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.
7
votes
3answers
220 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
60 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
$$\...
1
vote
0answers
37 views
Canonical Jordan form contradiction
I am faced with the following problem:
Given endomorphism $f$ whose characteristic polynomial is $$P_c(x) = (x+1)^{10} (x-1)^{10} x^{10}$$ and whose minimal polynomial is $$P_m (x) = (x+1)^5 (x-1)^...
-1
votes
1answer
29 views
let J(A) be the Jordan form of A. and let f be some polynomial. is it true that $\det(xI-f(A))=\det(xI-f(J(A))$ [closed]
I tried a couple of examples and it turned out to be true, but I couldn't prove it..
1
vote
1answer
31 views
Convergence of powers of matrices in Jordan Canonical Form (Jordan Normal Form)
I've actually been stuck on this for a bit while studying for an exam, so would appreciate any help.
The problem involves testing whether $\lim\limits_{m \to \infty}$ $A^m$ exists.
From lecture ...
0
votes
0answers
41 views
Invariant subspaces and eigenvalues [closed]
I am trying to solve this question:
Let $V$ be vector space over the complex field. Let $T:V\rightarrow V$ be a linear operator, and $W$ a T-invariant subspace of $V$, such that $W \neq V$.
...
1
vote
0answers
29 views
Finding possible Jordan forms of a matrix
Find the Jordan forms of a matrix $A$ subject to the following conditions:
the characteristic polynomial is $(x-1)^4(x+3)^5$.
matrix $A-I$ has nullity $4$ and matrix $A+3I$ has nullity $1$. ...
2
votes
1answer
38 views
How to find a Jordan basis and a Jordan matrix for a nilpotent matrix?
I am trying to find a general step-by-step "easy" / "intuitive" solution to finding Jordan basis and Jordan matrix (based on the basis) for a nilpotent matrix.
If you can add an intuition for the ...
2
votes
1answer
40 views
How to prove that the following matrices in $M_p(\Bbb F_p)$ is similar
How to prove that the following matrices in $M_p(\Bbb F_p)$ is similar:
Consider two matrices $$(a_{ij})=
\begin{pmatrix}
1 & 1 & 0 & \cdots & 0 & 0 \\
0 & 1 &...
2
votes
0answers
41 views
How can I find the Jordan form of this upper triangular Toeplitz matrix?
Given an $n \times n$ matrix $A$ whose $(i,j)$ entry is
$$a_{ij} = \begin{cases} n-j+i & \text{if } j \geq i\\ 0 & \text{otherwise}\end{cases}$$
find its Jordan form.
I know that ...
0
votes
1answer
32 views
Is this a correct solution? Jordan form of a marix
I am given a matrix $A$ that is defined the following way:
the element in row $i$ and column $j$ is $\alpha$ if $i=j$, $1$ if $j=i+2$ and $0$ otherwise.
I need to find Jordan Form. This is my ...
0
votes
1answer
23 views
If $V_j = \ker(A-\lambda I)^j$, Why does $\dim(V_j)=r_i$ if and only if $V_j = V_{j+1}$?
If we have a matrix $A$ and the characterisic polynomial is $(x-\lambda_1)^{r_1} \cdot...\cdot(x-\lambda_m)^{r_m}$ and we define for each $\lambda_i$, $V_{j} = \ker(A-\lambda I)^j$. Why does there ...
0
votes
0answers
17 views
Ordering eigenvector basis to produce diagonal matrix J
In order to solve the system
\begin{equation}
\begin{pmatrix}
y_{1}'\\
y_{2}'\\
y_{3}'
\end{pmatrix}=
\begin{pmatrix}
1 &-1&0 &\\1&3&0\\-2&1&-1
\end{pmatrix}\begin{...
0
votes
1answer
21 views
Can I show that for any matrix A = ST, S & T Both symmetric and invertible matrices
In my linear algebra class, my professor gave us an exercise, which is the following.
If A is an nxn matrix over the complex, one can show that A = ST for S and T both invertible and symmetric.
I ...
0
votes
0answers
24 views
How to find the invariant factors of a matrix given its Jordan canonical form?
Suppose we have a matrix $A \in M_{12}(\mathbb{C})$ whose Jordan canonical form is
$$J_1(0) \oplus J_1(0) \oplus J_2(0) \oplus J_2(0) \oplus J_1(\sqrt{2}) \oplus J_1(\sqrt{2}) \oplus J_1(-\sqrt{2}) ...
1
vote
1answer
36 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/...
2
votes
2answers
50 views
Jordan normal as transformation with respect to the basis of eigenvectors
I have the following matrix
$$A = \begin{pmatrix}
2 & 0 & 1 & -3 \\
0 & 2 & 10 & 4 \\
0 & 0 & 2 & 0 \\
0 & 0 & 0 & 3 \\
...
1
vote
1answer
34 views
Hoffman and Kunze ,Linear algebra Sec 7.4 exercise 4
Construct a linear operator $T$ with minimal polynomial $ x^2(x-1)^2 $ and characteristic polynomial $x^3(x-1)^4$. Describe the primary decomposition of the vector space under $T$ and find the ...
1
vote
1answer
30 views
Find the Jordan Canonical Form that is similar with the idempotent matrix A
Find the Jordan Canonical Form that is similar to the idempotent matrix $A$.
I know that since $A=A^2$ then $A(A-I)=0$ so the minimal polynomial is $m_A(\lambda)=\lambda(\lambda-1)$.
I also know ...
4
votes
2answers
204 views
Proof involving the spectral radius and the Jordan canonical form
Let $A$ be a square matrix. Show that if $$\lim_{n \to \infty} A^{n} = 0$$ then $\rho(A) < 1$, where $\rho(A)$ denotes the spectral radius of $A$.
Hint: Use the Jordan canonical form.
I am ...
1
vote
1answer
37 views
all 2 dimensional invariant subspaces
How we can find all 2 dimensional invariant subspaces of
\begin{pmatrix}
2 & 1 & 0 \\
0 & 2 & 0 \\
0 & 0 & 8
\end{pmatrix}
I know that there are at least 2 such subspaces, ...
0
votes
2answers
46 views
Computing the matrix exponential for a Jordan matrix
How can I compute $e^{At}$ where $A = J_{3}(5)$? That is,
$$A = \begin{pmatrix} 5 & 1 & 0 \\ 0 & 5 & 1 \\ 0 & 0 & 5 \end{pmatrix} $$
Using this, how can I write down a basis ...
1
vote
0answers
46 views
Given a Jordan canonical basis, how to find out to which generalized eigenspace picked generalized eigenvector belongs
Suppose we have finite-dimensional linear operator $A:V\to V$ , that has eigenvalues $\lambda_1 ,\lambda_2, ... \lambda_n$ . It is known that we can decompose $V$ into direct sum of generalized ...
0
votes
1answer
13 views
$Ch_A =(x+1)^6(x-2)^3 $ y $min_A = (x+1)^3(x-2)^2 $, List the possible Jordan forms for $A$
Let $A$ be a complex matrix such that $Ch_A =(x+1)^6(x-2)^3 $ y $min_A = (x+1)^3(x-2)^2 $, List the possible Jordan forms for $A$. And in each case write the corresponding rational
I do not know ...
0
votes
1answer
38 views
Deducing the additive Jordan decomposition
In $M_n(\Bbb C)$, I could prove that the additive Jordan decomposition of $X=D+N$ with $D$ diagonalizable and $N$ nilpotent gives a multiplicative Jordan decomposition $e^X=e^De^N$.
Is that true the ...
1
vote
1answer
50 views
When $A^i=A^j$ for $i,j\geq 0$ such that $i\neq j$ for matrix $A$ over algebraic closed field?
Let $A\in M_n(K)$ be a matrix over algebraic closed field $K$, where $n>1.$
When $A^i=A^j$ for $i,j\geq 0$ such that $i\neq j$?
I tried solve it by Jordan form of matrix $A;$ is is sufficient ...
1
vote
1answer
49 views
Converting Jordan Normal Form into Real Jordan Form
Given the matrix
$$\begin{bmatrix}
0 & 0 & 0 & -8\\
1 & 0 & 0& 16 \\
0 & 1 & 0 & -14 \\
0 & 0 & 1 & 6 \\
\end{bmatrix}$$
...
1
vote
0answers
20 views
Is Jordan cardinal form the matrix having the most zeros in the equivalent class of similarity? [duplicate]
I am concerned on this interesting question
Given matrix $A$, does the Jordan cardinal form have the most zeros, in other word, it has the least nonvanishing indices, among the equivalent class of ...
0
votes
0answers
94 views
How do I complete the steps of finding the Jordan of this $5\times 5$ matrix (with Octave)?
I know how to begin the procedure but I don't know how to finish it. Let's start with an example (sorry for it being so unwieldy).
Let
$$A =\begin{pmatrix}
177& 548& 271& -548& -...
0
votes
1answer
43 views
Generalised Eigenvectors Issue
Ok so i have been doing a few questions on 'Diagonalising' defective matrices, the method I've been using to find generalized Eigenvectors is to make the previous Eigenvector the subject. However i ...
1
vote
1answer
35 views
Decompose an invertible matrix into an exchangeable product of diagonalizable matrix and a matrix with all the eigenvalues equal to $1$
Let $ g $ be an invertible $ n\times n $ complex matrix. Show that $ g $ can be written as
$$ g=su=us ,$$
where $ s $ is diagonalizable and all eigenvalues of $ u $ are equal to $ 1 $.
My ...
0
votes
1answer
15 views
Can there be just one eigenvalue on the invariant subspace (generalized eigenspace) associated with an eigenvalue?
In the proof of Jordan decomposition here, once I know that an indecomposable subspace $V$ is of the form $V=Ker((f-\lambda Id)^n)$, can there be an other eigenvalue $\mu$ for $f\vert_V$?
2
votes
3answers
160 views
Intuition of generalized eigenvector.
I was trying to get an intuitive grasp about what the the generalized eigenvector intuitively is. I read this nice answer, so I understand that in the basis given by the generalized eigenvectors, a ...
0
votes
0answers
23 views
The theory behind linear recurrence relations solving (or - why does it work?)
tl;dr - a recommendation for a good book that explains the theory behind the auxiliary polynomial/companion matrix methods to solve linear recurrence relations with constant coefficients?
I've bumped ...
1
vote
1answer
51 views
Show that matrix $A$ is similar to a matrix $B$ with elements on diagonal $(0, …, 0, \operatorname{Tr(}A))$ respectively.
Let $A$ be a matrix $n \times n, n \geq 2 $. Let's assume that not all entries outside of the diagonal are zeros (we don't know what entries are on the diagonal). Show that matrix $A$ is similar to a ...
2
votes
1answer
55 views
Let's assume that $ XA = AX $. Show that there is such a matrix $M$ that $ p_A(X) = M(A-X), MA=AM$ and $ MX=XM $.
Let $ A, X \in M_{nxn}(K) $. Let $ p_A(t) $ be a characteristic polynomial of matrix A. Let's assume that $ XA = AX $. Show that there is such a matrix $M$ that $ p_A(X) = M(A-X), MA=AM$ and $ MX=XM $....
0
votes
1answer
32 views
Exponential of a Jordan Block using Cayley-Hamilton Theorem
For the sake of simplicity, I will only consider $2\times2$ matrices. The Cayley-Hamilton theorem allows us to conclude that
$$e^{At} = \alpha_0I + \alpha_1 A$$
where $\alpha_0$ and $\alpha_1$ can ...
6
votes
1answer
75 views
Find Jordan normal form and basis
Let $$A=M(\varphi)^{st}_{st}={\begin{bmatrix}0&1&1\\-4&-4&-2\\0&0&-2\end{bmatrix}}$$ and $ \varphi: \mathbb R^{3} \rightarrow \mathbb R^{3}$. Find the Jordan normal form $J_{A}$...
0
votes
1answer
48 views
How do I find the bases of the Jordan Canonical Form of $C$?
Let $$C = \left[ {\begin{array}{cccc}
0 & -1 & -2 & 3 \\
0 & 0 & -2 & 3 \\
0 & 1 & 1 & -1 \\
0 & 0 & -1 & 2
\end{array} } \right].$$ What ...
0
votes
1answer
26 views
Power of a matrix in REAL jordan form
Given a $2\times 2$ matrix in Jordan canonical form, whose eigenvalues are a couple of complex conjugate values
$$ J = \left[ \begin{array}{cc} \sigma+j\omega & 0 \\ 0 & \sigma - j\omega \end{...
0
votes
0answers
32 views
Jordan form of a matrix and finitely generated modules over PIDs.
We know that any square matrix of order n over Complex numbers is similar to a Jordan form,I am told that this relates to the structure theorem for finitely generated modules over PIDs. Can anyone ...
2
votes
1answer
29 views
Exericse about linear map $T\in L(V)$, where $\dim V=n\geq2$, with $\operatorname{null}T^{n-1}\neq\operatorname{null}T^n$
I have this problem that I am attempting, and am struggling with (b).
--
Assume $\dim V = n \geq 2$ and that $T \in L(V)$ such that $\operatorname{null}T^{n-1}\neq\operatorname{null}T{^n}$
--
(a) ...
1
vote
1answer
48 views
Calculate the Jordan normal form
I have the matrix $A=\begin{bmatrix} -2 & -3 & 6 \\ 1 & 2 & -2\\ -1 & -1 &3 \end{bmatrix}$ and I have to find the transformation matrix and its Jordan normal form. This is ...
1
vote
2answers
63 views
Why does $(A- \lambda I)^2 =0$ if A has two repeated eigenvalues?
This statement appears in my textbook as part of an introduction of the method for finding the Jordan form of a $2 \times 2$ matrix. I understand what it says but I'd really like to know where is it ...
1
vote
0answers
31 views
Proving that a matrix is nonnegative if its powers are nonnegative
I am working on a problem involving doubly stochastic matrices where I must prove that $P$ is doubly stochastic if and only if $P^k$ is doubly stochastic for $k = 1, 2, ...$ It is easy to show that if ...
1
vote
1answer
29 views
Convergence of powers of matrix given convergence of the powers of its absolute value.
I have a matrix A and a matrix B such that $B_{i, j} = |A_{i, j}|$. I am given that all of the eigenvalues of B have magnitude less than 1, and therefore: $ \displaystyle \lim_{k \to \infty} B^k = 0$ ...
0
votes
0answers
18 views
Computing transformation matrix between similar matrices
If I have the matrix $A = \begin{pmatrix}
2 & 0 & 1 & -3 \\
0 & 2 & 10 & 4 \\
0 & 0 & 2 & 0 \\
0 & 0 & 0 & 3
\end{pmatrix}$ and I've calculated its ...
0
votes
1answer
42 views
In the Jordan-Chevalley decomposition $M=D+N$, how obtaining $D$ and $N$ as polynomials in $M$?
The Jordan-Chevalley expresses a linear operator $M$ as
$$
M = D + N,
$$
where $D$ is semisimple (diagonalizable), $N$ is nilpotent and $DN=ND$. Although it is stated in many sources that $D$ and $N$ ...
2
votes
0answers
23 views
For which values the matrix is diagonalizable
For which values of $a$ matrix $A$ is diagonalizable?
$$A = \pmatrix{0&i\\i&a}$$
in the case that it is not diagonalizable determine a base of Jordan
Attempt: The minimal polynomial ...
1
vote
1answer
33 views
$A$ and $B$ be $n \times n$ matrices over the field $\mathbb F$ which have the same characteristic polynomial
Lemma: Let $N_1$ and $N_2$ be $3 \times 3$ nilpotent matrices over field $\mathbb F$. Then, $N_1$ and $N_2$ are similar if and only if they have the same minimal polynomial.
Use the result above ...
