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.
131
questions
33
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2
answers
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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} & \...
8
votes
1
answer
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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 correct....
45
votes
3
answers
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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 ...
17
votes
3
answers
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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.
13
votes
2
answers
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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 ...
4
votes
2
answers
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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 ...
7
votes
5
answers
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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
$$\...
25
votes
2
answers
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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 ...
7
votes
0
answers
804
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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\...
3
votes
2
answers
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Jordan form, number of blocks. [closed]
Suppose I have an eigenvalue $\lambda$, now I want to determine the number of Jordan blocks corresponding to that eigenvalue, as well as size of each block. I know that:
number of blocks is equal to ...
17
votes
3
answers
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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
...
11
votes
2
answers
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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 ...
9
votes
3
answers
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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 ...
8
votes
1
answer
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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.
4
votes
1
answer
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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/...
3
votes
3
answers
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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.
...
1
vote
1
answer
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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 $...
1
vote
1
answer
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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 ...
0
votes
1
answer
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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)...
223
votes
6
answers
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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 ...
13
votes
1
answer
4k
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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 ...
12
votes
2
answers
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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?
11
votes
2
answers
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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{...
9
votes
1
answer
6k
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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 ...
8
votes
3
answers
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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 ...
6
votes
2
answers
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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 \...
6
votes
2
answers
486
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 ...
6
votes
4
answers
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A linear map $T: \mathbb{R^3 \to \mathbb{R^3}}$ has a two dimensional invariant subspace.
Let $T: \mathbb{R^3 \to \mathbb{R^3}}$ be an $\mathbb{R}$-linear map. Then I want to show that $T$ has a $2$ dimensional invariant subspace of $\mathbb{R^3}.$
I considered all possible minimal ...
6
votes
3
answers
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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 $n \times n$ ...
5
votes
1
answer
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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 ...
5
votes
1
answer
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Proof for real Jordan canonical form
Let $A \in \operatorname{Mat}(n\times n, \mathbb{R})$ be a matrix that is diagonalizable in $\mathbb C$ with $k$ real eigenvalues of algebraic multiplicity $1$ and $(n-k)/2$ pairs of complex-...
5
votes
0
answers
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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 ...
5
votes
1
answer
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How do I find Jordan basis?
I have a matrix:
$$A=\begin{pmatrix}0&1&0\\-4&4&0\\-2&1&2\end{pmatrix}$$
solving $\det|A-\lambda{I}|$ I got characteristic polynom that equals to $(2-\lambda)^3 = 0$ for ...
4
votes
3
answers
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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:
$$\left(\begin{matrix}3&-1&1&7\\9&-3&-7&-1\\0&0&4&-8\\0&0&2&-4\end{matrix}\...
4
votes
1
answer
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The index of nilpotency of a nilpotent matrix
Let $A$ a matrix in $\mathcal{M}_5(\mathbb C)$ such that $A^5=0$ and $\mathrm{rank}(A^2)=2$, how prove that $A$ is nilpotent with index of nilpotency $4$? Thanks in advance.
3
votes
1
answer
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Inverse of the Jordan block matrix
There is the Jordan block matrix
$J_\lambda(n):=\begin{pmatrix} \lambda & 1 & & & \\ & \lambda & 1 \\ & & ... & ... \\ & & & \lambda & 1 \\ & ...
3
votes
2
answers
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Uniqueness of the Jordan decomposition
I have seen it said that a matrix $M$ (over $\mathbb{C}$, say) has a unique decomposition $M = D + N$ where $D$ is diagonal and $N$ is nilpotent. I'm having trouble seeing this, since the Jordan form ...
3
votes
2
answers
586
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If $(A-2I)^3(A+2I)^2=0$, then what are the possible Jordan canonical forms of $A$?
Here is the exercise:
Let $A$ be a $5\times5$ complex matrix such that $(A-2)^3(A+2)^2=0$, where we define $A-\mu:=A-\mu I$ for scalar $\mu$. Assume that $\lambda=2$ is an eigenvalue of $A$ and its ...
3
votes
2
answers
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prove that for any nonsingular matrix $A$ there exist $X$ such that $X^2=A$
Prove that given any matrix A, where $$\det(A)\neq0$$ $$A\in M_{n,n}(\mathbb C)$$
the following equation
$$X^2=A$$
always has a solution.
Should I do something with Jordan Normal form?
Any help will ...
3
votes
1
answer
299
views
If we know the eigenvalues of a matrix $A$, and the minimal polynom $m_t(a)$, how do we find the Jordan form of $A$?
We have just learned the Jordan Form of a matrix, and I have to admit that I did not understand the algorithm.
Given $A = \begin{pmatrix} 1 & 1 & 1 & -1 \\ 0 & 2 & 1 & -1 \\ 1 ...
2
votes
1
answer
256
views
Finding Jordan basis of a matrix ($3\times3$ example)
Our teacher didn't explain us how to find it so I've had to look up a bit by myself.
I have this matrix :
$$A = \begin{pmatrix} 9 & 4 & 5 \\ -4 & 0 & -3 \\ -6 & -4 & -2 \end{...
1
vote
2
answers
95
views
Jordan decomposition - help with calculation of transformationmatrices?
Let $$A=\begin{pmatrix}5&1&1 \\-1&3&1\\0&0&4\end{pmatrix}$$
The Jordan-decomposition is $$A=\begin{pmatrix}-1&-1&-1/2 \\1&0&0\\0&0&-1/2\end{pmatrix}\...
1
vote
1
answer
153
views
$2\times2$ 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 ...
1
vote
1
answer
197
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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 ...
1
vote
1
answer
1k
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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, ...
0
votes
1
answer
2k
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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 ...
0
votes
2
answers
312
views
Jordan normal form (Basis)
Define $A = \begin{pmatrix} -7 & -32 & -32 & -35 \\ 1 & 5 & 4 & 4 \\ 1 & 4 & 5 & 5 \\ 0 & 0 & 0 & 1 \end{pmatrix} \in \mathbb{C^{4x4}}$
I computed ...
0
votes
1
answer
360
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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 ...
60
votes
4
answers
12k
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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 ...
15
votes
1
answer
2k
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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....