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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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How do I complete the steps of finding the Jordan of this $5\times 5$ matrix (with Octave)?

I know how to begin the procedure but I don't know how to finish it. Let's start with an example (sorry for it being so unwieldy). Let $$A =\begin{pmatrix} 177& 548& 271& -548& -...
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Generalised Eigenvectors Issue

Ok so i have been doing a few questions on 'Diagonalising' defective matrices, the method I've been using to find generalized Eigenvectors is to make the previous Eigenvector the subject. However i ...
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Decompose an invertible matrix into an exchangeable product of diagonalizable matrix and a matrix with all the eigenvalues equal to $1$

Let $ g $ be an invertible $ n\times n $ complex matrix. Show that $ g $ can be written as $$ g=su=us ,$$ where $ s $ is diagonalizable and all eigenvalues of $ u $ are equal to $ 1 $. My ...
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Can there be just one eigenvalue on the invariant subspace (generalized eigenspace) associated with an eigenvalue?

In the proof of Jordan decomposition here, once I know that an indecomposable subspace $V$ is of the form $V=Ker((f-\lambda Id)^n)$, can there be an other eigenvalue $\mu$ for $f\vert_V$?
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Intuition of generalized eigenvector.

I was trying to get an intuitive grasp about what the the generalized eigenvector intuitively is. I read this nice answer, so I understand that in the basis given by the generalized eigenvectors, a ...
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If N is elementary nilpotent matrix, show that N Transpose is similar to N

If $N$ is a $k \times k$ elementary nilpotent matrix, i.e. $N^k = 0$ but $N^{k-1} \ne 0$, then show that $N^\top$ is similar to $N$. Now use the Jordan form to prove that every complex $n \times n$ ...
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The theory behind linear recurrence relations solving (or - why does it work?)

tl;dr - a recommendation for a good book that explains the theory behind the auxiliary polynomial/companion matrix methods to solve linear recurrence relations with constant coefficients? I've bumped ...
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Show that matrix $A$ is similar to a matrix $B$ with elements on diagonal $(0, …, 0, \operatorname{Tr(}A))$ respectively.

Let $A$ be a matrix $n \times n, n \geq 2 $. Let's assume that not all entries outside of the diagonal are zeros (we don't know what entries are on the diagonal). Show that matrix $A$ is similar to a ...
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Let's assume that $ XA = AX $. Show that there is such a matrix $M$ that $ p_A(X) = M(A-X), MA=AM$ and $ MX=XM $.

Let $ A, X \in M_{nxn}(K) $. Let $ p_A(t) $ be a characteristic polynomial of matrix A. Let's assume that $ XA = AX $. Show that there is such a matrix $M$ that $ p_A(X) = M(A-X), MA=AM$ and $ MX=XM $....
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Exponential of a Jordan Block using Cayley-Hamilton Theorem

For the sake of simplicity, I will only consider $2\times2$ matrices. The Cayley-Hamilton theorem allows us to conclude that $$e^{At} = \alpha_0I + \alpha_1 A$$ where $\alpha_0$ and $\alpha_1$ can ...
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Find Jordan normal form and basis

Let $$A=M(\varphi)^{st}_{st}={\begin{bmatrix}0&1&1\\-4&-4&-2\\0&0&-2\end{bmatrix}}$$ and $ \varphi: \mathbb R^{3} \rightarrow \mathbb R^{3}$. Find the Jordan normal form $J_{A}$...
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How do I find the bases of the Jordan Canonical Form of $C$?

Let $$C = \left[ {\begin{array}{cccc} 0 & -1 & -2 & 3 \\ 0 & 0 & -2 & 3 \\ 0 & 1 & 1 & -1 \\ 0 & 0 & -1 & 2 \end{array} } \right].$$ What ...
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Jordan form exercise

What am I doing wrong? I've been learning how to put matrices into Jordan canonical form and it was going fine until I encountered this $4 \times 4$ matrix: $A=\begin{bmatrix} 2 & 2 & 0 &...
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Is a Jordan block not further block diagonalizable?

We can always find a Jordan canonical form of a matrix A. It is a block diagonal matrix. Is it true that each block cannot be reduced to a matrix with more blocks in diagonal? In other words, for a ...
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How to calculate the matrix exponential explicitly for a matrix which isn't diagonalizable?

How can I compute an expression for $(\exp(Qt))_{i,j}$ for some fixed $i, j$ and matrix $Q$? When $Q$ is diagonalizable, we can diagonalize, but what can be done otherwise? Thanks.
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Exericse about linear map $T\in L(V)$, where $\dim V=n\geq2$, with $\operatorname{null}T^{n-1}\neq\operatorname{null}T^n$

I have this problem that I am attempting, and am struggling with (b). -- Assume $\dim V = n \geq 2$ and that $T \in L(V)$ such that $\operatorname{null}T^{n-1}\neq\operatorname{null}T{^n}$ -- (a) ...
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Finding $P$ in $A = P^{-1}JP$ (Jordan Form)

I'm having a lot of trouble understanding the process of finding a basis for the Jordan canonical form (the "algorithm"). My textbook (Friedberg 4E) isn't very clear, and I can't seem to find anything ...
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Power of a matrix in REAL jordan form

Given a $2\times 2$ matrix in Jordan canonical form, whose eigenvalues are a couple of complex conjugate values $$ J = \left[ \begin{array}{cc} \sigma+j\omega & 0 \\ 0 & \sigma - j\omega \end{...
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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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How does one obtain the Jordan normal form of a matrix $A$ by studying $XI-A$?

In our lecture notes, there's the following example problem. Find a Jordan normal form matrix that is similar to the following. $$A=\begin{bmatrix}2 & 0 & 0 & 0\\-1 & 1 & 0 &...
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Finding Jordan normal form using Smith normal form

An endomorphism $T:V\to V$ of a finite dimensional $\mathbb{C}$-vector space endows $V$ with a $\mathbb{C}[X]$-module structure defined by $X\cdot v = T(v)$. From the Structure theorem for finitely ...
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Proving that a matrix is nonnegative if its powers are nonnegative

I am working on a problem involving doubly stochastic matrices where I must prove that $P$ is doubly stochastic if and only if $P^k$ is doubly stochastic for $k = 1, 2, ...$ It is easy to show that if ...
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756 views

Jordan form of a specific 2x2 matrix

I'm trying to follow an algorithm for finding the Jordan form for the matrix: $$ \begin{pmatrix} 0 & -1 \\ 4 & 4 \\ \end{pmatrix} $$ Its eigenvalue is 2 with ...
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Convergence of powers of matrix given convergence of the powers of its absolute value.

I have a matrix A and a matrix B such that $B_{i, j} = |A_{i, j}|$. I am given that all of the eigenvalues of B have magnitude less than 1, and therefore: $ \displaystyle \lim_{k \to \infty} B^k = 0$ ...
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Find the invariant factors and elementary divisors from the relations matrix.

Let $V$ be a finite dimensional vector space over $\Bbb C$, and $T$ be a linear operator on $V$. Consider $V$ as an $\Bbb C[x]$-module by defining $xv = T(v)$ for each $v \in V$. Let $$A = \begin{...
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Let $A$ be an $n \times n$ matrix with real entries such that $A^2 + I = 0$ then $n$ is even.

Let $A$ be an $n \times n$ matrix with real entries such that $A^2 + I = 0$ then $n$ is even. And if $n = 2k$, then $A$ is similar over the field of real numbers to a matrix of the block form $$\...
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Computing transformation matrix between similar matrices

If I have the matrix $A = \begin{pmatrix} 2 & 0 & 1 & -3 \\ 0 & 2 & 10 & 4 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 3 \end{pmatrix}$ and I've calculated its ...
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In the Jordan-Chevalley decomposition $M=D+N$, how obtaining $D$ and $N$ as polynomials in $M$?

The Jordan-Chevalley expresses a linear operator $M$ as $$ M = D + N, $$ where $D$ is semisimple (diagonalizable), $N$ is nilpotent and $DN=ND$. Although it is stated in many sources that $D$ and $N$ ...
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$A$ and $B$ be $n \times n$ matrices over the field $\mathbb F$ which have the same characteristic polynomial

Lemma: Let $N_1$ and $N_2$ be $3 \times 3$ nilpotent matrices over field $\mathbb F$. Then, $N_1$ and $N_2$ are similar if and only if they have the same minimal polynomial. Use the result above ...
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For which values ​the matrix is ​diagonalizable

For which values ​​of $a$ matrix $A$ is ​​diagonalizable? $$A = \pmatrix{0&i\\i&a}$$ in the case that it is not diagonalizable determine a base of Jordan Attempt: The minimal polynomial ...
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Finding characteristic and minimal polynomials and the Jordan normal form of $f$, knowing some relations for $f$.

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<...
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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 & *...
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Finding ch. polynomial and Jordan normal form of $f$ 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 ...
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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 ...
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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 $\...
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Yet Another Question Regarding Jordan Form [duplicate]

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 ...
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$J={J_r}({\lambda})$ 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 ...