Newest 'jordan-normal-form' Questions

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Is it possible for a generalized eigenvector to have two different eigenvalues?

It may be a quite silly question but I am having trouble with this. My question is that if a nonzero vector $v\in V$ is a generalized eigenvector for a linear operator $T: V\to V$ such that $(T-\...

linear-algebra jordan-normal-form

asked Sep 17 at 9:43

Henry Choi

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Number of possible Jordan canonical form without constructing the matrix

Problem: Let $\text A$ be a square matrix of order $9$ and $1$ and $-2$ are Eigenvalues with algebraic multiplicity $5$ and $4$ respectively. And minimal polynomial of the matrix $A$ is $m(x) = (𝑥 − ...

asked Sep 14 at 17:02

nmasanta

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Some steps in the proof of Jordan Normal Form

I've been trying to understand the proof that every matrix in a algebraically closed field can be written in JCF with respect to a basis. It's been hard to me because the reference that I was ...

linear-algebra matrices linear-transformations jordan-normal-form

asked Sep 14 at 16:08

Bruno Reis

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Jordan Normal Form of $T$ with $\ker(T) = \text{im}(T^3)$

I was able to get the Jordan Normal Forms of: $T$ with $\ker(T) = \text{im}(T)$, which looks like this: $\left( \begin{array}{rrrr}J(2,0) & & & 0 \\ & \ddots & & \\ & &...

linear-algebra jordan-normal-form

asked Sep 14 at 13:11

LinearAlgebruh

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Jordan cononical form of nilpotent matrix

I know that a nilpotent operator with index of nilpotency equal to dimension of the vectorspace, then, has Jordan Canonical form. Does any nilpotent operator whose index of nilpotency is less than ...

linear-algebra vector-spaces jordan-normal-form

asked Sep 6 at 15:37

Esha

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Jordan form of linear operator

Given the linear operator $T: \mathbb{R}^4 \rightarrow \mathbb{R}^4: T^3=-4T^2$ and $\dim \text{Im}\ T=\dim\ker T$. Prove that exists Jordan Form of $T$ in $\mathbb{R}$ and find all of the possible ...

linear-algebra jordan-normal-form

asked Sep 1 at 13:07

sustainable

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Bound on the max norm of a matrix based on its Jordan canonical form

Here is a lemma that is part of Threorem $142$C in J. C. Butcher's book on numerical methods of ODEs for which I do not understand the proof. The lemma asserts that a) $\Rightarrow $ b), where A is an ...

proof-explanation upper-lower-bounds jordan-normal-form

asked Aug 29 at 11:45

Guillermo BCN

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Different linear maps with the same Jordan form

Imagine I have the following matrix in Jordan form \begin{align} \begin{pmatrix} 3&1&0\\ 0&3&0\\ 0&0&2 \end{pmatrix} \end{align} By defining this matrix to be a representation ...

linear-algebra matrices linear-transformations jordan-normal-form

asked Aug 27 at 5:39

Ramin

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How can I prove that the linear transformation has at most one non-zero eigen value?

My Attempt: If $0$ was not an eigenvalue of $T$ then the eigenvalues of $T^k$ would have been non-zero also, $\ker(T^n) = \ker(T^{n-1}) = \mathbf{0}$. Am I correct? Can anyone please help me with (b)...

linear-algebra eigenvalues-eigenvectors linear-transformations jordan-normal-form

asked Aug 25 at 10:21

sani

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Does the Jordan form of the matrix depend on where the Jordan blocks are placed?

Let's say that we have the matrix \begin{equation*} A = \begin{pmatrix} 1 & -1 & 0 \\ -1 & 4 & -1 \\ -4 & 13 & -3 \end{pmatrix}. \end{equation*} Now, the characteristic ...

linear-algebra jordan-normal-form

asked Aug 19 at 23:09

John Kameas

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Finding all possible Jordan Normal Forms

I need help with finding all Jordan Normal Forms with following infos: $F$ is an endomorphism of $V$, $\:\dim(V) = 8$, $\:\operatorname{rank}(F) = 5$, $\:\operatorname{rank}(F^2) = 4$, $\:\...

linear-algebra jordan-normal-form

asked Aug 17 at 20:03

Michael Maier

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$n \ge 2, (p,n) \ne (2,2).$ Prove that for any $v \in \mathbb{F}_p^n,$ the sequence $v, (I+M)v, (I+M+M^2)v, \dots$ has period $<p^n.$

Let $p$ be prime and $n \ge 2, (p,n) \ne (2,2), M \in \mathbb{M}_{n \times n}(\mathbb{F}_p).$ Prove that for any $v \in \mathbb{F}_p^n,$ the sequence $v, (I+M)v, (I+M+M^2)v, \dots$ has period $<p^n....

linear-algebra matrices jordan-normal-form

asked Aug 10 at 0:25

Display name

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Find a Jordan Canonical Form

Let $A$ and $B$ be matrices over the real numbers, such that $A$ is $3 \times 5$ and $B$ is $5 \times 3$, and the product $AB$ is $$ \left( \begin{array} \\ 1& 1 & 0 \\ 0 & 1 & 0 \\ 0 &...

linear-algebra abstract-algebra jordan-normal-form

asked Aug 5 at 16:22

LinearGuy

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Jordan normal form of sum of two commuting nilpotent matrices over a finite field (variant on a linear matrix pencil problem)

This question comes up with trying to construct Lie subalgebras of (large) Lie algebras that are invariant under a finite group $H$. I have two isomorphic $H$-invariant nilpotent subalgebras and am ...

linear-algebra matrices lie-algebras finite-fields jordan-normal-form

asked Aug 4 at 8:24

David A. Craven

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What does subspace A-matrix invariance tells me in terms of A Jordan canonical form.

I am asked to show that the semi group $(e^{tA})_{t\geq0}$ for $A \in M_n(\mathbb{C})$ is hyperbolic i.e. there exists direct decomposition $\mathbb(C)^n=X_s \oplus X_u$ in to A-invariant subspaces $...

linear-algebra operator-theory jordan-normal-form semigroup-of-operators

asked Jul 29 at 11:55

Kran

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How to find the Jordan form of an anti-diagonal matrix?

How to find the Jordan form of an anti-diagonal matrix? $$\begin{bmatrix} &&&{}a_{1}\\ &&\ddots &\\ &a_{\text{} n-1}&&\\ a_{n}&&& \end{bmatrix}$$ It ...

linear-algebra matrices jordan-normal-form

asked Jul 29 at 1:59

Tornado David

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Matrix whose square is in Jordan normal form

Let $$A = \begin{bmatrix}J_0^2 \\ & J_0^2 & \\ && J_{1/4}^3\end{bmatrix}\in M_7(\mathbb{Q})$$ Find, with proof, a matrix $B$ so that $B^2 = A$. I'm not sure how to find this matrix. ...

linear-algebra matrices systems-of-equations matrix-equations jordan-normal-form

asked Jul 28 at 15:40

user763400

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Jordan normal form of $\;\begin{pmatrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & a & b \end{pmatrix},\; a,b\in\mathbb{R}$

If possible, compute the Jordan normal form of $\begin{pmatrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & a & b \end{pmatrix}\in\mathbb{R}^{3\times 3}$ with $a,b\in\mathbb{R}$. In the ...

linear-algebra matrices jordan-normal-form

asked Jul 21 at 9:13

user

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Jordan normal form of powers of Jordan normal form

Previous related question: Jordan normal form powers Let $A$ be a $n\times n$ Matrix such that $A=PBP^{-1}$ where $B$ is in Jordan normal form with $\lambda_i(k)_j$ Where $i$ is the size, $k$ is the ...

linear-algebra matrices matrix-decomposition jordan-normal-form

asked Jul 19 at 9:54

razivo

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If $T:\mathbb{C}^n \to \mathbb{C}^n$ is diagonalizable when restricted to any two-dimensional invariant subspace, then $T$ is diagonalizable

Suppose $n\geq 2$. Let $T:\mathbb{C}^n\to \mathbb{C}^n$ be linear. Prove that the following are equivalent: (i) $T$ is diagonalizable. (ii) For every two-dimensional subspace $W\subseteq \mathbb{C}^n$ ...

linear-algebra jordan-normal-form

asked Jul 19 at 0:29

LAGC

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Jordan normal form powers

Let $A$ be a $n\times n$ such that $A=PBP^{-1}$ where $B$ is in Jordan normal form with $\lambda_i(k)_j$ Where $i$ is the size, $k$ is the eigenvalue and $j$ the order. If $A$ was diagonal($i=1$) ...

linear-algebra matrices matrix-decomposition jordan-normal-form

asked Jul 18 at 9:59

razivo

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Function (Taylor series) of Jordan canonical form about arbitrary point

On the Wikipedia page for Jordan matrices, under the section on functions $f$ of matrices $A = PJP^{-1}$ (that being the Jordan Canonical Form), the following can be found (I've paraphrased it a tiny ...

linear-algebra matrices power-series taylor-expansion jordan-normal-form

asked Jul 17 at 16:07

Mew

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Chain of kernels, generalised eigenvector

Given $A\in M_{n,n}(\mathbb{R})$ and $\lambda$ an eigenvalue, a generalized eigenvector of rank $i$ is defined as $v \in ker(A-\lambda E)^i\setminus ker(A-\lambda E)^{i-1}$. Why does such vector ...

jordan-normal-form generalized-eigenvector

asked Jul 13 at 12:10

kern711

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$W$ is a $T$-invariant subspace of $V$, prove a Jordan form of $T|_W$ contained the Jordan form of $T$.

The meaning of the title is to show that each block of the Jordan form of $T|_W$ corresponds to a block in the Jordan form of $T$ of equal or greater size. I know that each Jordan block in the form of ...

linear-algebra jordan-normal-form invariant-subspace

asked Jul 12 at 6:31

Adgorn

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From generalized eigenvector to Jordan form

I can't figure out the following part of Chen's Linear Systems book. How does he "readily obtain" $Av_2=v_1+\lambda v_2$?

linear-algebra matrices eigenvalues-eigenvectors jordan-normal-form generalized-eigenvector

asked Jul 5 at 14:11

João Pedro

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Jordan Normal Form eigenvalues

I need to find the Jordan Normal Form of a square matrix $\Phi$ such that the entries in the diagonal (or near diagonal) matrix has its elements ordered by absolute value. In particular, I need to ...

linear-algebra jordan-normal-form

asked Jun 30 at 22:01

MHall

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A linear operator on a $5$ dimensional complex vector space

A linear operator $T$ on a complex vector space $V$ has the characteristics polynomial $x^3(x-5)^2$ and the minimal polynomial $x^2(x-5)$ .Choose all correct options. $(a)$ The Jordan Form of $T$ is ...

linear-algebra eigenvalues-eigenvectors solution-verification jordan-normal-form

asked Jun 30 at 12:01

user710290

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Multiplicity of each eigenvalue in a minimal polynomial of a matrix

It is well known that for a $n \times n$ matrix $A$ , the charicteristic polynomial $p(x)$ satisfies $p(x)=\prod_{\lambda : eigenvector} (x-\lambda)^{a(\lambda)}$ where $a(\lambda )$ is the ...

linear-algebra matrices eigenvalues-eigenvectors jordan-normal-form minimal-polynomials

asked Jun 29 at 23:59

Junsukim

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Proving Jordan chain for nilpotent matrix is linearly independent

I am looking for a proof for the Jordan chain $\{L^ix, L^{i-1}x, \dots, x\}$ being independent, where $L$ is a nilpotent matrix of index $k$, $i<k$ and $x \neq 0$. I have tried the following. ...

independence jordan-normal-form nilpotent-groups

asked Jun 29 at 9:52

learner

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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. ...

lie-groups lie-algebras algebraic-groups jordan-normal-form reductive-groups

asked Jun 28 at 22:07

unknownymous

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Proof about diagonalizable Matrix [duplicate]

The proof said the following, let $A \in M_{n \times n}(\mathbb{C})$ such that $A^r=I_{n \times n}$ for some $r \in \mathbb{N},r>0$ then $A$ is diagonalizable. Mi attempt is the next, since $A$ is ...

linear-algebra algebra-precalculus linear-transformations jordan-normal-form

asked Jun 26 at 17:48

Brien Navarro

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Stuck on finding the canonical form an endomorphism.

When asked to find the Jordan form of an endomorphism I'm usually given a matrix associated with the endomorphism from which I can compute the Jordan, yet this isn't the case with this excercise; ...

linear-algebra jordan-normal-form

asked Jun 22 at 7:31

Leo

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Is the Jordan normal form uniquely determined by the characteristic and minimal polynomial?

I was looking into this answer to a question about obtaining the Jordan normal form given the characteristic and minimal polynomials of a matrix. In this answer, it is stated that "The ...

linear-algebra matrices jordan-normal-form minimal-polynomials characteristic-polynomial

asked Jun 20 at 7:52

Robert Lee

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Minimum annihilating polynomial of matrix $A$ of order $n$ is equal to $(λ + 1)^2(λ - 2)$.

Minimum annihilating polynomial of matrix $A$ of order $n$ is equal to $(λ + 1)^2(λ - 2)$. What can be said about the minimal annihilating polynomial of matrix $A^2$? Justify the answer. While working ...

linear-algebra matrices jordan-normal-form characteristic-polynomial

asked Jun 17 at 11:50

Maria White

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possible Jordan normal forms of endomorphism

Let $f:\mathbb{C}^6\to\mathbb{C}^6$ be a linear map with characteristic polynomial $ch_f(X)=(X+2)^4(X-1)^2$ and $rank(f+2id)>rank(f+2id)^2=1$, as well as $rank(f-id)=5$. Assignment: Find the ...

linear-algebra jordan-normal-form

asked Jun 17 at 11:24

user

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Minimal polynomial = chratacteristic polynomial $\iff$ distinct eigenvalues associated with distinct Jordan blocks?

"Let M be the given matrix of order n and its Jordan Canonical Form be J . Prove that the minimal and characteristic polynomial of M are same, if and only if, distinct eigenvalues of M ...

matrices jordan-normal-form minimal-polynomials characteristic-polynomial

asked Jun 15 at 13:57

Hououin Kyouma

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Prove that the Jordan Canonical Form of $T$ contains the Jordan Canonical Form of $T|_W$ for any $T$-invariant $W.$

Let $V$ be a complex vector space with a linear operator $T : V \to V$ and a $T$-invariant subspace $W \subseteq V.$ Prove that the Jordan Canonical Form of $T$ contains the Jordan Canonical Form of $...

linear-algebra jordan-normal-form invariant-subspace

asked Jun 9 at 17:25

paxtibimarce

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Calculating matrix exponential for $n \times n$ Jordan block [duplicate]

I want to calculate exponential of the matrix which on diagonal has some $a \in \mathbb{R}$ and ones above. The $n\times n$ matrix looks like following $$ A = \left( \begin{matrix} a & 1 & 0 ...

linear-algebra matrices eigenvalues-eigenvectors jordan-normal-form matrix-exponential

asked Jun 7 at 8:56

John

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Minimal polynomial of a diagonalizable linear operator on a finite dimensional vector space is a product of distinct linear polynomials. [duplicate]

Prove that the minimal polynomial of a diagonalizable linear operator on a finite dimensional vector space is a product of distinct linear polynomials

jordan-normal-form minimal-polynomials

asked Jun 6 at 7:25

user796897

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Find nondiagonalizable matrix A

Find nondiagonalizable matrix A such that matrix $A^2-6A$ is diagonizible. Simpliest nondiagonizable matrix is a Jordan block 2 by 2 $\begin{pmatrix} a & 1\\ 0 & a \end{pmatrix}$. $\begin{...

matrices jordan-normal-form

asked Jun 3 at 18:19

Hrackadont

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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^(-...

linear-algebra jordan-normal-form

asked Jun 3 at 12:13

Alex_Lesley

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Relation between Jordan Normal Form and cyclic modules

I've just started reading about the relation between cyclic modules and Jordan Normal Form and, being honest, I've quite a doubt. The text I am using says that "clearly", the following assumption is ...

linear-algebra abstract-algebra vector-spaces modules jordan-normal-form

asked May 31 at 16:24

Anyway142

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Jordan matrix form and polynomial proof.

let $ f\in F[x] $ be a polynomial. and prove that the matrix $ f\left(J_{n}\left(\lambda\right)\right) $ satisfies $ [f\left(J_{n}\left(\lambda\right)\right)]_{ij}=\begin{cases} \frac{1}{\left(j-i\...

linear-algebra matrices polynomials jordan-normal-form

asked May 30 at 13:32

FreeZe

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Question involving eigenspaces and generalized eigenspaces

I need to solve the question given above; however, I am unsure of how to proceed exactly; am I required to use the invertibility of $U$ to somehow show the two equalities? Any help would be ...

eigenvalues-eigenvectors jordan-normal-form generalized-eigenvector

asked May 30 at 0:11

Einstein the troll

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Prove that $U(E_{\lambda})=E_{\lambda}$ and $U(K_{\lambda})=K_{\lambda}$.

Let T be a linear map on a finite-dimensional vector space V , and let $\lambda$ be an eigenvalue of T with corresponding eigenspace and generalized eigenspace $E_{\lambda}$ and $K_{\lambda}$. Let U ...

linear-algebra jordan-normal-form

asked May 29 at 17:02

Beacon

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Alll the matrices $A\in M_{7x7}\left(\mathbb{C}\right)$, with characteristic polynomial is: $\left(x-1\right)^3\left(x-2\right)^4$, …

I need to find all the matrices $A\in M_{7x7}\left(\mathbb{C}\right)$, all I know is the characteristic polynomial is: $$\left(x-1\right)^3\left(x-2\right)^4$$ $$\dim\:\ker\:\left(A-2I\right)=3$$ $$\...

matrices jordan-normal-form minimal-polynomials characteristic-polynomial cayley-hamilton

asked May 28 at 18:45

Ilya.K.

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find all the matrices (which are not similiar) which fulfill this formula

I need to find all the matrices $A\in M_{4x4}\left(\mathbb{C}\right)\:$ such that: $$A^4-2A^2+I\:=\:0$$ which means $\left(A^2-I\right)^2=0$ So I see that there is a few groups of which can give ...

jordan-normal-form minimal-polynomials characteristic-polynomial cayley-hamilton similar-matrices

asked May 28 at 18:24

Ilya.K.

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finding Jordan Normal Form of two possibilities

$$\begin{pmatrix}7&1&2&2\\ \:\:1&4&-1&-1\\ \:\:-2&1&5&-1\\ \:\:1&1&2&8\end{pmatrix}$$ I found the characteristic polynomial to be $\left(x-6\right)^4$ ...

eigenvalues-eigenvectors jordan-normal-form minimal-polynomials characteristic-polynomial

asked May 28 at 15:03

Ilya.K.

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Show that there exist $a_1,\ldots, a_{2n-1}$ such that $ a_{2n-1}J^{2n-1}+\cdots+a_1 J=I_n,$ where $J$ is a Jordan matrix

Let $J\in\mathbb{C}^{n\times n}$ be a Jordan normal form and assume that ${\rm tr~}J<2n$. Prove or disprove that there exist $a_1,\ldots, a_{2n-1}\in\mathbb{R}$ such that \begin{equation} a_{2n-1}...

linear-algebra matrices spectral-theory jordan-normal-form

asked May 23 at 21:36

James Vestal

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Calculate power of a matrix using jordan form

I need to calculate: $$ \begin{bmatrix} 1&1\\ -1&3 \end{bmatrix}^{50} $$ The solution i have uses jordan form and get to: There are some points that i dont understand: $1.$ In the right ...

linear-algebra matrices diagonalization jordan-normal-form

asked May 21 at 23:27

Alon

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