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.
80
questions
18
votes
2answers
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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:
$$
J_k(\lambda)^n = \begin{bmatrix}
\lambda^n & \binom{n}{1}\lambda^{n-1} & \...
17
votes
3answers
10k views
How to calculate the matrix exponential explicitly for a matrix which isn't diagonalizable?
How can I compute an expression for $(\exp(Qt))_{i,j}$ for some fixed $i, j$ and matrix $Q$?
When $Q$ is diagonalizable, we can diagonalize, but what can be done otherwise?
Thanks.
3
votes
2answers
1k views
All nilpotent $2\times 2$ matrices
I want to find all nilpotent $2\times 2$ matrices.
All nilpotent $2 \times 2$ matrices are similar($A=P^{-1}JP$) to $J = \begin{bmatrix} 0&1\\0&0\end{bmatrix}$
But how do I find all of these ...
11
votes
2answers
3k views
Why does the largest Jordan block determine the degree for that factor in the minimal polynomial?
Let $A$ be a square matrix, so $A$ has some Jordan Normal form. Then $A$ has a minimal polynomial, say $m(X)=\prod_{i=1}^k (t-\lambda_i)^{m_i}$.
Wikipedia says
The factors of the minimal ...
11
votes
2answers
7k views
What is the purpose of Jordan Canonical Form?
I don't claim at all to be an expert on this topic. In many (advanced) linear algebra textbooks for undergraduates, I usually find something about the "Jordan Canonical Form" of a matrix.
What is ...
8
votes
3answers
3k views
Prove that $ND = DN$ where $D$ is a diagonalizable and $N$ is a nilpotent matrix.
Let $A$ be an $n \times n$ complex matrix. Prove that there exist a diagonalizable matrix $D$ and a nilpotent matrix $N$ such that
a. A = D + N
b. DN = ND
and show that these matrices are uniquely ...
5
votes
3answers
1k views
Let $A$ be an $n \times n$ matrix with real entries such that $A^2 + I = 0$ then $n$ is even.
Let $A$ be an $n \times n$ matrix with real entries such that $A^2 + I = 0$ then $n$ is even. And if $n = 2k$, then $A$ is similar over the field of real numbers to a matrix of the block form
$$\...
11
votes
2answers
527 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 ...
3
votes
3answers
433 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.
...
1
vote
1answer
2k 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 $...
1
vote
1answer
395 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 ...
-1
votes
1answer
4k 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)...
12
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
...
11
votes
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{...
9
votes
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 ...
5
votes
2answers
692 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 \...
3
votes
2answers
188 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 ...
4
votes
3answers
2k 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 $n \times n$ ...
4
votes
2answers
5k 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?
4
votes
1answer
3k 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.
3
votes
2answers
333 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 ...
1
vote
1answer
379 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, ...
1
vote
1answer
1k 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 $...
5
votes
0answers
192 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 ...
4
votes
2answers
207 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 ...
2
votes
1answer
6k views
Matrix exponential using the Jordan form
How do I calculate the matrix exponential $\Bbb e^{At}$ for $A = \left( \begin{matrix} 1 & 0 & 0 \\ 0 & 2 & 3 \\ 0 & 0 & 2 \end{matrix} \right)$ using the Jordan form of $A$? I ...
2
votes
2answers
2k views
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 ...
2
votes
1answer
139 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{...
2
votes
1answer
233 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
vote
1answer
87 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 ...
0
votes
1answer
726 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 ...
0
votes
1answer
192 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 ...
45
votes
4answers
7k 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 ...
8
votes
1answer
1k 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 ...
4
votes
1answer
1k views
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.
0
votes
1answer
4k 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 ...
1
vote
1answer
811 views
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 ...
8
votes
4answers
1k views
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/...
5
votes
3answers
176 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
79 views
Can we reduce finding matrix roots to finding roots of Jordan blocks?
I just found some interesting question about matrix square roots and I came to think of one way to find them, or at least reduce them to a set of simpler problems.
Assume we have a matrix $\bf A$ and ...
5
votes
1answer
242 views
Jordan normal form over $\mathbb{C}$
Let there be $T:\mathbb{C}^8\rightarrow \mathbb{C}^8$
Such that $
T\left(\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \\ x_{5} \\ x_{6} \\ x_{7} \\ x_{8} \end{array}\right)=\left(\begin{...
4
votes
0answers
154 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\...
4
votes
1answer
438 views
Is the matrix norm of a matrix equal to the maximum of the norms of its Jordan block?
Let $J$ be a Jordan block matrix with blocks $J_1,\cdots,J_n$. I came up with some examples of $J$ and noticed that $\|J\|=\max_{i=1,\cdots,n}\|J_i\|$. Does this result always hold?
The norm I use ...
4
votes
1answer
356 views
Jordan Form Superdiagonal
How do you know how many of the super diagonal entries in the Jordan Form are zeros and how many are ones, and where they are placed?
Thanks.
3
votes
1answer
601 views
How to find the 'real' jordan canonical form of a matrix
Given that the the Jordan normal form of a matrix is,
$J=\begin{bmatrix}2&1&0&0\\0&2&0&0\\0&0&1-i&0\\0&0&0&1+i\end{bmatrix}$
How do you find the 'real'...
2
votes
1answer
161 views
How to turn this matrix to Jordan normal form?
Matrix $A$ is $
\left( \begin{array}{ccc}
3 & 0 & 8 \\
3 & -1 & 6 \\
-2 & 0 & -5 \end{array} \right)$ and I need to find a matrix P such that $P^{-1} A P = J$ where $J$ is a ...
2
votes
1answer
85 views
Prove that $A$ is similar to $A^n$ based on A's Jordan form
Let $A =
\begin{bmatrix}1&-3&0&3\\-2&-6&0&13\\0&-3&1&3\\-1&-4&0&8\end{bmatrix}$,
Prove that $A$ is similar to $A^n$ for each $n>0$.
I found that ...
2
votes
2answers
277 views
On the rank inequality $\operatorname{rank}(A)+\operatorname{rank}(A^3)\geq2\operatorname{rank}(A^2)$
So, I have to show that for all square matrices $A$,
$$\newcommand{\rank}{\operatorname{rank}}\rank(A)+\rank(A^3)\geq2\rank(A^2).$$
My attempt at it so far:
So, if I try to use this theory:
Let $A$ ...
2
votes
0answers
546 views
Complex eigenvalues Jordan real matrix
As I posted here and here I'm studying Jordan forms and similar concepts.
I've got a problem with complex eigenvalues in jordan real matrices. I know (at least I think so) how to compute the Jordan ...
1
vote
1answer
183 views
Matrix reduction trigonalisaton
Let
$
\mathbf{A}=\begin{bmatrix}
2 & -1 & -1 \\
2 & 1 & -2\\
3 & -1 & -2
\end{bmatrix}
$
Trigonalise a matrix
in process of ...
