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.

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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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\...
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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 ...
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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): $...
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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 ...
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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 ...
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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{...
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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 ...
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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 ...
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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),...
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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 ...
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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 ...
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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 $$(\...
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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}} &...
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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 ...
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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 ...
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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 \...
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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,...,...
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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 ...
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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
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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
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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
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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, ...
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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 ...
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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
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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 ...
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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 $\phi$?
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1answer
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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 ...
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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 ...
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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 $...
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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}&\...
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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
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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(\...
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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 $\lambda=0$ ...
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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 $Tf(x,y)=f(x+1,y+1)$. ...
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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 ...
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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, ...
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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
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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-...
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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
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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 ...
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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\\ ...
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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
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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....
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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
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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 ...

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