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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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An alternative algorithm to find the Jordan form/basis for a complex matrix.

I am currently studying System Theory, and the exam involves a lot of finding Jordan forms/bases for state transition matrices. I know there is an algorithm for doing so which involves generalized ...
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Jordan normal form over $\mathbb{C}$

Let there be $T:\mathbb{C}^8\rightarrow \mathbb{C}^8$ Such that $ T\left(\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \\ x_{5} \\ x_{6} \\ x_{7} \\ x_{8} \end{array}\right)=\left(\begin{...
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Why two possibles Jordan Canonical forms of a matrix cannot be similar?

Consider an matrix $A_{5x5}$ with only one eigenvalue $\lambda$. If the dimension of the eigenspace $\lambda$ is two, then we can have two possibilities of the Jordan blocs here: I didnt ...
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Have I found the Jordan form correctly?

I am given that the minimal polynomial and characteristic polynomial of a matrix are both $(x-1)^2(x+1)^2$. I have found the Jordan form to be $$\begin{bmatrix}1&1&0&0\\0&1&0&0\...
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Can basis of kernel be extended to a Jordan basis?

Let $A\in\mathbb C^{n\times n}$ be nilpotent. A Jordan basis of $A$ is a basis of $\mathbb C^n$ with respect to which $A$ has Jordan normal form. Assume that we do not know the Jordan structure of $A$....
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Generalization of the Jordan form for infinite matrices

Under what conditions is it the case that for a matrix $M$ whose rows and columns are indexed by a countably infinite set $S$ one has a Hamel basis consisting of generalized eigenvectors (i.e. $v \in \...
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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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Jordan Canonical Form of matrix

I am having trouble figuring out computing Jordan Canonical Form. Can someone explain how to get there with this example matrix? $A=\begin{bmatrix}1&1&1\\0&2&0\\0&0&2\end{...
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Jordan form of a power of Jordan block?

How, in general, does one find the Jordan form of a power of a Jordan block? Because of the comments on this question I think there is a simple answer.
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Similar Matrices and their Jordan Canonical Forms [duplicate]

Let $A$ and $B$ be two matrices in $M_n$. Is the following ture: $A$ and $B$ are similar $\iff$ $A$ and $B$ have the same jordan canonical form. Could someone explain?
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Finding the Jordan canonical form of A and Choose the correct option

Let $$ A = \begin{pmatrix} 0&0&0&-4 \\ 1&0&0&0 \\ 0&1&0&5 \\ 0&0&1&0 \end{pmatrix}$$ Then a Jordan canonical form of A is Choose the correct ...
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Jordan normal form theorem - a question about the proof I've found

I've been reading this proof of Jordan's theorem: http://www.cs.uleth.ca/~holzmann/notes/jordan.pdf and there are a few questions I hope you could answer for me. Firstly, why $(A_{\lambda})_{\mu _i} ...
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'Diagonalization' of Jordan block

We know the Jordan block $$J = \begin{bmatrix} \lambda & 1 & & \\ & \lambda & \ddots & \\ & & \ddots & 1\\ & & & \lambda \end{bmatrix} $$ ...
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When does a matrix admit a Jordan canonical form?

If a matrix over the field $\mathbb R$ has as elementary divisors: $x-4$, $x^2 + 2$, does it then admit a Jordan canonical form? Am I right in thinking that a matrix has a Jordan canonical form ...
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Misconception on Jordan Canonical Form

Say we have matrix $$M= \begin{pmatrix} 2 & 0 & 1 & -3\\ 0 & 2 & 4 & 8\\ 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 3 \end{pmatrix}\DeclareMathOperator{\Id}{Id}$$ ...
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Is the matrix norm of a matrix equal to the maximum of the norms of its Jordan block?

Let $J$ be a Jordan block matrix with blocks $J_1,\cdots,J_n$. I came up with some examples of $J$ and noticed that $\|J\|=\max_{i=1,\cdots,n}\|J_i\|$. Does this result always hold? The norm I use ...
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Matrix functions of a non-diagonalizable matrix

Let $A$ be the following $3 \times 3$ matrix: $$ A = \begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \\ \end{pmatrix} $$ I'm supposed to calculate $A^n$, where $n \in \...
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Matrix exponential for Jordan canonical form

Let $X$ be a real $n \times n$ matrix, then there is a Jordan decomposition such that $X = D+N$ where $D$ is diagonalisable and $N$ is nilpotent. Then, I was wondering whether the following is ...
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$3 \times 3$ matrices completely determined by their characteristic and minimal polynomials

How do you show that two $3 \times 3$ matrices with the same characteristic and minimal polynomials both conjugate to the same Jordan normal form, assuming no knowledge of the eigenspaces? I know ...
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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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Jordan's decomposition

I have a matrix $A\in R^{n,n}$. $A= \begin{bmatrix} 1&0&-2&0&0&\dots&0\\ 0&1&0&-6&0&\dots&0\\ 0&0&1&0&-12&\dots&0\\ \vdots&\...
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Jordan Normal Form

I'm asked to find the Jordan Normal form of $A\in M_5(\mathbb{C}^{5x5})$ with the characteristic polynomial: $p(A)=(\lambda-1)^3(\lambda+1)^2$ and minimum polynomial $m(A)=(\lambda-1)^2(\lambda+1)$ ...
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Jordan Normal Form and eigenvalue 0

I understand the processes of putting a matrix into Jordan normal form and forming the transformation matrix associated to "diagonalizing" the matrix. So here's my question: Why is it that when you ...
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Jordan form and an invertible $P$ for $A = \begin{pmatrix} 1&1&1 \\ 0 & 2 & 2 \\ 0 & 0 & 2 \end{pmatrix}$

$A = \begin{pmatrix} 1&1&1 \\ 0 & 2 & 2 \\ 0 & 0 & 2 \end{pmatrix}$, find the jordan form and the invertible $P$ such that: $A = P J P^{-1}$. Now I found the characteristic ...
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Can we reduce finding matrix roots to finding roots of Jordan blocks?

I just found some interesting question about matrix square roots and I came to think of one way to find them, or at least reduce them to a set of simpler problems. Assume we have a matrix $\bf A$ and ...
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Proving if a certain matrix exists or not

The question is: Is there a natural $n$ and $A\in M_{n}(\mathbb C)$ (complex $n\times n$ matrices) such that the conditions \begin{align} \operatorname{rank}(\,A\,) &= 10\\ \operatorname{...
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unitarily similar characterization

Today at lesson, the professor brought up the notion of unitary similarity when talking about Schur theorem. After lesson, we talked about complete characterizations of this similarity (not only for ...
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All Possible Jordan Canonical Forms Given Characteristic Polynomial

I am given the characteristic polynomial $x^2(x^2-1)$ and am asked to find all possible jordan canonical forms. What I have so far is: Possible elementary divisors are: 1) $x,x,(x+1),(x-1)$, 2) $x,x,...
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Jordan Form Superdiagonal

How do you know how many of the super diagonal entries in the Jordan Form are zeros and how many are ones, and where they are placed? Thanks.
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The nonexistence of a Jordan normal form over a finite field

Since finite fields are not algebraically closed, this suggests to me that there may be matrices over finite fields whose characteristic polynomials don't split over that finite field, and thus do not ...
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Two different definitions of Jordan canonical form

I am currently reading two linear algebra books. One is Hoffman/Kunze's and the other one is Friedberg/Insel/Spence's. They define Jordan canonical form of linear operator in different ways. In ...
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Find all possible Jordan Canonical forms of $A^2$

Finding Jordan Canonical forms seems pretty straightforward mostly, but this one threw me off: Let $A\in M_n(\Bbb{F}) $ be a matrix with a minimal polynomial $m_A(t)=(t-\lambda)^n$. Find all the ...
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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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What is wrong with this Jordan normal form computation?

The question I am working on is to compute the Jordan normal form of $$A := \begin{pmatrix} 2 & 1 & 5 \\ 0 & 1 & 3\\ 1 & 0 & 1\end{pmatrix}.$$ The characteristic polynomial and ...
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The index of nilpotency of a nilpotent matrix

Let $A$ a matrix in $\mathcal{M}_5(\mathbb C)$ such that $A^5=0$ and $\mathrm{rank}(A^2)=2$, how prove that $A$ is nilpotent with index of nilpotency $4$? Thanks in advance.
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Finding a Jordan basis of a $3\times 3$ matrix

Find a Jordan basis for the following matrix: $$A= \begin{pmatrix} 1 & -3 & 4 \\ 4 & -7 & 8 \\ 6 & -7 & 7 \\ \end{pmatrix} $$ Hey everyone. First I have ...
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Interpreting the Jordan Normal Form

What's the best way to interpret Jordan Normal Form (e.g. in terms of a linear map)? For instance, how should we interpret those $1$'s?
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Finding all possible Jordan forms of an $ 8\times 8$ matrix given the minimal polynomial

Find all possible Jordan forms of an $ 8\times 8$ matrix given that $$t^2(t-1)^3$$ is the minimal polynomial. I don't really know where to start so all help would be appreciated
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AB and BA have identical nonsingular Jordan blocks

If A and B are square matrices of the same size I know how to show that AB and BA have the same eigenvalues and characteristic polynomials. But I want to show that they have identical nonsingular ...
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Inferring Jordan Block structure from dimension of null spaces

Suppose $T- \lambda I$ is a nilpotent linear transformation on vector space $V$ with index of nilpotency $m < \mathrm{dim}V$. Assume we know $n_k = \mathrm{dim}(T - \lambda I)^k$ for $k = 1, 2, \...
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Jordan Block of Kronecker Product

Let $A$ be a $(p\times p$)-Jordan block of generalized eigenvalue $\lambda$. Let $B$ be a $(q\times q$)-Jordan block of generalized eigenvalue $\mu$. Then what is the Jordan canonical form for $A\...
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Why block-diagonal form for nilpotent matrices?

I am currently reading Jim Hefferon's Linear Algebra. In chapter 5, nilpotence, strings, he goes through the process of finding a string basis of a map, and proves that there exists a string basis ...
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Inequality about modified Jordan form

I'm trying to prove an inequality, which in turn I think is equivalent to this inequality, $$(|\lambda_1|-\epsilon)^2\bar{y}^Ty\le\bar{y}^T\left(\overline{(\lambda_rI_r+\epsilon F_r)_r}^T(\...
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Prove the direct sum of generalized eigenspaces is the whole vector space

Given a $n\times n$ matrix $A$ over an algebraically closed field, let $\lambda_1,...,\lambda_k$ be its eigenvalues, and let $V_{\lambda_i}$ be the generalized eigenspace of $\lambda_i$. The question ...
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Lost on rational and Jordan forms

I'm having a lot of trouble trying to understand rational canonical form, primary rational canonical form, and Jordan form. I've looked at the posts about this, but I haven't been able to understand ...
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relation between minimal polynomial and jordan normal form

I just solved some exercises on minimal polynomials and i remember that there is a relation between the minimal polynomial and the jordan normal form. But my question is the following : knowing the ...
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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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How to show that the matrix exponential is invertible for non-diagonalizable matrix A,

I have shown the easy case, when A is diagonalizable. But I am stuck on the case when A is not. So, I put A in its Jordan canonical form. Then say A = $SJS^{-1}$. Then $e^A = Se^JS^{-1}$, where $...
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All nilpotent $2\times 2$ matrices

I want to find all nilpotent $2\times 2$ matrices. All nilpotent $2 \times 2$ matrices are similar($A=P^{-1}JP$) to $J = \begin{bmatrix} 0&1\\0&0\end{bmatrix}$ But how do I find all of these ...
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Matrix exponential using the Jordan form

How do I calculate the matrix exponential $\Bbb e^{At}$ for $A = \left( \begin{matrix} 1 & 0 & 0 \\ 0 & 2 & 3 \\ 0 & 0 & 2 \end{matrix} \right)$ using the Jordan form of $A$? I ...