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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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 ...
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1answer
43 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 ...
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1answer
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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 $....
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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 ...
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1answer
63 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 $J_{A}$...
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1answer
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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 ...
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1answer
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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{...
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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
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1answer
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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) ...
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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 ...
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2answers
53 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 ...
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0answers
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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 ...
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1answer
24 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$ ...
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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 ...
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1answer
36 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,
$$
where $D$ is semisimple (diagonalizable), $N$ is nilpotent and $DN=ND$. Although it is stated in many sources that $D$ and $N$ ...
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For which values the matrix is diagonalizable
For which values of $a$ matrix $A$ is diagonalizable?
$$A = \pmatrix{0&i\\i&a}$$
in the case that it is not diagonalizable determine a base of Jordan
Attempt: The minimal polynomial ...
1
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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 ...
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1answer
52 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 ...
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1answer
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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
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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<...
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1answer
58 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 ...
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$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
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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
46 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 ...
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1answer
54 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}
.
$$
Find the Jordan Form, $J$, of this matrix, and an invertible matrix $Q$ such that $A = QJQ^{-1}$. I have ...
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vote
1answer
35 views
Matrix Exponential Jordan Form Linear System
Given the Linear System $\dot{x}(t)=A x(t)$ with $x_0=(x_{01},x_{02})$ as initial state and $A=\begin{pmatrix} 0 & 1 \\ -k/M & -h/M \end{pmatrix}$, when $h^2=4Mk$ the matrix A has a single ...
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votes
1answer
27 views
Power of a Jordan Normal Form
In my notes I have that the Jordan normal form of $B^2$ is $$\begin{bmatrix}
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 &1 \\
0 & 0 & 0 &...
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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+...
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2answers
53 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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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 ...
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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 ...
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3answers
55 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) = ...
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0answers
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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 ...
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1answer
46 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
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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
70 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 ...
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1answer
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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 ...
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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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2answers
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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 ...
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0answers
74 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
51 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 &...
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2answers
63 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 ...
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1answer
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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
24 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
104 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
votes
1answer
112 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 ...
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0answers
52 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&...
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1answer
24 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).
...
