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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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Using span of T to find Jordan

Looking for help on all parts of the question below: Let $U$ be the complex vector space of polynomials of at most degree 6. Define $D, T: U \rightarrow U$ by $D(f) = f'$ and $T = D^2 + D^3$ Set $V =...
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428 views

Finding the Jordan canonical form a characteristic polynomial

Q:Given $$ A = \pmatrix{6 & 1 & 1\\ -1 & 4 & 2\\ 0 & 0 & 5} $$ Find the characteristic polynomial of $A$ And Let $f : U → U$ be a linear map represented by the matrix $A$ ...
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39 views

Multiplicity of Jordan blocks between $B$ and $-B$

Let $B$ and $-B$ be square complex matrices such that they are similar. If there is $m$ Jordan block $J_k(\lambda)$ in $B$, the Jordan block $J_k(-\lambda)$ also appears $m$ times in $B$. This is my ...
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Size of Jordan blocks [duplicate]

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 ...
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82 views

Change of basis help turning a sub-matrix from complex to real

I have the following matrix: $$ \Lambda=\begin{bmatrix} \Lambda_1 \\ & \Lambda_2 \\ & & \ddots \\ & & & \Lambda_m \end{bmatrix}\in\mathbb R^{n\times n} $$ Where $\Lambda_1\...
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Decomposing a set of complex matrices into orbits of the operation of conjugation

I need some assistance with the proof for part (b) of the following problem statement: Problem Statement: Decompose the set $\mathbb{C}^{2\times2}$ of $2\times2$ complex matrices into orbits for ...
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296 views

Finding the Jordan Normal Form for a General Linear Transformation

Hey everyone here's the problem: Let V be a vector space with dim(V)=n For a particular linear transformation,f, we are given that there are two distinct eigenvalues, λ1 and λ2, with corresponding ...
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110 views

Jordan Form of generic matrix

Say $ A\in\mathbb{C}^{6\times6} $ and has eigenvalues $\lambda_1$ and $\lambda_2$ of multiplicity $ 3$ both of them. And for $\kappa=1,2,3$ the echelon form of the matrix $$ (A-\lambda_1I)^\kappa $$ ...
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106 views

Show that every Jordan matrix has a cyclic vector

Is my following reasoning correct? Since an $n\times n$ Jordan matrix has rank $n-1$ (because we can only make the last row the zero row), its geometric multiplicity is 1, which means the matrix has ...
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46 views

find Jordan form [duplicate]

Determine the jordan form of $A = \begin{pmatrix} 1 & 2 & 3\\ 0 & 4 & 5\\ 0 & 0 & 4 \end{pmatrix} $ First, I find the characteristic polynomial. $C_A(x)=(x-1)(x-4)^2$. ...
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2answers
63 views

Trouble finding Jordan Normal form for $4 \times $ 4 matrix

$M = \left(\begin{array}{cccc}0 & 1 & 0 & 0 \\-3 & 4 & 0 & 0 \\2 & -1 & 2 & 0 \\-1 & 1 & 1 & 2\end{array}\right)$. I find the eigenvalues to be $\...
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78 views

Norm bound for the Jordan basis matrix

Suppose I have a matrix $A$. We can assume, if it will be relevant, that it is stochastic with entries in $\{0,\frac{1}{2},\frac{3}{4},1\}$ and eigenvalues with magnitudes in $(0,1]$. $A$ has a ...
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1answer
56 views

Dissimilar Jordan matrices

So I need to find all dissimilar Jordan matrices J with the minimal polynomial: $x^3(x^2-1)^3$ and characteristic polynomial: $x^4(x^2-1)^4(x+1)^2$. So my question is, since the minimal polynomial ...
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1answer
44 views

How to quickly compute the Jordan Matrix for this $4\times 4$ matrix

Now I've learnt how to compute the Jordan matrix of a $2\times 2$ block. I wonder how you can do this fast without numerical tools. Let my matrix be: $A = \begin{bmatrix} 0 & −1 &1 & 0\...
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3answers
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No two 2x2 matrices in Jordan form are similar?

Let $S$ be the set of 2x2 matrices in Jordan Normal form $\begin{pmatrix}x&a\\ 0&y\end{pmatrix}$ with $a=0$ or $1$ and $x \leq y$. How do I show that no two matrices in $S$ are similar? Thank ...
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1answer
243 views

If $T^k = Id$ for $k\ge 1$ then $T$ is diagonalizable [duplicate]

Let $V$ a finite dimension space over $\mathbb{C}$ and $T:V\to V$, a linear transformation such that $T^k = Id$ for $k\ge 1$. Prove that $T$ is diagonalizable. I'd be glad for an hint. How do I ...
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How can same number of linearly independent vectors exist in smaller power nullspaces of a matrix?

Suppose I have a $n \times n$ matrix $A$. Let the dimension of null-space of $A^k$(denoted by $\mathcal{N}_{A^k}$) be $2$. This implies there can be at most $2$ linearly independent vectors $u,v$ in $...
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1answer
296 views

Jordan Canonical Form transition matrix

I have this matrix $M$ $M = \begin{bmatrix} 1 & 1 & 1\\ 2 & 1 & -1\\ 0 & -1 & 1 \end{bmatrix}$ And I was asked to put it into Jordan Canonical ...
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1answer
571 views

Rational Canonical Form of 2x2 Matrix

I have a 2x2 matrix where I need to find the rational canonical form over the field of rational numbers and real numbers. The matrix given is $$A=\begin{pmatrix}2 &-1 \\ 1 & -1\end{pmatrix}.$...
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Help with constructing a jordan form basis

My question is concerning Lemma 2.3 in the following paper , "On the Irreducibility of Commuting Varieties of Nilpotent Matrices " Specifically, I'm trying to understand the second half of the proof ...
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1answer
830 views

jordan canonical form of 6x6 matrix

Let Nl and N2 be 6 X 6 nilpotent matrices over the field F. Suppose that N1 and N2 have the same minimal polynomial and the same nullity. Prove that N1 and N2 are similar. Show that this is not true ...
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1answer
143 views

JCF given characteristic polynomial

A $15 \times 15$ matrix $M$ with complex entries has characteristic polynomial equal to $(x-1)^7(x-2)^8$. Find all possible minimal polynomials for $M$ such that the characteristic and minimal ...
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264 views

Why the similarity transformation matrix in jordan block decomposition can not be chosen unitary?

We know that any matrix can be transformed into its jordan form by similarity transformation. But why can't we choose a Matrix S with its inverse $S^-1$ as unitary matrices for non diagonilizable ...
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69 views

Find characteristic polynomial

Suppose $A$ and $B$ are $n \times n$ complex matrices such that $$AB-BA=aI+A,$$ where $a \in \mathbb{C}.$ Find the characteristic polynomial of $A$. If $A$ happens to be a Jordan block, this would be ...
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549 views

Norm of a nilpotent matrix

A proof I was reading used the claim that $\vert\vert{N}\vert\vert_2$ = 1 for a nilpotent matrix $N$. I tried to prove it, and have a couple of questions on it. First, my "proof": We know that there ...
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1answer
123 views

Invariant subspaces for endomorphisms with associated Jordan matrices

I would like to know which are the invariant subspaces for the endomorphisms $f1$, $f2$, $f3$, $f4$, $f5$ from vector space $V$ that have the next associated Jordan matrices: $J1 = \left( \begin{...
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0answers
750 views

Prove matrix is nilpotent, find its invariants and Jordan form

Prove that the matrix $A = \begin{pmatrix} 1&1&1 \\ -1&-1&-1 \\ 1&1&0 \end{pmatrix}$ is nilpotent, and find its invariants and Jordan form. So far, I've verified directly that ...
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478 views

Decomposition of a linear map to nilpotent and diagonalizable matrices.

Question Let $V$ be a vector space over the complex field $C$ , and let $T$ be an linear map $V\rightarrow V$. Show that exists a pair of linear maps $D$ and $N$ such that $D+N=A$, $D$ is ...
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1answer
129 views

JCF of matrices $A^2$ and $B^2$

Going through a past paper and I've come across this True or False question: If $A$ and $B$ have the same Jordan Canonical Form (JCF), $A^2$ and $B^2$ have the same JCF. I thought it was true, and ...
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1answer
44 views

Similarity of $3\times3$ matrices via Jordan canonical form

Let matrices $$A=\begin{bmatrix} -3&3&-2\\ -7&6&-3\\ 1&-1&2 \end{bmatrix},\quad B=\begin{bmatrix} 0&1&-1\\ -4&4&-2\\ -2&1&1 \end{bmatrix},\quad C=\...
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1answer
129 views

Reordering Jordan Matrix Sub-Blocks

Let $J$ and $K$ be $n × n$ Jordan matrices which can be made equal after a reordering of sub-blocks. Show that $J= S^{-1}KS$ by explaining how to construct $S$. I understand that these two matrices ...
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1answer
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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)...
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1answer
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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
32 views

Jordan normal form options from minimal polynomial

what is the Jordan normal form options from this minimal polynomial 𝑚𝐴(𝑥) = $𝑥^3 − 2x^2$. 4X4 matrix I know of course that the 2 eigenvalues are 0,2.
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2answers
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Are they all similar to each other?

I have a confusion. I have read one statement."It can be shown that the Jordan normal form of a given matrix A is unique up to the order of the Jordan blocks". But I could not understand. I have taken ...
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1answer
94 views

Let $I \neq A ∈ M_{n×n}(\mathbb{R})$ be an involutory matrix. Show that the Jordan canonical form of $A$ is a diagonal matrix.

Let $I \neq A ∈ M_{n×n}(\mathbb{R})$ be an involutory matrix. Show that the Jordan canonical form of $A$ is a diagonal matrix. I'm not sure how to do this, any solutions/hints are greatly appreciated....
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1answer
375 views

Find the characteristic & minimal polynomials, eigenvectors, and dimension of the eigenspace for this 6x6 jordan matrix?

3 1 0 0 0 0 0 3 1 0 0 0 0 0 3 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 1 I think the characteristic polynomial is: ((x-3)^3)((x-1)^3) Found by taking the ...
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1answer
52 views

$T:f(x)\to f(x-1)+x^3f'''(x)/3$ Find the Jordan normal form and a Jordan basis for $T$.

Let $T\in \mathcal{L}(\mathcal{P_3}(\mathbb{C})$ be the operator $$T:f(x)\to f(x-1)+\frac{x^3f'''(x)}{3}$$ Find the Jordan normal form and a Jordan basis for $T$.
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1answer
190 views

Inverse Jordan Decomposition Matlab code?

I did Jordan decomposition of a matrix by using this code: A = [1 -3 -2; -1 1 -1; 2 4 5]; [V, J] = jordan(A) Now I need to do inverse Jordan decomposition to ...
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1answer
40 views

Find the Jordan canonical form [closed]

$N$ is a nilpotent $15\times15$ matrix over $\mathbb{R}$ such that $$\dim(\ker N) = 5, \quad \dim (\ker{N^2}) =8, \quad \dim(\ker{N^3})= 11,$$ $$\dim (\ker{N^4}) = 13, \quad \dim(\ker{N^5}) =15$$ ...
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1answer
77 views

Jordan matrix two looks

i have question. I have something like this: $\begin{bmatrix} -2 & 2 \\ 1 & 3 \\ \end{bmatrix}$ $\lambda_{1} = -1$ $\lambda_{2} = -4$ When jordan matrix looks like this: $\begin{...
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1answer
426 views

Is every Jordan block diagonalisable? [closed]

Is every Jordan block diagonalisable? I need to also give a short justification. Can anyone lend a hand?