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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29 votes
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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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  • 675
7 votes
1 answer
11k views

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....
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43 votes
3 answers
7k 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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17 votes
3 answers
12k 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.
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12 votes
2 answers
5k 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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  • 121
3 votes
2 answers
7k views

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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3 votes
2 answers
2k views

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 ...
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  • 139
6 votes
0 answers
542 views

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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  • 2,327
18 votes
2 answers
11k views

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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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 ...
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15 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 ...
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6 votes
3 answers
8k 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 $$\...
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11 votes
2 answers
1k 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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  • 1,871
4 votes
1 answer
3k views

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/...
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  • 1,423
1 vote
1 answer
928 views

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 ...
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  • 1,496
3 votes
3 answers
581 views

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. ...
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  • 141
1 vote
1 answer
3k views

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 $...
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0 votes
1 answer
5k views

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)...
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215 votes
6 answers
15k views

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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7 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.
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  • 369
5 votes
1 answer
11k views

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 ...
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  • 6,064
9 votes
1 answer
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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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  • 1,097
11 votes
2 answers
5k 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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9 votes
2 answers
10k views

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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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-...
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  • 635
2 votes
1 answer
3k views

Inverse of the Jordan block matrix

There is the Jordan block matrix $J_\lambda(n):=\begin{pmatrix} \lambda & 1 & & & \\ & \lambda & 1 \\ & & ... & ... \\ & & & \lambda & 1 \\ & ...
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  • 395
6 votes
2 answers
1k views

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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  • 1,732
6 votes
4 answers
988 views

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 ...
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  • 2,241
5 votes
2 answers
825 views

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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  • 51
3 votes
2 answers
485 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 ...
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3 votes
2 answers
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 ...
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3 votes
2 answers
271 views

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 ...
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  • 439
5 votes
3 answers
3k views

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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4 votes
1 answer
4k views

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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4 votes
3 answers
3k views

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}\...
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  • 155
1 vote
1 answer
898 views

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, ...
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5 votes
2 answers
365 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 ...
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5 votes
0 answers
268 views

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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  • 482
3 votes
1 answer
280 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 ...
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  • 4,755
2 votes
1 answer
209 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{...
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  • 1,714
1 vote
1 answer
126 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 ...
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  • 1,067
1 vote
1 answer
186 views

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 ...
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  • 2,173
0 votes
1 answer
272 views

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 ...
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  • 7,534
0 votes
1 answer
35 views

Jordan canonical form and basis for 4 by 4 matrix with two eigenvalues

So, I am given the matrix in standard basis $$A =\begin{bmatrix}-3&1&3&3\\-10&2&9&9\\-4&0&5&4\\2&1&-3&-2\end{bmatrix}$$ characteristic polynomial is $(λ−...
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  • 1
0 votes
1 answer
2k views

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 ...
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53 votes
4 answers
10k views

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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12 votes
1 answer
3k 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 ...
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11 votes
2 answers
3k 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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  • 4,488
4 votes
2 answers
6k views

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 \...
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  • 8,444
4 votes
2 answers
3k views

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 ...
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  • 2,753