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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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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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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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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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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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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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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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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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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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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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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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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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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$T:f(x)\to f(x-1)+x^3f'''(x)/3$ 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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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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Find the Jordan canonical form [closed]

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