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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4
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2answers
218 views

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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0answers
1k views

Jordan Canonical Form and Minimal Polynomial

I was wondering what the relationship between the minimal polynomial and the Jordan Canonical Form is. Given a matrix, all one needs to do is to compute the characteristic polynomial to determine the ...
3
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1answer
533 views

Jordan basis of $A$ when $A$ is the companion matrix?

The actual question: when $A$ is the companion matrix, why the general form of $M_i$ (the group of columns of the Jordan matrix $M$ that belongs to the block associated to $\lambda_i$) is: $$ M_i^{h,...
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2answers
3k views

How do I write this matrix in Jordan-Normal Form

I have the matrix $A=\begin{pmatrix}2&2&1\\-1&0&1\\4&1&-1\end{pmatrix}$, I want to write it in Jordan-Normal Form. I have $x_1=3,x_2=x_3=-1$ and calculated eigenvectors $v_1=\...
3
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2answers
2k views

Uniqueness of the Jordan decomposition

I have seen it said that a matrix $M$ (over $\mathbb{C}$, say) has a unique decomposition $M = D + N$ where $D$ is diagonal and $N$ is nilpotent. I'm having trouble seeing this, since the Jordan form ...
22
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2answers
10k views

Why does the $n$-th power of a Jordan matrix involve the binomial coefficient?

I've searched a lot for a simple explanation of this. Given a Jordan block $J_k(\lambda)$, its $n$-th power is: $$ J_k(\lambda)^n = \begin{bmatrix} \lambda^n & \binom{n}{1}\lambda^{n-1} & \...
1
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1answer
521 views

Index of nilpotency Jordan block

If $T$ is an endomorphism, there exists a basis, according to which $T$ will be a block-diagonal matrix. Because if $V$ is the domain of $T$, $V$ will be the direct sum of the generalized eigenspaces, ...
2
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2answers
592 views

Eigenvalues and Jordan form

I have a $5\times 5$ matrix and I need to find the Jordan form and its inverse. I know how to find the inverse. But for the Jordan form I am screwed. The matrix is $$\begin{bmatrix}3 & 0 & 0 ...
9
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1answer
5k views

Possible Jordan Canonical Forms Given Minimal Polynomial

I was supposed to find all possible Jordan canonical forms of a $5\times 5$ complex matrix with minimal polynomial $(x-2)^2(x-1)$ on a qualifying exam last semester. I took the polynomial to mean ...
0
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1answer
93 views

Practice with another $e^{At}$

Given the matrix: $A=\begin{pmatrix}-2&0&0\\4&-2&0\\1&0&-2\end{pmatrix}$, find $e^{At}$. I found the eigenvalues to be $-2,-2,-2$. I need to use the Jordan form to solve it. I'...
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3answers
976 views

Jordan Canonical form of a matrix over rationals whose all entries are 1.

How to compute the Jordan canonical form for the $n \times n$ matrix over $\mathbb{Q}$ whose entries equals to $1$.
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2answers
4k views

Finding the Jordan canonical form of this upper triangular $3\times3$ matrix

I am supposed to find the Jordan canonical form of a couple of matrices, but I was absent for a few lectures. \begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 2 \\ 0 & 0 & 3 \end{...
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2answers
1k views

Showing companion matrix is similar to Jordan block using Jordan-Chevalley decomposition

The Jordan-Chevalley decomposition says that given a linear operator $L$, you can decompose it as $L = S + N$, where $S$ is diagonalizable and $N$ is nilpotent. My textbook (Linear Algebra by ...
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1answer
758 views

Jordan blocks and the characteristic polynomial

For $A \in \mathbb{C}^{n,n}$ and $\{ \lambda_1, \dots , \lambda_r\}$ are the eigenvalues of $A$. Then my lecture notes say that the characteristic polynomial of $A$ is $$(-1)^n\prod_{i=1}^r(x-\...
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2answers
2k views

Jordan block and Jordan chain question?

I don't understand why it's apparently 'clear' that the matrix of $T$ with respect to the basis $v_1, \dots, v_n$ is a Jordan block of degree $n$ if and only if $v_1, \dots ,v_n$ is a Jordan chain for ...
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1answer
1k views

Jordan normal form for complex matrices

Suppose we are given the  characteristic polynomial and minimal polynomial of a matrix, say, $(x-a)^4(x-b)^2$ and $(x-a)^2(x-b)$. Then, I can tell what the largest Jordan blocks are, and hence work ...
0
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1answer
123 views

Polynomials and Jordan form

Problem Let A ($n\times n$ matrix) be a single Jordan block and let $C$ be an $n\times n$ matrix that commutes with $A$. Prove that $C = f(A)$ for some polynomial $f$.
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2answers
3k views

Why does the largest Jordan block determine the degree for that factor in the minimal polynomial?

Let $A$ be a square matrix, so $A$ has some Jordan Normal form. Then $A$ has a minimal polynomial, say $m(X)=\prod_{i=1}^k (t-\lambda_i)^{m_i}$. Wikipedia says The factors of the minimal ...
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1answer
227 views

Alternative proof for Jordan form statement

I am trying to understand why given a nilpotent matrix $L,$ rank ($L^{k-1}$) - rank ($L^{k}$) is the number of Jordan blocks sized $\geq k \times k$ in the Jordan representation of $L.$ There is a ...
3
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1answer
214 views

Jordan Normal Form of a $2 \times 2$ matrix

I wish to prove that any $2\times 2$ matrix $T$ with only one eigenvalue $ \lambda$ of geometric multiplicity 1 is similar to one of the form $\left[\begin{array}{cc} \lambda & 1 \\ 0 &\...
2
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1answer
2k views

Determining the Jordan form of a matrix given the invariant factors

I am trying to recover the Jordan normal form of a matrix given a list of invariant factors and was wondering if I am proceeding correctly in constructing the Jordan blocks. Let $F = \mathbb{C}$ and ...
37
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3answers
6k views

When can two linear operators on a finite-dimensional space be simultaneously Jordanized?

IN a comment to Qiaochu's answer here it is mentioned that two commuting matrices can be simultaneously Jordanized (sorry that this sounds less appealing then "diagonalized" :P ), i.e. can be brought ...
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2answers
344 views

Jordan form and base for a $n \times n$ Matrix

Given this matrix in size $n \times n$ $\begin{pmatrix} 1 &1 &1&\cdots & 1 & 1 & \\ 0 &1 & & & \\ . & &.& & & \\ . &...
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3answers
10k views

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.
3
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1answer
564 views

Finding Jordan form base for $\left(\begin{smallmatrix} 2 &1 &2 \\ -1 &0 &2 \\ 0&0 & 1 \end{smallmatrix}\right)$

I need your help for understanding how to compute jordan base for this matrix: $\begin{pmatrix} 2 &1 &2 \\ -1 &0 &2 \\ 0&0 & 1 \end{pmatrix}$ This is what I tried to do:...
3
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1answer
487 views

Jordan normal form

Good evening, I would love your help with this. I want to know what's jordan normal form of matrix that it's $Characteristic$ $ $ $polynomial$ : $(t-3)^{4}\cdot (t-5)^{4}$ and it's $Minimal$ $ $ $ ...

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