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.
1,144
questions
1
vote
1answer
54 views
$V/\ker(T-5I)$ is nilpotent.
A linear operator $T$ on a complex vector space $V$ has characteristic polynomial $x^3(x-5)^2$ and minimal polynomial $x^3(x-5)$. Choose correct options.
The operator induced by $T$ on quotient ...
3
votes
2answers
67 views
Prove: $\det(I+A) = 2^{\text{rank}(A)}$ if $A$ is a square idempotent matrix. Find $(I+A)^{-1}$ such that the expression doesn't have inverses.
To prove:
$\det(I+A)$ = $2^{\operatorname{rank}(A)}$ if $A \in$ $\mathbb{R}^{n\times n}$ and $A^{2}=A$.
Find an expression for $(I+A)^{-1}$ such that it does not involve inverses.
Is there any way I ...
0
votes
3answers
56 views
Jordan decomposition of Idempotent matrix.
Matrix A $\in$ $\mathbb{R}^{n\times n}$ is idempotent if $A^{2} = A$. Describe the Jordan form of A.
How do I do this?
I am able to decompose a matrix to its Jordan form given that the matrix contains ...
0
votes
0answers
37 views
Prove than a square matrix $A$, with complex entries, is diagonalizable if and only if the minimal polynomial of $A$ has distinct roots.
Question: Prove than a square matrix $A$, with complex entries, is diagonalizable if and only if the minimal polynomial of $A$ has distinct roots.
In this answer Prove that T is diagonalizable if and ...
0
votes
1answer
32 views
Let $A$ be nilpotent such that $A^{n-1}\neq 0$. Show that $A$ has exactly one Jordan Block.
Question: Let $A$ be $n\times n$ and nilpotent such that $A^{n-1}\neq 0$. Show that $A$ has exactly one Jordan Block.
My Attempt: Since $A$ is nilpotent, $A^k=0$ some $1\leq k\leq n$. So, the ...
1
vote
1answer
18 views
Suppose $T^2=0$. Prove that the Jordan canonical form of $T$ consists of $\dim(\ker T)$ Jordan blocks, $\dim(Im T)$ of which are $2\times 2$ blocks.
Question: Let $T:V\rightarrow V$ be a linear transformation satisfying $T^2=0$. Prove that the Jordan canonical form of $T$ consists of $\dim(\ker T)$ Jordan blocks, $\dim(Im T)$ of which are $2\...
2
votes
1answer
31 views
Question about relating the minimal polynomial, characteristic polynomial, and Jordan Canonical form of a matrix together.
Question: I want to determine the JCF for a matrix with characteristic polynomial $(x-3)^5$ and minimal polynomial $(x-3)^3$. So, I know we have a $5\times 5$ matrix by the degree of char poly, and I ...
1
vote
0answers
15 views
Making Off-Diagonals of Jordan Normal Form Arbitrarily Small
I know that the off-diagonals of the Jordan blocks for a matrix can be made arbitrarily small, i.e. we can find a similarity transformation such that the Jordan blocks look like this (as an example):
$...
1
vote
1answer
36 views
Determining the transformation matrix for Jordan normal form.
Let $$A=
\begin{pmatrix}
2 & 0 & -4 & -4 \\
0 &4&2&3\\
2&0&8&4\\
-1&0&-2&2\\
\end{pmatrix}$$
I want to find Jordan normal form of $A$ and the ...
1
vote
0answers
40 views
About Jordan normal form and stability in GIT
I am preparing linear algebra exam and I met a problem about Jordan normal form.
Suppose $V$ is a vector space over field $\mathbb K$, $\psi$ is a linear transform on $V$, the char polynomial and ...
0
votes
1answer
37 views
Why is the algebraic multiplicity the dimension of the kernel of $(A-\lambda I)^q$, where $\ker(A-\lambda I)^q=\ker(A-\lambda I)^{q+1}$?
$\DeclareMathOperator{\Ker}{Ker}\DeclareMathOperator{\id}{Id}$I know the proof of the existence of Jordan normal form. A large part of it rests on the following:
$$\exists q\in\Bbb{N}:\forall r\in\Bbb{...
1
vote
0answers
33 views
What is the link between this article's "relative bases" and the Jordan basis?
$\DeclareMathOperator{\Ker}{Ker}\DeclareMathOperator{\id}{Id}\DeclareMathOperator{\N}{\mathcal{N}}$
I make reference to this paper, which I have noticed, as I've been taking notes from it, struggles ...
1
vote
4answers
92 views
Why do successive compositions of nilpotent operators strictly decrease their rank?
$\DeclareMathOperator{\N}{\mathcal{N}}\DeclareMathOperator{\Ker}{Ker}\DeclareMathOperator{\id}{Id}\DeclareMathOperator{\rk}{Rank}$It is important in my reading of a formal proof of the existence of ...
1
vote
1answer
20 views
Question about finding 2 non-similar matrices
I need to find two 8x8 matrices $A,B$ with the same minimal & characteristic polynomials and same algebraic multiplicity for every eigenvalue.
I was thinking about something like $A=J_3(0), J_2(0),...
3
votes
0answers
31 views
What is the fastest way to find the characteristic polynomial of $4\times4$ matrix and its change of basis matrix?
Let:
$$A=\begin{pmatrix}
3 & 0 & -2 & -3\\
4 & -8 & 14 & -15\\
2 & -4 & 7 & -7\\
0 & 2 & -4 & 3
\end{pmatrix}$$
Find a change of basis matrix $P$ such ...
2
votes
1answer
37 views
Obtaining the Jordan canonical form from the characteristic and minimal polynomials
Let's say $A$ is a $6\times6$ matrix with characteristic polynomial $p(x)=(x-\lambda_1)^2(x-\lambda_2)^4$, for example, when $\lambda_1\neq\lambda_2$. I have a doubt about obtaining all possibilities ...
0
votes
1answer
39 views
Diagonal Matrix and Minimal Polynomial [closed]
If A is diagonalizable matrix , is it true that the Minimal polynomial is like $$(x-\lambda_1)^1\cdot(x-\lambda_2)^1\cdots(x-\lambda_n)^1$$ when $$(\lambda_i)$$ must be different ?
because if $$(\...
0
votes
1answer
68 views
A question about basis for $\ker \varphi ^{k}$
Let $V$ be a finite-dimensional complex vector spaces , and let $\varphi:V\rightarrow V$ be a linear operator such that
$$\begin{matrix}
\mathbf{a_{1}},& \mathbf{a_{2}}, & \mathbf{a_{3}} &...
0
votes
1answer
40 views
Jordan canonical form with more than 1 eigenvector
I need to solve: $$ v'=\left(\begin{array}{ccc}
5 & -1 & -2\\
-4 & 5 & 4\\
4 & -2 & -1
\end{array}\right)v $$
After finding the characteristic polynomial and finding the ...
3
votes
0answers
35 views
Jordan form with -1 in block
I'm having troubles finding the right base, I keep getting a weird Jordan form.
I need to find the Jordan form and basis of $$ \left[\begin{array}{cc}
1 & 1\\
-1 & 3
\end{array}\right] $$
this ...
0
votes
2answers
43 views
find value of $x$ in a way that the matrix has square root with real entries
I want to find all possible real values of $x$ in a way that $X=\begin{pmatrix}x & -x & -1 & 0 \\x & -x & 0 & -1 \\ 1 & 0 & x & -x\\ 0 & 1 & x & -x
\...
1
vote
2answers
82 views
Generalized Jordan canonical form
Suppose $\mathbb F_q$ is the finite field of order $q$. Let $f(x)=x^d-a_{d-1}x^{d-1}-\cdots-a_{1}x-a_0\in\mathbb F_q[x]$ be irreducible with $\deg (f(x))=d$.
Prove that we can find a basis $\{e_1,...,...
2
votes
0answers
32 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 ...
1
vote
0answers
27 views
Jordan canonical form and rational canonical form for $n \times n$ matrix $A$ such that $A^2 = A$.
I am trying to solve the following problem using both Jordan canonical form and rational canonical form:
let $A$ and $B$ be $n\times n$ matrices with entries in a field. Suppose that $A^2 = A$ and $B^...
3
votes
1answer
62 views
Show $A^k$ doesn't converge to $0$
Let $A \in \mathbb C^{n \times n}$. Let $r=\{\max \lvert \lambda \rvert \text{ such that }\lambda \in \mathbb C \text { is an eigenvalue of } A\}$. If $r\geq 1$ show that $A^k$ doesn't converge to the ...
3
votes
2answers
108 views
What is wrong with my derivation of Jordan normal form?
EDIT:
A key error in my working is that it allows $M$ to be singular! I tried to solve $AM=MJ$, not realising that $M^{-1}AM=J$ requires $M^{-1}$ to exist and be completely linearly independent!
OP:
...
2
votes
1answer
42 views
Help finding the Jordan normal form in a university problem - how to factor a cubic characteristic polynomial?
EDIT: I ran the polynomial in Wolfram Alpha, and it gave the roots as decimal approximations, with its "exact form" presentation being a hugely complex expression with many nested radicals, ...
1
vote
1answer
74 views
Characteristic polynomial of an endomorphism over 2x2 matrices
I am given the endomorphism over the vector space of 2x2 matrices Mat$_2(\mathbb{R})$ defined by
$$ f(X) = \begin{pmatrix}2&2\\0&2\end{pmatrix}X.$$
I am intending to find the characteristic ...
0
votes
1answer
41 views
Does a nilpotent operator always have a Jordan representation?
(Supposing the Jordan representation exists) Since a Jordan block associated to $\lambda$ has the form $J= \lambda I + N$, where $I$ is the identity matrix and $N$ a nilpotent matrix. One could find a ...
2
votes
1answer
79 views
Matrix power of $e$ with complex e-values
Calculate $e^A$ where $$A = \begin{bmatrix}1&0&3\\-1&2&0\\0&1&-1\end{bmatrix}$$
I knew how to do it if it was diagonalizable and real eigenvalues. How can I calculate when the ...
1
vote
1answer
32 views
How many possible Jordan decompasitions for an Endomorphism?
Let V be an vector space and $\phi \in End_\mathbb{C}(V)$ a linear map with the characteristic polynomial $P_\phi(X) = (x-2)^3(x-5)^2$. How many possible Jordan normal form are there for $\phi$?
0
votes
1answer
36 views
If we know the size of Jordan blocks of a matrix, what else can we derive? [closed]
If we have a 4x4 matrix with two distinct eigenvalues and if all Jordan blocks are of size 1x1, can we say that the matrix must be diagonal?
I am struggling to correlate the number non-diagonal ...
0
votes
1answer
23 views
rank$({A-\lambda I_n})^k$=rank$(B-\lambda I_n)^k$ iff $A$ is similar to $B$
Let $A,B \in M_n(\mathbb{C})$. Prove that rank$({A-\lambda I_n})^k$=rank$(B-\lambda I_n)^k$ for every $k \in \mathbb{N}$ and $\lambda \in \mathbb{C}$ iff they are similar.
I know that every matrix has ...
1
vote
1answer
54 views
Jordan Normal Form Proof in Lang
In Lang’s third edition of Linear Algebra on page $264$, I don’t understand the first line of his proof that any nonzero vector space over the complex numbers can be decomposed into the direct sum of $...
0
votes
1answer
35 views
Finding a basis where the rational canonical form holds
The matrix I have to consider is
$A=
\begin{bmatrix}x-\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&-\frac{1}{2}\\-\frac{1}{2}&x-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}\\\frac{1}{2}&\...
1
vote
2answers
51 views
Finding Jordan Canonical form given the minimal polynomial and the dimension of the kernel
How can we find the Jordan Canonical form given the minimal polynomial and the dimension of the kernel? I know how we could find it if either the matrix or the characteristic polynomial was given, but ...
2
votes
1answer
41 views
How to determine Jordan form, by knowing dimension of Kernel
I have the following information:
$p_A(\lambda) = (\lambda - 1)^6(\lambda + 2)^4$ and $m_A(\lambda) = (\lambda - 1)^3(\lambda + 2)^2$
and also $\dim(\operatorname{Ker}(A-I)^2) = 5$ and $\dim(\...
0
votes
1answer
61 views
How do I compute the matrix exponential for a non diagonalizable matrix?
I am trying to compute the matrix exponential $e^{At}$ for the matrix $A=\begin{pmatrix}
0 &0 \\
1&0
\end{pmatrix}$. In this case, I have computed the eigenvalues, which are in $\lambda=0$ ...
6
votes
2answers
92 views
The Jordan Canonical Form of linear operator in two variables polynomial
I've been trying to solve the following exercise,
In the space of bivariate polynomials of the form $f(x,y)=\sum_{n,m=0}^2a_{n,m}x^ny^m$, the lineal operator $T$ is defined by $Tf(x,y)=f(x+1,y+1)$. ...
1
vote
0answers
41 views
Show that $N$ and $NT$ has the same Jordan's form
Let $V$ be a finite complex vector space. Let $T$ and $N$ be
commutative operators in $V$ s.t $T$ is invertible and $N$ is
nilpotent. Show that $N$ and $NT$ has the same Jordan's form.
My attempt was ...
0
votes
0answers
25 views
Jordan normal form when no eigenvalues
Is there a Jordan normal form, when the matrix has no eigenvalues at all (over a certain field)?
For example if the characteristic polynomial is like $h(x)=(x^2 + x + 1)^2$ over the reals.
And if yes, ...
0
votes
0answers
19 views
Sketch Jordan canonical form given characteristic polynomial and its defect
I am trying to solve the following problem
Let $T$ be a linear operator on a 4-dimensional real vector space $V$ with characteristic polynomial $p(x)=(x-2)^3(x+3)$. Suppose that the eigenvalue $\...
4
votes
2answers
76 views
Minimal Polynomial Exercise
$\newcommand{\Ker}{\operatorname{Ker}}$Let $E$ be a vector space. We are given a matrix $A$ that has a characteristic polynomial $q(t) = -(t-2)^5$. We know that $\dim\Ker(A-2I)^2 = 3$ and $\dim\Ker(A-...
1
vote
0answers
24 views
Possible Jordan and Rational Canonical Forms ... over $\mathbb{F}_5$
Sorry for the long question and attempt, but I want to really make sure I understand and I'm not doing anything stupid, and that I've done everything I can.
I am asked to find all the the possible ...
1
vote
1answer
54 views
Find the Jordan form of a $7 \times 7$-real valued matrix - Help last step
I'm currently working on a linear algebra problem concerning Jordan forms. Indeed I'm given the matrix:
$$A = \left(\begin{array}{ccccccc}
1 & 0 & 1 & 0 & 0 & 0 & 0\\
0 & 1 ...
0
votes
1answer
33 views
Form of Jordan block
I'm doing an exercice about Jordan matrix, and I have to write the Jordan matrix of :
$$A =\begin{pmatrix}
3 & 0 & 1& 0&0 &0 &0\\
0 & 3 & 0& 1& 0&0 &0\\
...
1
vote
0answers
39 views
Number of Jordan boxes of size j for eigenvalue $\lambda$ is $2\dim\ker(A-\lambda I)^j-\dim\ker(A-\lambda I)^{j+1}-\dim \ker(A-\lambda I)^{j-1}$ [duplicate]
My question is about the computation of the Jordan normal form. Let $K$ be field. Let $A \in K^{n \times n}$ be a matrix, that characteristic polynomial splits into linear factors over $K$. I've ...
1
vote
2answers
82 views
Jordan form of 3 x 3 repeated eigenvalue
Consider the matrix
\begin{pmatrix}
1 & -3 &1 \\
1 & 5 & -1 \\
\
2 & 6 &0 \\
\end{pmatrix}
This has eigenvalue 2 with 3 multiplicity....
0
votes
1answer
51 views
Jordan Normal Form Exercise
I have been given a problem a while ago with no given solution so I thought I'd check with the folks at mathstackexchange to see if my method is correct. It goes like this:
I have been given a random ...
-1
votes
1answer
30 views
Find jordan from of matrix
I' am trying to find Jordan form of given matrix:
\begin{bmatrix}
1 & 2 & 0 \\
-1 & -1 & -1 \\
0 & 0 & 1
\end{bmatrix}
So far i founD characteristic polynomial : $(1 - \lambda ...
