from the Characteristic polynomial
Let A be a 7 x 7 matrix with characteristic polynomial $(t − 2)^4(3 − t)^3$. It is
known that in the Jordan form of A, the largest blocks for both the eigenvalues are of order 2. Show that there are ...
2
votes
0
answers
50
views
Finding all invariant subspaces of $T_A:\mathbb{C}^n \to \mathbb{C}^n$ where $A=J_{n_1}(\lambda_1)\oplus ... \oplus J_{n_k}(\lambda_k)$
Let $A=J_{n_1}(\lambda_1)\oplus ... \oplus J_{n_k}(\lambda_k)\in \text{Mat}_n(\mathbb{C})$ when $\lambda_i\neq\lambda_j$ for every $i\neq j$. I need to describe all the $T$-invariant subspaces of $T=...
2
votes
0
answers
46
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 ...
2
votes
0
answers
37
views
Switch from 1 to random real in Jordan Decomposition
Context : Let's suppose $L$ is a linear map from $\mathbb{R^k}\rightarrow \mathbb{R}^k$ , $k$ strictly positive integer. Let's suppose $\epsilon$ is a strictly positive real.
In an exercice , i have ...
2
votes
0
answers
66
views
Finding generalized eigenvectors of a matrix
I would like to know how to find the generalized eigenvectors to the following matrix $A$, so that I can express $A$ as $PJP^{-1}$. $$ A = \begin{bmatrix} 1 & -3 & 1\\ 1 & 5 & -1\\2 &...
2
votes
0
answers
104
views
Connection between uniqueness of Jordan normal form and Jordan–Chevalley decomposition
Let $f$ be an endomorphism of a finite-dimensional vector space $V$.
Definition.
A Jordan–Chevalley decomposition of $f$ is a decompositon $f = d + n$ where $d$ and $n$ are endomorphisms of $V$ such ...
2
votes
0
answers
175
views
Which of the following options are correct about $X = \left \{C \in GL_3 (\Bbb R)\ |\ CAC^{-1}\ \text {is triangular} \right \}\ $?
Let $A \in M_3 (\Bbb R)$ and let $X = \left \{C \in GL_3 (\Bbb R)\ |\ CAC^{-1}\ \text {is triangular} \right \}.$ Then
$(1)$ $X \neq \varnothing$
$(2)$ If $X = \varnothing,$ then $A$ is not ...
2
votes
0
answers
136
views
Looking for an intuition of the definition of Generalized Eigenspaces
The eigenspace of (a square matrix) $A$ corresponding to $\lambda$ is the collection of all vectors $\mathbf{x}$ that satisfy $A\mathbf{x}=\lambda\mathbf{x}$, or equivalently, $(A-\lambda I)\mathbf{x}=...
2
votes
0
answers
92
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. ...
2
votes
1
answer
103
views
Is it true that for jordan block with zero eigenvalue we can choose basis where all diagonal elements are non zero?
Is it true that for jordan block with zero eigenvalue we can choose basis where all diagonal elements are non zero?
if there is a proper number 0, then you can try to find a matrix in the form of J^(-...
2
votes
0
answers
280
views
Bounding 2-norm of powers of a matrix
Suppose that $A$ is a $n \times n$ matrix with $\rho(A) \leq 1$ and $\|A\|_2 \leq R$, where $R>1$. How can I show an upper bound on $\|A^k\|_2$ that is polynomial in $k$? A trivial upper bound is ...
2
votes
0
answers
174
views
Jordan normal form of upper bidiagonal matrix
I am trying to find the Jordan normal form of a general matrix form, which is an $N \times N$ real upper bidiagonal matrix with non-zero diagonal entries and rows that sum to one:
$$\mathbf{M} = \...
2
votes
0
answers
149
views
Jordan Canonical Form of Nilpotent matrix
Let $A\in \mathbb{C}^{n\times n}$ be a nilpotent matrix. I want to find a nonsingular matrix $P$, s.t. $P^{-1}AP$ is in Jordan canonical form. Denote $F_i=N(A^i).$ (where $N(A)$ is the null space of $...
2
votes
0
answers
81
views
Properties of blocks in blockmatrix A if the matrix pair (E,A) is regular
I am working on a problem from the book "Differential-Algebraic Equations: Analysis and Numerical Solution" by Kunkel, Mehrmann. It is the Exercise 3 from Page 53 concerning matrix pairs $(E,A)$ and ...
2
votes
0
answers
98
views
How many similarity classes are zeroes of a given polynomial.
I am looking for an easy way of calculating the number of similarity classes of complex matrices that satisfy some polynomial $p(t)$.
As an example consider $p(t)=(t-1)^3(t+1)^4$ and $5\times 5$ ...
2
votes
0
answers
75
views
Analogy of Jordon Normal Form for Antilinear Maps
Given complex vector spaces $V$, and antilinear $T:V \rightarrow V$, then if we fix a basis of $V$, we can represent $T$ by the matrix of the linear $T \circ J$, where $J$ is complex conjugation.
I ...
2
votes
0
answers
135
views
How can I find the Jordan form of this upper triangular Toeplitz matrix?
Given an $n \times n$ matrix $A$ whose $(i,j)$ entry is
$$a_{ij} = \begin{cases} n-j+i & \text{if } j \geq i\\ 0 & \text{otherwise}\end{cases}$$
find its Jordan form.
I know that all the ...
2
votes
0
answers
38
views
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 ...
2
votes
0
answers
339
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 ...
2
votes
0
answers
235
views
Uniqueness of Jordan-Chevalley-Decomposition - why do the nilpotent matrices commute?
The Jordan-Chevalley-Theorem states that for a given endomorphism $f\in \mathrm{End}_K(V)$ of a $K$-vector space $V$, such that the characteristic polynomial $\chi_f$ splits into factors, there exist ...
2
votes
0
answers
173
views
Square root of a matrix by looking at the eigenspaces
A problem that is well known in linear algebra is the existence of square root of a matrix, where square root of matrix $A \in M_n(\mathbb F)$ is defined to be $K \in M_n(\mathbb F)$ such that $K^2=A$
...
2
votes
0
answers
81
views
Trouble in proof of Jordan Canonical Theorem
Jordan Canonical Theorem stated that:
Let $K$ be an algebra closed field. Let $V$ be a nonzero, finite dimensional vector space over $K$, and let $\psi \in \operatorname{End}_K(V)$. Then there ...