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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1answer
36 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{...
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1answer
29 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
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1answer
70 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 = ...
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2answers
185 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_{...
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2answers
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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 ...
5
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1answer
4k 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 ...
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1answer
49 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 $\mathbb{C}[X]$ ...
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2answers
131 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}& \\ ...
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1answer
33 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?
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1answer
100 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 ...
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1answer
69 views

Can a matrix be similar to a block matrix with Jordan Block or companion matrix of the non-linear irreducible factors in its diagonal block?

Let $A$ be $3 \times 3$ real matrix with minimal polynomial $f(X)=(X-1)(X^2 +1)=X^3-X^2+X-1.$ Then By Rational Canonical Form we know that $A$ is similar to the Companion matrix of $f(X)$ which is $\...
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1answer
78 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 & \\ & & \...
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0answers
40 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 ...
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1answer
37 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 ...
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2answers
55 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 $A$...
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1answer
6k views

Exponential of a Jordan block [duplicate]

I am having difficulties with calculating exponential of a Jordan block, I cannot understand the method, can please someone help me, I have an exam on Monday. 'J' is my Jordan matrix and 'P' is my ...
2
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2answers
74 views

A relation satisfied by the real Jordan form of a $2 \times 2$ matrix with complex eigenvalues

Let $A$ be a $2 \times 2$ matrix with complex eigenvalues, $\lambda_1 = \alpha + i \beta$ and $\lambda_2 = \alpha - i \beta$ ($\alpha, \beta \in \mathbb{R}$). We know that $u + iv$ is an eigenvector ...
2
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1answer
921 views

What is the connection between Jordan Canonical Form and minimal polynomial?

I saw a bunch of examples of Jordan Canonical Form and how it is related to the minimal polynomial. I have noticed the following patterns: Let $A$ be a matrix and $\lambda_1,\dots,\lambda_k$ be the ...
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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
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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 ...
2
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1answer
440 views

What is the easiest proof you know for the Jordan Canonical Form

I read numerous demonstration of the existence of the Jordan Canonical Form, but all of them involve more than 2 pages of demonstration with numerous lemmas in between. I'm writing some notes for ...
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2answers
62 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 $2^n$...
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2answers
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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 ...
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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} ...
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1answer
49 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 ...
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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 \...
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2answers
56 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. ...
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0answers
34 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 ...
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1answer
88 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 ...
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81 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 ...
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0answers
21 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 ...
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0answers
30 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 ...
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1answer
41 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
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1answer
82 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 ...
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1answer
106 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 ...
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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& -...
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1answer
148 views

Jordan canonical form for a matrix

How do I find the Jordan canonical form and its transitions matrix of this matrix? \begin{pmatrix}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\end{pmatrix} The ...
2
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1answer
67 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&...
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0answers
45 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 ...
3
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1answer
52 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
35 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}{...
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0answers
19 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
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4answers
103 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 ...
6
votes
3answers
415 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 ...
2
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3answers
59 views

Two possible cases for JNF of a matrix

I'm trying to find the Jordan Normal Form of the following matrix: $\pmatrix{2 & 0 & 1 & 1 \\0 & 2 & 1 & 1\\0 & 0 & 2 & 1\\0 & 0 & 0 & 2\\}$. Now since ...
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0answers
41 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)^...
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1answer
100 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
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1answer
31 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..
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1answer
58 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 ...
1
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0answers
33 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 $1$. ...