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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21 views

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-\...
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1answer
18 views

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 − ...
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29 views

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 ...
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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 & & \\ & &...
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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 ...
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1answer
37 views

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 ...
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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 ...
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33 views

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 ...
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1answer
45 views

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)...
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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 ...
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1answer
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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 $V$, $\:\dim(V) = 8$, $\:\operatorname{rank}(F) = 5$, $\:\operatorname{rank}(F^2) = 4$, $\:\...
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$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....
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3answers
78 views

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 &...
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2answers
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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 ...
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1answer
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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 $...
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1answer
65 views

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 ...
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2answers
47 views

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. ...
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1answer
65 views

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 ...
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1answer
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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 ...
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1answer
63 views

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 $W\subseteq \mathbb{C}^n$ ...
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1answer
31 views

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($i=1$) ...
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0answers
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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 ...
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1answer
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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 ...
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1answer
31 views

$W$ 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 ...
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1answer
20 views

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" $Av_2=v_1+\lambda v_2$?
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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 ...
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45 views

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 ...
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1answer
37 views

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 ...
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1answer
30 views

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. ...
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42 views

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. ...
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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 ...
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1answer
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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; ...
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2answers
63 views

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 ...
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43 views

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 ...
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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 ...
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1answer
46 views

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 ...
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1answer
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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 $...
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1answer
48 views

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 ...
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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
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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{...
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1answer
39 views

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^(-...
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1answer
31 views

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 ...
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1answer
39 views

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\...
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19 views

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 ...
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1answer
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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 ...
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1answer
34 views

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: $$\left(x-1\right)^3\left(x-2\right)^4$$ $$\dim\:\ker\:\left(A-2I\right)=3$$ $$\...
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1answer
27 views

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 ...
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0answers
23 views

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 $\left(x-6\right)^4$ ...
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1answer
21 views

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}...
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3answers
48 views

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 ...

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