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.
875
questions
0
votes
1answer
19 views
Obtaining the change of basis matrix to the Jordan matrix
Introduction and description of my problem
I have trouble when finding the matrix change of base $P$ that allows me to obtain the Jordan form from the matrix $A$, in other words, find $P$ that ...
0
votes
1answer
43 views
Is there a way to calculate exponent $n$ in matrix vector product: $w=M^nv$
Find $n$ for given square matrix $M$ and vectors $v,w$ in $$w=M^nv$$
Trial (updated)
(as vujazzman suggested)
Jordan normal form:
$$ w = (A J^n A^{-1})v$$
$$ A^{-1}w = J^n A^{-1}v$$
After this ...
0
votes
0answers
5 views
Proof connecting Jordan canonical basis and cycles of generalized eigenvectors
Let T be an operator on a finite dimensional vector space V. Let B be an ordered basis for V. Prove that B is a Jordan canonical basis if and only if B is the disjoint union of cycles of generalized ...
2
votes
2answers
64 views
Proof that SN (or Jordan-Chevalley) Decomposition is unique?
Let $M$ be a matrix with entries in $\mathbb C$. The SN (or Jordan-Chevalley) decomposition theorem states that we can find unique matrices $S$ and $N$ such that:
$M=S+N$
$S$ is diagonalizable
$N$ is ...
1
vote
1answer
49 views
Eigenvalues of matrix exponential and its Jordan form
Given a matrix $A$, we can write the Jordan decomposition as $$A=SJS^{-1}$$
My question is whether the followings now holds:
$$\text{eig}(e^{At})=\text{eig}(e^{Jt})$$
I've tried relating the ...
2
votes
2answers
68 views
Linear maps satisfying $T^2 = I_n$ and Jordan Form of $T(X) = AX - XA$
Let $V$ be any $n$-dimensional vector space and $W = M_2(\mathbb{C})$.
(a) Construct all linear maps $T: V \to V$ such that $T^2 = I_n$.
For $T^2 = I_n$, taking $p(X) = X^2 - 1$ we must have $p(...
1
vote
1answer
21 views
Jordan Normal Form of $A_{\pi}: \mathbb{C}^n \to \mathbb{C}^n$ given by $A_{\pi}(v) = A_{\pi}(v_1,…,v_n) = (v_{\pi(1)},…,v_{\pi(n)})$.
Let $\pi \in S_n$ be a permutation. Prove that $A_{\pi}: \mathbb{C}^n \to \mathbb{C}^n$ given by $A_{\pi}(v) = A_{\pi}(v_1,...,v_n) = (v_{\pi(1)},...,v_{\pi(n)})$. Show that $A_{\pi}$ is linear and ...
2
votes
1answer
82 views
Is this a family of similar matrices $\left(\begin{smallmatrix} 0&x\\ 0&0 \end{smallmatrix}\right)$?
Is matrix $A = \begin{pmatrix} 0&1\\ 0&0 \end{pmatrix}$ similar to matrix $B=\begin{pmatrix} 0&2\\ 0&0 \end{pmatrix}$? If so, how do I prove this?
I came here from following the ...
1
vote
1answer
76 views
Matrix exponential using Jordan form
I'm trying to calculate the matrix exponential $e^{At}$ for $$A=\frac{1}{2}\begin{bmatrix}-1&1&-1\\2&-2&0\\1&-1&-1\end{bmatrix}$$
I found the eigenvalues $\lambda_1=\lambda_2=-...
2
votes
1answer
68 views
Finding the Jordan canonical form when the characteristic polynomial does not split?
A problem on the 2009 qualifying exam for Harvard is the following:
Suppose $\phi$ is an endomorphism of a 10-dimensional vector space over $\mathbb{Q}$ with the following properties:
The ...
4
votes
1answer
53 views
Understanding the proof of Jordan normal form, by matrix trick
I read a proof for JNF which I am unclear.
Proof:
For any complex matrix $A$. Assume $Av_1=\lambda v_1$ for some $v_1 \in \mathbb{C}^n$, then $A(v_1, \cdots, v_n)=(v_1, \cdots, v_n)\begin{pmatrix}\...
2
votes
1answer
33 views
Show exists a subspace $W \subseteq \mathbb{C}^{n}$ of dimension $1$ such that every Jordan basis of $ \mathbb{C}^{n}$ contains a generator of $W$
Let $n\geq 2$.
Given $f$ nilpotent endomorphism of $\mathbb{C}^{n}$ such that exists an integer $k \geq 1$ such that $dim \hspace{0.1cm} Kerf^{k+1} = dim \hspace{0.1cm} Kerf^{k}+1$.
$(1) \hspace{0....
3
votes
2answers
167 views
Find a Jordan basis for the linear operator $T$
Find a possible Jordan basis for the linear operator $T$ such that:
$T(x, y, z, t) = (2y, −2x + 4y, z + t, z + t)$
Is there an specific method to find a Jordan basis? Since I'm teaching myself I'm ...
1
vote
1answer
40 views
System of equations involving complex eigenvalues
Consider the following equation:
$$ x_{n+2}-2ax_{n+1}+x_n=0$$
a) Define a new auxiliary variable $y_n = x_{n+1}$ and rewrite the previous equation as a discrete, two-equation dynamical system.
b) What ...
2
votes
1answer
32 views
Finding the Jordan form given nullities
The question:
"Let $B$ be a $10 \times 10$ matrix and let $\lambda$ be a scalar. Suppose it is known that
$$
\text{nullity}(B - \lambda I) = 5, \\
\text{nullity}(B - \lambda I)^2 = 8, \\
\text{...
0
votes
1answer
25 views
Jordan canonical form with 2 is the dimension of eigenespace,
$$B= \begin{pmatrix}
0 & 1 & 0 \\
-4 & 4 & 0 \\
-2 & 1 & 2 \end{pmatrix}$$ I need to find Jordan decomposition of B.
My sketch:
We find Jordan's matrix ...
2
votes
1answer
66 views
Similarity class of $3 \times 3$ matrices with entries in $\mathbb{F}_3$
I've been trying to solve the following problem.
Find a representative for each similarity class of $3 \times 3$ matrices $A$ with entries in $\mathbb{F}_3 = \mathbb{Z}/3\mathbb{Z}$ such that $A^4 = ...
3
votes
2answers
89 views
Can't find the Jordan form of this 3x3
I have the matrix
$$\begin{pmatrix}
2 & 2 & -1 \\
-1 & -1 & 1 \\
-1 & -2 & 2
\end{pmatrix}$$
and need to find its Jordan canonical form. I can find that the only eigenvalue ...
6
votes
2answers
176 views
Show The Jordan Normal Form Of $\varphi$.
Fix a nonnegative integer $n$, and consider the linear space
$$\mathbb{R}_n\left [x,y \right ] := \left\{
\sum_{\substack{
i_1,i_2;\\
i_1+i_2\leq n}}a_{i_1i_2}x^{i_1}y^{i_2}\quad\Big|{}_{\quad}a_{...
0
votes
1answer
46 views
Jordan Normal form as consequence of the Classification theorem for finitely generated modules over PID
Let $V$ be a $n$-dimensional $\mathbb{C}$-vector space, so $V\cong \mathbb{C}^n$. Let further $T:\mathbb{C}\to \mathbb{C}$ be a $\mathbb{C}$-linear transformation. We consider $V$ as a $\mathbb{C}[X]$ ...
1
vote
2answers
123 views
Finding $\bf{P}$ such that $\bf{P^{-1}AP=B}$ for both fixed $\bf{A}，\bf{B}$.
How can I find a matrix $\bf{P}\in \mathbb{{R}^{n\times n}}$, such that $\bf{P^{-1}AP=B}$，where
$$\bf{A}=\begin{bmatrix}
\bf{A_2}& \bf{C_2}& \\
& \bf{A_2}& \bf{C_2}& \\ ...
1
vote
1answer
30 views
Number of Orthogonal Matrices over R in Jordan normal form [closed]
Is there any way to find the number of Orthogonal Matrices over the real field in Jordan normal form?
2
votes
1answer
76 views
Possible Jordan Canonical Forms: Intuition
As I was reviewing linear algebra before I head off to grad school in the fall, I came across a question about Jordan Canonical Forms. It reads:
"Suppose that A is a square complex matrix with ...
1
vote
1answer
77 views
Jordan form of operator $X \mapsto AXA$ [closed]
Matrices $n \times n$ on complex field. Compute Jordan form of operator $X \mapsto AXA$:
$$
A = \begin{bmatrix}
0 & 1 & & \\
& 0 & \ddots & \\
& & \...
2
votes
0answers
37 views
Analogy of Jordon Normal Form for Antilinear Maps
Given complex vector spaces $V$, and antilinear $T:V \rightarrow V$, then if we fix a basis of $V$, we can represent $T$ by the matrix of the linear $T \circ J$, where $J$ is complex conjugation.
I ...
1
vote
1answer
35 views
When can we bring a matrix to its Jordan form within a subfield of $\mathbb C$?
Can the following matrix $A$ be brought into Jordan form over the field of rational numbers?
$$ A=\begin{pmatrix}-3&-1&-1\\6&4&1\\6&5&0\end{pmatrix} $$
My solution:
By ...
0
votes
2answers
52 views
Why is it not sufficient that geometric multiplicity is equal to algebraic multiplicity to imply that $A$ diagonalizable
I have been told that for a given matrix $A$: $A \operatorname{diagonalizable} \Rightarrow m_{a}(\lambda)=m_{g}(\lambda)$ for all $\lambda \in \sigma (A)$ where $\sigma (A)$ denotes the spectrum of $A$...
0
votes
0answers
51 views
Kronecker-Weierstrass problem, 3x6 matrices conjugacy or congruent classes?
I'm here again with a somewhat vague and hard question our teacher asked us, we have to check and proof that for all matrices $A$ and $B$ that by applying certain simultaneous transformations, we can ...
0
votes
1answer
35 views
Non-nilpotent and non-invertible matrices that have the same characteristic and minimal polynomials have the same Jordan-form
I've come across the following question and am not sure why the answer makes sense.
Let $f,g \in End(\mathbb{C}^4)$ be neither nilpotent nor invertible with their characteristic and minimal ...
1
vote
2answers
61 views
Relation between matrix power and Jordan normal form
(a) Assume $A\in\mathbb{C}^{n\times n}$ has $n$ distinct eigenvalues.
Prove that there are exactly $2^n$ distinct matrices $B$ such that
$B^2 = A$ (i.e., in particular, there are no more than $2^n$...
-1
votes
2answers
37 views
Find the permutation matrix
Let:
$$J=\begin{bmatrix} \lambda&1&0\\ 0&\lambda&1\\ 0&0&\lambda
\end{bmatrix}$$
Find a permutation matrix $M$ such that $$M J M^{-1} = J^{t}$$
I know that $J$ is a ...
0
votes
2answers
57 views
Jordan Block of a complex matrix, with $A^4=I$
The following statement is false or true:
If $A \in M(n, \mathbb{C})$ is a matrix with complex entries of order $n$ such that $A^4=I$ then
\begin{pmatrix}
i & 1\\
0 & i
\end{pmatrix}
...
0
votes
1answer
45 views
Jordan Canonical form with zero eigenvalue?
Can anyone tell me how to find the Jordan Canonical form of the matrix below?
$$A=\begin{pmatrix}
0 & 1 & 2\\
0 & 0 & 1\\
0 & 0 & 0
\end{pmatrix}$$
Obviously this matrix ...
-1
votes
1answer
62 views
If $A \in M_2(\mathbb{C})$ and $A^2 = 0$ then what are the possible forms of A?
Let $A \in M_2(\mathbb{C})$
If $A^2 = 0$ determine all of the JCF's possible.
If $A^2 = 0$ determine all of the possible A's.
Show that $A^2 = 0 \;\exists n \geq 2 \Leftrightarrow A^2 = 0 \...
0
votes
2answers
46 views
If $\text{tr}(A) = \text{rank}(A) = 1$, find the Jordan Canonical Form of $A$.
Let $A \in M_{n \times n}(\mathbb{C})$ with $n > 1$. If $\text{tr}(A) = \text{rank}(A) = 1$, find the Jordan Canonical Form of $A$.
Since $A$ is a complex matrix, it must have a Jordan Form. ...
0
votes
0answers
32 views
Using power of matrix to find JCF
Given a $5 \times 5$ matrix $A$, find any Jordan canonical form for $A$.
There is a hint, that you should calculate $A^3$ first (crucially, not $(A-\lambda E_5)^3$). What information does the power ...
0
votes
0answers
78 views
If $B$ is nilpotent and $AB=BA$ then $\det(A+B)=\det(A)$ (Asking for other method) [duplicate]
Let $K$ be some field and $A, B \in M_n(K)$. Prove that: If $B$ is nilpotent and $AB=BA$ then $\det(A+B)=\det(A)$.
I believe there is a nice solution here. However, it seems that this problem could ...
1
vote
0answers
19 views
Let $A, B, C$ be some complex matrices. If $AB - BA = C$ and $AC = CA$, then $C^k = 0$ for some $k$. [duplicate]
Let $A, B, C$ be some complex matrices. Suppose that $AB - BA = C$ and $AC = CA$. Prove that: $C^k = 0$ for some $k$.
It is an exercise in the section of "Jordan Canonical Form of Nilpotent ...
1
vote
1answer
65 views
Jordan normal form and spectral decomposition
In this post by Terence Tao (exercise 28, point vi), he proves the following theorem (called spectral decomposition):
Theorem:
Let $A$ be a complex square matrix. Then $A$ can be written as
$A=\sum ...
1
vote
0answers
29 views
Galois descent for a semisimple automorphism
Let $K$ be a perfect field and $\overline{K}$ be the algebraic/separable closure. Let $V$ be a finite dimensional $K$-vector space, and let $V_{\overline{K}} = V \otimes_K \overline{K}$. Given an ...
0
votes
1answer
39 views
Jordan Normal Form Question
I'm trying to properly get to know the Jordan normal form Theorem, and am confused as to why this proposition holds. I have read that if A is a matrix in Jordan normal form and $T:V\rightarrow V$ then ...
5
votes
1answer
77 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 ...
0
votes
0answers
27 views
Jordan basis of the matrix with the only one eigenvalue 0
Find the Jordan basis of the following matrix:
$$\begin{pmatrix}
1& 1& -2& 3& -1\\
0 & 0 &-1 &2 &0\\
2& 2& -6& 10& -2\\
1& 1& -3& 5& -...
1
vote
1answer
75 views
Why is Jordan Normal Form unique?
I know that the Jordan Normal Form of a matrix is unique (up to reordering the Jordan blocks), but I don't really see why. Say we're looking at a 3x3 case. Now, all we need to do to compute the Jordan ...
2
votes
1answer
66 views
Given the Jordan form $J$, find matrix $P$
In a question set in my linear algebra course, I'm asked the following:
Find $P$ such that $P^{-1}AP=J$, where $$A = \begin{pmatrix}6&5&-2&-3\\ -3&-1&3&3\\ 2&1&-2&...
0
votes
0answers
34 views
Find Jordan norm form of matrix using lambda-matrix techniques
I can't transform this matrix of lambda-matrix techniques. I don't know this method, but I should use this method for solve. Eigenvalues equal to "-1". Guys, help me, please.
$$A=\left(\begin{array}{...
1
vote
0answers
15 views
I want to find an explicit direct sum decomposition of the space into T-cyclic subspaces
I construct a linear operator $T \in \mathcal{L(\mathbf{C}^7)}$ , where the minimal polynomial is $m_T (x) = x^2(x-1)^2$ and the caraterisitic polynomial is $p_T (x) = x^3(x - 1)^4$.
The linear ...
6
votes
4answers
101 views
Find Jordan Decomposition of $\begin{pmatrix} 4 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 3 \end{pmatrix}$ over $\mathbb{F}_5$
Find the Jordan decomposition of
$$
A := \begin{pmatrix} 4 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 3 \end{pmatrix}
\in M_3(\mathbb{F}_5),
$$
where $\mathbb{F}_5$ is the field ...
7
votes
3answers
320 views
Show that the characteristic polynomial is the same as the minimal polynomial
Let $$A =\begin{pmatrix}0 & 0 & c \\1 & 0 & b \\ 0& 1 & a\end{pmatrix}$$
Show that the characteristic and minimal polynomials of $A$ are the same.
I have already computated ...
1
vote
1answer
80 views
Let $A$ be an $n \times n$ matrix with real entries such that $A^2 + I = 0$ then $n$ is even. Not duplicated
Let $A$ be an $n \times n$ matrix with real entries such that $A^2 + I = 0$ then $n$ is even. And if $n = 2k$, then $A$ is similar over the field of real numbers to a matrix of the block form
$$\...
