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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Why does this matrix give the derivative of a function?

I happened to stumble upon the following matrix: $$ A = \begin{bmatrix} a & 1 \\ 0 & a \end{bmatrix} $$ And after trying a bunch of different examples, I noticed the ...
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4answers
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An intuitive approach to the Jordan Normal form

I want to understand the meaning behind the Jordan Normal form, as I think this is crucial for a mathematician. As far as I understand this, the idea is to get the closest representation of an ...
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Why do we need a Jordan normal form? [duplicate]

My professor said that the main idea of finding a Jordan normal form is to find the closest 'diagonal' matrix that is similar to a given matrix that does not have a similar matrix that is diagonal. I ...
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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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Motivation for Jordan Canonical Form

I took linear algebra and understood the proof that linear operators on a vector space over an algebraically closed field have a Jordan Canonical Form. Why should I care about this theorem? I ...
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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} & \...
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1answer
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Why are there multiple Jordan Blocks corresponding to the same eigenvalue?

Though the title seems clear enough, I'd like to start with a discussion of how I personally came to derive the Jordan Normal Form, because my question is very specific to the details of my derivation....
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“Geometric” problems on the Jordan normal form of a particular operator

Assume you have a class of students more or less familiar with the notion of the matrix of a linear operator. They have seen and calculated lots of examples in various context: geometric ...
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Non-integral powers of a matrix

Question Given a square complex matrix $A$, what ways are there to define and compute $A^p$ for non-integral scalar exponents $p\in\mathbb R$, and for what matrices do they work? My thoughts ...
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2answers
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What is the purpose of Jordan Canonical Form?

I don't claim at all to be an expert on this topic. In many (advanced) linear algebra textbooks for undergraduates, I usually find something about the "Jordan Canonical Form" of a matrix. What is ...
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2answers
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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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3answers
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Find the Jordan normal form of a nilpotent matrix $N$ given the dimensions of the kernels of $N, N^2, N^3$

Let $N\in \text{Mat}(10 \times 10,\mathbb{C})$ be nilpotent. Furthermore let $\text{dim} \ker N =3 $, $\text{dim} \ker N^2=6$ and $\text{dim} \ker N^3=7$. What is the Jordan Normal Form? The only ...
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Finding Jordan Basis of a matrix

Having trouble finding the Jordan base (and hence $P$) for this matrix $A = \begin{pmatrix} 15&-4\\ 49&-13 \end{pmatrix}$ I know that the eigenvalue is $1$, this gives an eigenvector $\...
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2answers
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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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2answers
473 views

Prove that $V = \ker(\phi^n) \oplus \text{image}(\phi^n)$

Let $V$ be a $n$-dimensional complex vector space and $\phi:V\to V$ a linear mapping. Prove that $$V = \ker(\phi^n) \oplus \text{image}(\phi^n)$$ Here is my attempt: Since $\phi^n$ is also a linear ...
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4answers
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Where does the Jordan canonical form show up in more advanced mathematics?

My bounty for this question expires soon :) Edit: in regards to the bounty offered, what current research trends use the Jordan canonical form? If one takes a second course in Linear Algebra — or a ...
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3answers
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Criterion for deciding whether matrix is diagonalizable

Let $B \in$ GL$_n(\mathbb{C})$. In a paper I'm reading someone probably claims the following: Lemma: For showing that $B$ is diagonalizable it suffices to show the following: Let $\lambda$ be an ...
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2answers
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Order of invertible matrices

I recently came across an interesting problem in Artin which says: If $A \in GL_2(\mathbb{Z})$ is of finite order then it has order $1,2,3,4,6$. I was looking for a generalization of this problem. ...
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4answers
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Prove Why $B^2 = A$ exists?

Define $$A = \begin{pmatrix} 8 & −4 & 3/2 & 2 & −11/4 & −4 & −4 & 1 \\ 2 & 2 & 1 & 0 & 1 & 0 & 0 & 0 \\ −9 & 8 & 1/2 & −4 & 31/...
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3answers
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Prove that $ND = DN$ where $D$ is a diagonalizable and $N$ is a nilpotent matrix.

Let $A$ be an $n \times n$ complex matrix. Prove that there exist a diagonalizable matrix $D$ and a nilpotent matrix $N$ such that a. A = D + N b. DN = ND and show that these matrices are uniquely ...
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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 ...
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3answers
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Show that the characteristic polynomial is the same as the minimal polynomial

Let $$A =\begin{pmatrix}0 & 0 & c \\1 & 0 & b \\ 0& 1 & a\end{pmatrix}$$ Show that the characteristic and minimal polynomials of $A$ are the same. I have already computated ...
7
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1answer
975 views

Jordan-Chevalley vs Jordan normal decomposition

I am confused about a proof of the Jordan-Chevalley decomposition I was reading in Peterson's linear algebra book. Let $T : V \to V$ be an $n$-dimensional operator on a complex vector space. The ...
7
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1answer
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Conjugacy classes in $SU_2$

I'm trying to find all conjugacy classes in $SU_2$. Matrices in $SU_2$ are of the form: $M = \begin{bmatrix} \alpha & \beta \\ - \bar{\beta} & \bar{\alpha} \end{bmatrix}$...
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3answers
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What's the Jordan canonical form of this matrix?

given is the $6 \times 6$-matrix $A$: $A = \begin{pmatrix} 0 & 1 & 0 & -1 & 0 & 0 \\ 0 &0&1&1&-1&0\\ -1&0&0&0&-1&-1 \\ 1 & 0&0&...
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4answers
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Find Jordan Decomposition of $\begin{pmatrix} 4 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 3 \end{pmatrix}$ over $\mathbb{F}_5$

Find the Jordan decomposition of $$ A := \begin{pmatrix} 4 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 3 \end{pmatrix} \in M_3(\mathbb{F}_5), $$ where $\mathbb{F}_5$ is the field ...
6
votes
1answer
440 views

Let $A$ be a $5 \times 5$ matrix such that $A^2=0$. Then how to compute the maximum rank for such A?

Attempt : Suppose $A$ has a non-zero eigenvalue $\lambda$. Then corresponding to it's non-zero eigen vector $X$, we have $AX=\lambda X \Rightarrow A^2X=\lambda^2 X\Rightarrow 0=\lambda^2 X$. Which is ...
6
votes
2answers
81 views

Center of a non-abelian subgroup of $GL(2, \mathbb{C})$

I'm trying to do the following exercise: Let be $G$ a non-abelian subgroup of $GL(2, \mathbb{C})$. Prove that the center of $G$ is contained in the center of $GL(2, \mathbb{C})$ My (very partial) ...
6
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1answer
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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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1answer
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How to find the Jordan canonical form of tensor products.

Let $k$ be a field, $ A \in M_{m\times m}(k)$ be a single Jordan block with eigenvalue $a$, and $B \in M_{n\times n}(k)$ be a single Jordan block with eigenvalue $b$. $A$ and $B$ together define a ...
6
votes
1answer
218 views

Existence of Jordan decomposition over finite field

Prove that over finite field $\mathbb F$ exists additive Jordan-Chevalley decomposition: for all matrix $M$ there are semisimple matrix $M_{s}$ and nilpotent matrix $M_{n}$ such that $M=M_{s}+M_{n}$. ...
6
votes
0answers
62 views

Find a flag to transform a matrix to an upper triangular one

Consider $F: \mathbb{R^3} \to \mathbb{R^3}$ represented by: $ A= \begin{bmatrix} 1 & 1 & 2 \\ -2 & 5 & 6 \\ 1 & -2 & -2 \\ \end{bmatrix} $ , eigenvalues: $...
6
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0answers
807 views

An inverse of Jordan matrix - basis

Let $A\in M_{n\times n}$ be and invertible matrix over complex field and we assume it's already at Jordan form where $B=\{v_1,…,v_n \}$ is Jordan basis for A. Find Jordan form and Jordan basis for $...
6
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1answer
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An explanation for a Jordan normal form proof from the Kaye and Wilson book

In this proof of Jordan normal form in the Kaye and Wilson book, then for a transformation $T$ with minimal polynomial $m(x) = (x-e)^k$, they take a basis of $\texttt{ker}\;T$, extend it to a basis of ...
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2answers
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Help in finding the Jordan canonical form of a matrix

Determine the Jordan Canonical Form of the following matrix: $$A=\begin{bmatrix} 1 & 2 & 3\\ 0 & 4 & 5\\ 0 & 0 & 4\\ \end{bmatrix}$$ I am trying to determine the Jordan ...
5
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3answers
764 views

Matrices over a finite field with given Jordan normal form over the algebraic closure

Can one describe the (conjugacy classes of) square matrices over a finite field such that over the algebraic closure of this finite field their Jordan normal form consists of one Jordan block? (Such ...
5
votes
2answers
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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 &...
5
votes
3answers
109 views

Prove that there exists number $k\in \mathbb{N}$ such that $ V = \operatorname{Ker}A^{k} \dot{+} \operatorname{Im}A^{k}$

Problem: Let A be linear operator A $\in L(V)$. Prove that there exists number $k\in \mathbb{N}$ such that $ V = \operatorname{Ker}A^{k} \dot{+} \operatorname{Im}A^{k}$. Then prove that operator $\...
5
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2answers
137 views

Prove $\exp(\mathrm{Tr}(X))=\det(\exp(X))$

Show that $\exp(\mathrm{Tr}(X))=\det(\exp(X))$ where $X$ is a matrix using the concept of the Jordan normal form I realised this formula by considering that: $\det(\exp(X))=\exp(\lambda_1) \times\...
5
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1answer
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Proof for real Jordan canonical form

Let A $\epsilon$ Mat(nxn, $\mathbb R$) be a matrix that is diagonalizable in $\mathbb C$ with k real eigenvalues of algebraic multiplicity 1 and (n-k)/2 pairs of complex-conjugated eigenvalues of ...
5
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3answers
938 views

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 $$\...
5
votes
2answers
174 views

Find Jordan form of a $3\times 3$ matrix

$$\left( \begin{array}{ccc} 0 & 1 & 2 \\ -5 &-3 & -7 \\ 1 & 0 & 0 \end{array} \right) $$ I figured out the eigenvalues are all -1 from the characteristic polynomial, but I'm ...
5
votes
2answers
156 views

Prove that two operators have a common eigenvector

Suppose $S,T:\mathbb C^3\to\mathbb C^3$ are linear operators. The degree of the minimal polynomial of each of the operators is at most 2. Show that they share a common eigenvector. I tried to ...
5
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1answer
174 views

Jordan Normal Form - Number of Ones on Superdiagonal

I was reading notes on Jordan Normal Form and it says that for a given matrix $A$, the number of ones on the super-diagonal of its associated Jordan matrix is equal to $n-d$, however they seem to ...
5
votes
1answer
146 views

Text recommendations for linear algebra (tensors, jordan forms)

I'm having extreme difficulty trying to understand to topic of tensor products, freespaces, and jordan forms. Are there any text books that take an elementary approach to these topics that you may ...
5
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1answer
788 views

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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2answers
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What exactly determines the block-sizes for Jordan forms?

For instance, with $T \in \mathcal{L}$(Mat($2,2,\mathbb{C}$)) we are given that the minimal polynomial of $T$ is $p(z) = (z - 2i)(z + 7)^2$. I want to find the possible Jordan Forms pertaining to this ...
5
votes
3answers
173 views

$n$-th root of $3 \times 3$ invertible matrix

Yo, I couldn't solve this exercise after thinking for a while. For every $A \in GL_{3} (\mathbb{C})$ and $n$, there's a $B \in Mat_{3, 3}(\mathbb{C})$ such that $B^n = A$ The previous exercise was ...
5
votes
1answer
362 views

Finding the Jordan Canonical form of a $6 \times 6$ matrix

Find the Jordan Canonical Form of the following matrix $$\begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0\\ 1 & 1 & 0 & 0 & 0 & 0\\ 1 & 0 & 1 & 0 & 0 &...
5
votes
1answer
140 views

Describe the finite order integer matrices over complex field

I stuck so much on this question! I need to describe finite order integer matrices without 1 eigenvalues over $\mathbb C$. I need description in terms of classes of equivalent matrices(such that ...