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.
4
votes
2answers
180 views
Proof involving the spectral radius and the Jordan canonical form
Let $A$ be a square matrix. Show that if $$\lim_{n \to \infty} A^{n} = 0$$ then $\rho(A) < 1$, where $\rho(A)$ denotes the spectral radius of $A$.
Hint: Use the Jordan canonical form.
I am ...
1
vote
1answer
18 views
all 2 dimensional invariant subspaces
How we can find all 2 dimensional invariant subspaces of
\begin{pmatrix}
2 & 1 & 0 \\
0 & 2 & 0 \\
0 & 0 & 8
\end{pmatrix}
I know that there are at least 2 such subspaces, ...
0
votes
2answers
42 views
Computing the matrix exponential for a Jordan matrix
How can I compute $e^{At}$ where $A = J_{3}(5)$? That is,
$$A = \begin{pmatrix} 5 & 1 & 0 \\ 0 & 5 & 1 \\ 0 & 0 & 5 \end{pmatrix} $$
Using this, how can I write down a basis ...
1
vote
0answers
44 views
Given a Jordan canonical basis, how to find out to which generalized eigenspace picked generalized eigenvector belongs
Suppose we have finite-dimensional linear operator $A:V\to V$ , that has eigenvalues $\lambda_1 ,\lambda_2, ... \lambda_n$ . It is known that we can decompose $V$ into direct sum of generalized ...
0
votes
1answer
12 views
$Ch_A =(x+1)^6(x-2)^3 $ y $min_A = (x+1)^3(x-2)^2 $, List the possible Jordan forms for $A$
Let $A$ be a complex matrix such that $Ch_A =(x+1)^6(x-2)^3 $ y $min_A = (x+1)^3(x-2)^2 $, List the possible Jordan forms for $A$. And in each case write the corresponding rational
I do not know ...
0
votes
1answer
32 views
Deducing the additive Jordan decomposition
In $M_n(\Bbb C)$, I could prove that the additive Jordan decomposition of $X=D+N$ with $D$ diagonalizable and $N$ nilpotent gives a multiplicative Jordan decomposition $e^X=e^De^N$.
Is that true the ...
1
vote
1answer
49 views
When $A^i=A^j$ for $i,j\geq 0$ such that $i\neq j$ for matrix $A$ over algebraic closed field?
Let $A\in M_n(K)$ be a matrix over algebraic closed field $K$, where $n>1.$
When $A^i=A^j$ for $i,j\geq 0$ such that $i\neq j$?
I tried solve it by Jordan form of matrix $A;$ is is sufficient ...
1
vote
1answer
41 views
Converting Jordan Normal Form into Real Jordan Form
Given the matrix
$$\begin{bmatrix}
0 & 0 & 0 & -8\\
1 & 0 & 0& 16 \\
0 & 1 & 0 & -14 \\
0 & 0 & 1 & 6 \\
\end{bmatrix}$$
...
1
vote
0answers
20 views
Is Jordan cardinal form the matrix having the most zeros in the equivalent class of similarity? [duplicate]
I am concerned on this interesting question
Given matrix $A$, does the Jordan cardinal form have the most zeros, in other word, it has the least nonvanishing indices, among the equivalent class of ...
0
votes
0answers
76 views
How do I complete the steps of finding the Jordan of this $5\times 5$ matrix (with Octave)?
I know how to begin the procedure but I don't know how to finish it. Let's start with an example (sorry for it being so unwieldy).
Let
$$A =\begin{pmatrix}
177& 548& 271& -548& -...
0
votes
1answer
41 views
Generalised Eigenvectors Issue
Ok so i have been doing a few questions on 'Diagonalising' defective matrices, the method I've been using to find generalized Eigenvectors is to make the previous Eigenvector the subject. However i ...
1
vote
1answer
33 views
Decompose an invertible matrix into an exchangeable product of diagonalizable matrix and a matrix with all the eigenvalues equal to $1$
Let $ g $ be an invertible $ n\times n $ complex matrix. Show that $ g $ can be written as
$$ g=su=us ,$$
where $ s $ is diagonalizable and all eigenvalues of $ u $ are equal to $ 1 $.
My ...
0
votes
1answer
13 views
Can there be just one eigenvalue on the invariant subspace (generalized eigenspace) associated with an eigenvalue?
In the proof of Jordan decomposition here, once I know that an indecomposable subspace $V$ is of the form $V=Ker((f-\lambda Id)^n)$, can there be an other eigenvalue $\mu$ for $f\vert_V$?
2
votes
3answers
132 views
Intuition of generalized eigenvector.
I was trying to get an intuitive grasp about what the the generalized eigenvector intuitively is. I read this nice answer, so I understand that in the basis given by the generalized eigenvectors, a ...
0
votes
0answers
22 views
The theory behind linear recurrence relations solving (or - why does it work?)
tl;dr - a recommendation for a good book that explains the theory behind the auxiliary polynomial/companion matrix methods to solve linear recurrence relations with constant coefficients?
I've bumped ...
1
vote
1answer
51 views
Show that matrix $A$ is similar to a matrix $B$ with elements on diagonal $(0, …, 0, \operatorname{Tr(}A))$ respectively.
Let $A$ be a matrix $n \times n, n \geq 2 $. Let's assume that not all entries outside of the diagonal are zeros (we don't know what entries are on the diagonal). Show that matrix $A$ is similar to a ...
2
votes
1answer
54 views
Let's assume that $ XA = AX $. Show that there is such a matrix $M$ that $ p_A(X) = M(A-X), MA=AM$ and $ MX=XM $.
Let $ A, X \in M_{nxn}(K) $. Let $ p_A(t) $ be a characteristic polynomial of matrix A. Let's assume that $ XA = AX $. Show that there is such a matrix $M$ that $ p_A(X) = M(A-X), MA=AM$ and $ MX=XM $....
0
votes
1answer
24 views
Exponential of a Jordan Block using Cayley-Hamilton Theorem
For the sake of simplicity, I will only consider $2\times2$ matrices. The Cayley-Hamilton theorem allows us to conclude that
$$e^{At} = \alpha_0I + \alpha_1 A$$
where $\alpha_0$ and $\alpha_1$ can ...
6
votes
1answer
71 views
Find Jordan normal form and basis
Let $$A=M(\varphi)^{st}_{st}={\begin{bmatrix}0&1&1\\-4&-4&-2\\0&0&-2\end{bmatrix}}$$ and $ \varphi: \mathbb R^{3} \rightarrow \mathbb R^{3}$. Find the Jordan normal form $J_{A}$...
0
votes
1answer
48 views
How do I find the bases of the Jordan Canonical Form of $C$?
Let $$C = \left[ {\begin{array}{cccc}
0 & -1 & -2 & 3 \\
0 & 0 & -2 & 3 \\
0 & 1 & 1 & -1 \\
0 & 0 & -1 & 2
\end{array} } \right].$$ What ...
0
votes
1answer
22 views
Power of a matrix in REAL jordan form
Given a $2\times 2$ matrix in Jordan canonical form, whose eigenvalues are a couple of complex conjugate values
$$ J = \left[ \begin{array}{cc} \sigma+j\omega & 0 \\ 0 & \sigma - j\omega \end{...
0
votes
0answers
23 views
Jordan form of a matrix and finitely generated modules over PIDs.
We know that any square matrix of order n over Complex numbers is similar to a Jordan form,I am told that this relates to the structure theorem for finitely generated modules over PIDs. Can anyone ...
2
votes
1answer
29 views
Exericse about linear map $T\in L(V)$, where $\dim V=n\geq2$, with $\operatorname{null}T^{n-1}\neq\operatorname{null}T^n$
I have this problem that I am attempting, and am struggling with (b).
--
Assume $\dim V = n \geq 2$ and that $T \in L(V)$ such that $\operatorname{null}T^{n-1}\neq\operatorname{null}T{^n}$
--
(a) ...
1
vote
1answer
42 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 ...
1
vote
2answers
57 views
Why does $(A- \lambda I)^2 =0$ if A has two repeated eigenvalues?
This statement appears in my textbook as part of an introduction of the method for finding the Jordan form of a $2 \times 2$ matrix. I understand what it says but I'd really like to know where is it ...
1
vote
0answers
31 views
Proving that a matrix is nonnegative if its powers are nonnegative
I am working on a problem involving doubly stochastic matrices where I must prove that $P$ is doubly stochastic if and only if $P^k$ is doubly stochastic for $k = 1, 2, ...$ It is easy to show that if ...
1
vote
1answer
26 views
Convergence of powers of matrix given convergence of the powers of its absolute value.
I have a matrix A and a matrix B such that $B_{i, j} = |A_{i, j}|$. I am given that all of the eigenvalues of B have magnitude less than 1, and therefore: $ \displaystyle \lim_{k \to \infty} B^k = 0$ ...
0
votes
0answers
18 views
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 ...
0
votes
1answer
37 views
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 $N$ ...
2
votes
0answers
20 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 ...
1
vote
1answer
33 views
$A$ 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 ...
1
vote
1answer
55 views
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 ...
0
votes
1answer
30 views
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 ...
1
vote
1answer
34 views
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<...
1
vote
1answer
59 views
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 ...
0
votes
0answers
22 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 ...
1
vote
1answer
13 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 ...
2
votes
1answer
50 views
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 ...
0
votes
1answer
64 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
48 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
votes
1answer
28 views
Power of a Jordan Normal Form
In my notes I have that the Jordan normal form of $B^2$ is $$\begin{bmatrix}
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 &1 \\
0 & 0 & 0 &...
0
votes
1answer
34 views
Complement space of a finite dimensional space over a finite field
Let $V$ be a finite dimensional space over the field $\mathbb{F}_q$ of $q$ elements and let $U\subset V$ a subspace of $V$. How many subspaces $W\subset V$ are there such that $W\cap U = 0 $ and $V=W+...
1
vote
2answers
58 views
Finding the Jordan Form of a matrix…
I know that this type of question has been asked on here before but I am still having a hard understanding what is going on. The text that I am learning from is "Linear Algebra Done Right by Sheldon ...
0
votes
0answers
19 views
Invariant factors and Jordan reduction : how to find the adapted basis?
I have a question about the Jordan reduction using the module theory and especially the invariant factors. If we have a vector space $E$ of dimension $n$ over a field $k$, and $f \in End(E)$ then it's ...
1
vote
1answer
43 views
Calculation of Jordan Normal form to determine similarity
I wanted to find a way to tell whether or not two matrices are similar.
Of course, first you check that similarity isn’t ruled out by matrices having different determinants, eigenvalues, trace.
Is ...
2
votes
3answers
58 views
find the Jordan basis of a matrix
I'm trying to find the Jordan basis of the matrix $$A =\begin{bmatrix} 8 & 1 & 2 \\ -3 & 4 & -2\\ -3 & -1 & 3\end{bmatrix}$$ I've got the characteristic equation to be $CA(x) = ...
3
votes
0answers
42 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 ...
0
votes
1answer
50 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 $\...
1
vote
2answers
52 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
83 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 ...
