# 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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### Jordan form step by step general algorithm

So I am trying to compile a summary of the procedure one should follow to find the Jordan basis and the Jordan form of a matrix, and I am on the lookout for free resources online where the algorithm ...
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### Proof involving the spectral radius and the Jordan canonical form

Let $A$ be a square matrix. Show that if $$\lim_{n \to \infty} A^{n} = 0$$ then $\rho(A) < 1$, where $\rho(A)$ denotes the spectral radius of $A$. Hint: Use the Jordan canonical form. I am ...
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### Computing the matrix exponential for a Jordan matrix

How can I compute $e^{At}$ where $A = J_{3}(5)$? That is, $$A = \begin{pmatrix} 5 & 1 & 0 \\ 0 & 5 & 1 \\ 0 & 0 & 5 \end{pmatrix}$$ Using this, how can I write down a basis ...
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### all 2 dimensional invariant subspaces

How we can find all 2 dimensional invariant subspaces of \begin{pmatrix} 2 & 1 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 8 \end{pmatrix} I know that there are at least 2 such subspaces, ...
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### Given a Jordan canonical basis, how to find out to which generalized eigenspace picked generalized eigenvector belongs

Suppose we have finite-dimensional linear operator $A:V\to V$ , that has eigenvalues $\lambda_1 ,\lambda_2, ... \lambda_n$ . It is known that we can decompose $V$ into direct sum of generalized ...
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### $Ch_A =(x+1)^6(x-2)^3$ y $min_A = (x+1)^3(x-2)^2$, List the possible Jordan forms for $A$

Let $A$ be a complex matrix such that $Ch_A =(x+1)^6(x-2)^3$ y $min_A = (x+1)^3(x-2)^2$, List the possible Jordan forms for $A$. And in each case write the corresponding rational I do not know ...
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### Deducing the additive Jordan decomposition

In $M_n(\Bbb C)$, I could prove that the additive Jordan decomposition of $X=D+N$ with $D$ diagonalizable and $N$ nilpotent gives a multiplicative Jordan decomposition $e^X=e^De^N$. Is that true the ...
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### When $A^i=A^j$ for $i,j\geq 0$ such that $i\neq j$ for matrix $A$ over algebraic closed field?

Let $A\in M_n(K)$ be a matrix over algebraic closed field $K$, where $n>1.$ When $A^i=A^j$ for $i,j\geq 0$ such that $i\neq j$? I tried solve it by Jordan form of matrix $A;$ is is sufficient ...
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### Converting Jordan Normal Form into Real Jordan Form

Given the matrix $$\begin{bmatrix} 0 & 0 & 0 & -8\\ 1 & 0 & 0& 16 \\ 0 & 1 & 0 & -14 \\ 0 & 0 & 1 & 6 \\ \end{bmatrix}$$ ...
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### Is Jordan cardinal form the matrix having the most zeros in the equivalent class of similarity? [duplicate]

I am concerned on this interesting question Given matrix $A$, does the Jordan cardinal form have the most zeros, in other word, it has the least nonvanishing indices, among the equivalent class of ...
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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 ...
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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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### Classify up to similarity all $n\times n$ complex matrices such that $A^n=I$

Classify up to similarity all $n\times n$ complex matrices such that $A^n=I$. I've seen this question for $n=3$. But I was wondering how to generalize this result. First $A^n=I$ is diagonalizable....
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### 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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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 $f(v_4)\in\left<... 1answer 61 views ### How many realizations are there then in a structured set of matrices yielding characteristic polynomial$(t+1)^4$? Let us consider a subset$S$of$M_4(\mathbb R)which has following form \begin{align*} \begin{pmatrix} 0 & * & 0 & * \\ 1 & * & 0 & * \\ 0 & * & 0 & * \\ 0 & *... 1answer 30 views ### Finding ch. polynomial and Jordan normal form off$knowing$\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 ... 1answer 55 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 ... 1answer 50 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$\...
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 ...
### $J={J_r}({\lambda})$ 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 ...