Frequent 'jordan-normal-form' Questions

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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} & \...

asked Mar 10 '13 at 18:34

user34295

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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 ...

linear-algebra linear-transformations jordan-normal-form

asked Jul 4 '11 at 9:45

Mark

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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.

matrices jordan-normal-form matrix-exponential

asked Apr 19 '11 at 16:39

Ben Derrett

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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 ...

linear-algebra matrices jordan-normal-form minimal-polynomials

asked Nov 16 '11 at 7:20

Boma

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votes

1answer

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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 ...

linear-algebra matrices exponential-function matrix-calculus jordan-normal-form

asked Sep 25 '15 at 17:37

user167575

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votes

2answers

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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 ...

linear-algebra abstract-algebra matrices jordan-normal-form nilpotence

asked Mar 22 '15 at 11:43

user142198

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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 ...

linear-algebra matrices jordan-normal-form

asked Nov 23 '14 at 5:44

Clarinetist

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votes

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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 ...

linear-algebra matrices diagonalization jordan-normal-form

asked May 10 '15 at 2:12

rackne

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votes

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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 $$\...

linear-algebra cyclic-groups jordan-normal-form cyclic-decomposition

asked Nov 10 '16 at 7:40

User8976

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votes

2answers

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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 ...

linear-algebra jordan-normal-form

asked Sep 4 '14 at 11:51

rehband

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vote

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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 ...

linear-algebra matrices jordan-normal-form

asked Dec 14 '16 at 4:16

Max

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votes

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Finding Jordan form

Find Jordan form of the following matrix: $$\left(\begin{matrix}4&-5&2 \\ 5&-7&3\\ 6&-9&4 \end{matrix}\right)$$ So I got stuck pretty much trying to find the eigenvalues. ...

linear-algebra matrices eigenvalues-eigenvectors jordan-normal-form

asked Dec 9 '13 at 15:49

err

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Jordan normal form for a characteristic polynomial $(x-a)^5$

Write down all the possible Jordan normal forms for matrices with characteristic polynomial $(x-a)^5$. In each case, calculate the minimal polynomial and the geometric multiplicity of the eigenvalue $...

linear-algebra matrices jordan-normal-form

asked Nov 21 '13 at 16:21

user101293

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Finding Jordan Canonical form for 3x3 matrix

I was looking at http://www.math.hkbu.edu.hk/~zeng/Teaching/math3407/Jordan_Form.pdf (section 2) $A =\left(\begin{array}{ccc}4 & 0 & 1 \\2 & 3 & 2 \\1 & 0 & 4\end{array}\right)...

linear-algebra jordan-normal-form

asked Apr 15 '15 at 17:33

larry

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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 ...

linear-algebra matrices derivatives matrix-calculus jordan-normal-form

asked Nov 22 '15 at 21:43

ASKASK

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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.

linear-algebra matrices exponentiation jordan-normal-form

asked Jul 5 '16 at 14:58

linalg

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Finding Jordan Canonical form given the minimal and characteristic polynomial.

I have the following information: the characteristic polynomial of $A$ is $p_A(t)=(t-4)^3(t+6)^2$ and the minimal polynomial is $q_A(t)=(t-4)^2(t+6).$ I'm having problems seeing how one would work ...

linear-algebra matrices jordan-normal-form

asked Mar 31 '15 at 2:16

emka

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votes

3answers

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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 ...

linear-algebra complex-numbers eigenvalues-eigenvectors exponentiation jordan-normal-form

asked Dec 22 '13 at 14:40

MvG

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10

votes

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{...

linear-algebra matrices jordan-normal-form

asked Nov 1 '12 at 8:11

N3buchadnezzar

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votes

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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 ...

linear-algebra abstract-algebra matrices jordan-normal-form

asked Jan 6 '13 at 0:11

Frank White

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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?

linear-algebra jordan-normal-form

asked Aug 23 '15 at 18:27

MATH

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votes

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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 \...

matrices functional-analysis eigenvalues-eigenvectors markov-chains jordan-normal-form

asked May 26 '14 at 2:15

Rookatu

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votes

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prove that for any nonsingular matrix $A$ there exist $X$ such that $X^2=A$

Prove that given any matrix A, where $$\det(A)\neq0$$ $$A\in M_{n,n}(\mathbb C)$$ the following equation $$X^2=A$$ always has a solution. Should I do something with Jordan Normal form? Any help will ...

linear-algebra matrices jordan-normal-form

asked Mar 16 '18 at 0:10

Igor

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votes

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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 ...

linear-algebra matrices matrix-decomposition jordan-normal-form

asked Mar 21 '13 at 6:50

Dorothy Kimmel

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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 $n \times n$ ...

linear-algebra matrices jordan-normal-form

asked Dec 13 '13 at 7:56

GeorgeShi

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votes

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Finding Jordan basis of a matrix $(4\times 4)$

I'm facing a problem finding a Jordan basis for this ($4 \times 4$) matrix: $$\left(\begin{matrix}3&-1&1&7\\9&-3&-7&-1\\0&0&4&-8\\0&0&2&-4\end{matrix}\...

linear-algebra jordan-normal-form

asked Jan 5 '16 at 18:58

Misha

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votes

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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.

linear-algebra matrices jordan-normal-form

asked Jun 27 '13 at 19:42

user66407

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2answers

375 views

If $(A-2I)^3(A+2I)^2=0$, then what are the possible Jordan canonical forms of $A$?

Here is the exercise: Let $A$ be a $5\times5$ complex matrix such that $(A-2)^3(A+2)^2=0$, where we define $A-\mu:=A-\mu I$ for scalar $\mu$. Assume that $\lambda=2$ is an eigenvalue of $A$ and its ...

linear-algebra matrices eigenvalues-eigenvectors jordan-normal-form

asked Dec 5 '13 at 14:58

user9464

2

votes

1answer

2k views

Jordan canonical form of an upper triangular matrix

Find the Jordan canonical form of the matrix. Justify your answer. $A=\begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 4 \end{bmatrix} $ My Try: The eigenvalues are $...

linear-algebra matrices jordan-normal-form

asked Jul 31 '15 at 6:33

Extremal

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Jordan normal form and invertible matrix of generalized eigenvectors proof

Struggling to find a place to start with this proof- just began learning about Jordan normal. Given a 2-by-2 matrix $A$ and a Jordan normal form matrix $J_{\lambda}$, there exists a matrix $S = [v1, ...

linear-algebra matrices jordan-normal-form

asked Oct 24 '15 at 1:34

sir_thursday

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votes

2answers

296 views

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 ...

linear-algebra jordan-normal-form

asked Jul 3 '16 at 8:38

quicksilver

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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 ...

linear-algebra matrices polynomials jordan-normal-form

asked Dec 30 '15 at 11:00

Who knows

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Size of Jordan block

Imagine that I'm writing the Jordan form of a matrix and I know that the eigenvalue needs to appear 4 times in the diagonal (algebraic multiplicity is 4) and we need 2 Jordan blocks (geometric ...

linear-algebra jordan-normal-form

asked Jan 13 '16 at 13:01

Granger Obliviate

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Finding Jordan basis of a matrix ($3\times3$ example)

Our teacher didn't explain us how to find it so I've had to look up a bit by myself. I have this matrix : $$A = \begin{pmatrix} 9 & 4 & 5 \\ -4 & 0 & -3 \\ -6 & -4 & -2 \end{...

linear-algebra jordan-normal-form

asked Jun 2 '16 at 14:04

Desura

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If we know the eigenvalues of a matrix $A$, and the minimal polynom $m_t(a)$, how do we find the Jordan form of $A$?

We have just learned the Jordan Form of a matrix, and I have to admit that I did not understand the algorithm. Given $A = \begin{pmatrix} 1 & 1 & 1 & -1 \\ 0 & 2 & 1 & -1 \\ 1 ...

linear-algebra matrices jordan-normal-form

asked May 19 '13 at 18:21

TheNotMe

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$2\times2$ matrix $A$ such that $A$ has one independent eigenvector while $A^{2}$ has two independent eigenvectors

Give an example of $2\times2$ matrix $A$ such that $A$ has one independent eigenvector while $A^{2}$ has two independent eigenvectors. I would like to know a systematic answer of how to get this. My ...

linear-algebra matrices eigenvalues-eigenvectors determinant jordan-normal-form

asked Nov 10 '14 at 17:01

Silver moon

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How to determine the Jordan form and give a Jordan base for a matrix?

given is $\begin{pmatrix} 3&0&-1&0&0 \\ 1&3&0&1&0 \\ 0&0&3&0&0 \\ 0&0&0&3&0 \\ 0&0&0&0&-3 \end{pmatrix}$ I have to ...

linear-algebra matrices jordan-normal-form

asked Oct 21 '14 at 12:39

Vazrael

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The exponential of a Jordan block

Is it true that the exponential of a Jordan block is an upper triangular matrix? I tried two examples and got just diagonal matrices which may be a coincidence, as diagonal matrices are also upper/...

linear-algebra matrices jordan-normal-form matrix-exponential

asked Apr 27 '19 at 12:28

Leif

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Superdiagonal for the Jordan form of a Jordan block power

The question is an extension of the Prove that $A$ is similar to $A^n$ based on A's Jordan form. Let $J$ be Jordan block of any form. In what circumstances Jordan form of power $J^n$ has the ...

linear-algebra jordan-normal-form

asked Jul 12 '17 at 13:45

Widawensen

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Computation of transformation matrix for jordan normal form: how to choose eigenvectors

During this semester at university we we're introduced to the jordan normal form of a matrix. While we never wrote down an explicit algorithm of how to find the matrix $B$, such that $B^{-1}AB$ is a ...

linear-algebra matrices jordan-normal-form

asked Jul 12 '15 at 15:36

Stefan Perko

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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 ...

linear-algebra matrices intuition jordan-normal-form

asked Jun 5 '13 at 10:08

user66906

12

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1answer

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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 ...

linear-algebra matrices jordan-normal-form

asked Sep 10 '17 at 7:06

Ethan Alwaise

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4

votes

2answers

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Jordan canonical form of a squared matrix

I was wondering: if I were given a matrix $A$, I could calculate its Jordan canonical form. If I considered then $A^2$, I could say that if $\lambda$ is an eigenvalue of $A$, then $\lambda^2$ is an ...

linear-algebra jordan-normal-form

asked Sep 3 '16 at 14:02

tommy1996q

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Find all possible Jordan Canonical forms for a nilpotent matrix

$A$ is a $10 \times 10$ nilpotent matrix of order $4$ ($A^4=0$) over $\mathbb C$ with $\operatorname{rank} (A)=6$. Find all possible Jordan Canonical forms The nullity of $A$ is $4$ so there are $4$ ...

linear-algebra matrices jordan-normal-form nilpotence

asked Jul 4 '16 at 9:03

asaf92

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Why is the geometric multiplicity of an eigen value equal to number of jordan blocks corresponding to it?

Geometric multiplicity of an eigen value is $$ \dim \mathrm{null} (A -\lambda I)\tag 1.$$ Suppose $A$ is in jordan normal form and has two Jordan forms with eigen value $\lambda$, one of size $2 \...

linear-algebra matrices eigenvalues-eigenvectors jordan-normal-form

asked Oct 31 '13 at 11:58

MathMan

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votes

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What commutes with a matrix in Jordan canonical form?

The question I would like answered is the following: Given a matrix $G$ and that $G$ commutes with another matrix $X$, that is $[G, X] = 0$, what is $X$? Or more generally, what properties of $X$ may ...

linear-algebra matrices matrix-equations jordan-normal-form

asked Feb 20 '14 at 15:39

P. Plowman

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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 ...

linear-algebra matrices jordan-normal-form minimal-polynomials characteristic-polynomial

asked Apr 7 '13 at 19:49

mathnoob

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Inverse of the Jordan block matrix

There is the Jordan block matrix $J_\lambda(n):=\begin{pmatrix} \lambda & 1 & & & \\ & \lambda & 1 \\ & & ... & ... \\ & & & \lambda & 1 \\ & ...

matrices inverse jordan-normal-form

asked May 12 '18 at 11:12

Gurterz

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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/...

linear-algebra matrices jordan-normal-form

asked May 17 '16 at 22:27

AriNubar

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$n$-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 ...

linear-algebra jordan-normal-form nilpotence unipotent-matrices

asked Oct 13 '16 at 19:39

user40276

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