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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.

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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 ...
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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 $$\...
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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)^...
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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..
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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 ...
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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 $W \neq V$. ...
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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 $1$. ...
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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 ...
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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 &...
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0answers
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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 ...
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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 ...
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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 ...
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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{...
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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 ...
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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}) ...
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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/...
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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 \\ ...
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1answer
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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 ...
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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 ...
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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 ...
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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, ...
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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 ...
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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 ...
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1answer
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$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 ...
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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 ...
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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 ...
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1answer
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Converting Jordan Normal Form into Real Jordan Form

Given the matrix $$\begin{bmatrix} 0 & 0 & 0 & -8\\ 1 & 0 & 0& 16 \\ 0 & 1 & 0 & -14 \\ 0 & 0 & 1 & 6 \\ \end{bmatrix}$$ ...
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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 ...
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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& -...
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1answer
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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 ...
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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 ...
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1answer
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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 $f\vert_V$?
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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 ...
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0answers
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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 ...
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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 ...
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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 $....
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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 ...
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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 $J_{A}$...
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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 ...
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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{...
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0answers
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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 ...
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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) ...
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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 ...
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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 ...
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0answers
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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 ...
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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$ ...
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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
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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 $N$ ...
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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 ...
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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 ...