Questions tagged [jordan-normal-form]

This tag is for questions relating to the Jordan normal form, also known as a Jordan canonical form or JCF of a matrix which is a similar block matrix having diagonal blocks when the matrix is diagonalizable and diagonal + nilpotent blocks more generally.

The Jordan normal form of a matrix is a canonical form given by a similarity transformation. The Jordan normal form consists of diagonal blocks corresponding to its generalized eigenvectors, with blocks that are themselves constant diagonal when an eigenvalue has a basis of eigenvectors and otherwise blocks that are constant diagonal + nilpotent. For more information, see this Wikipedia article.

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Let $A$ be a $5 \times 5$ matrix such that $A^2=0$. Then how to compute the maximum rank for such A?

Attempt : Suppose $A$ has a non-zero eigenvalue $\lambda$. Then corresponding to it's non-zero eigen vector $X$, we have $AX=\lambda X \Rightarrow A^2X=\lambda^2 X\Rightarrow 0=\lambda^2 X$. Which is a contradiction. Hence $\lambda=0$ with algebraic…
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Jordan form of a power of Jordan block?

How, in general, does one find the Jordan form of a power of a Jordan block? Because of the comments on this question I think there is a simple answer.
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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 correct. $$ e^{tX}(x) = \sum_{k=0}^{m} \frac{t^k…
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An explanation for a Jordan normal form proof from the Kaye and Wilson book

In this proof of Jordan normal form in the Kaye and Wilson book, then for a transformation $T$ with minimal polynomial $m(x) = (x-e)^k$, they take a basis of $\texttt{ker}\;T$, extend it to a basis of $\texttt{ker}\;T^2$, ..., extend it to a basis…
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Generalization of the Jordan form for infinite matrices

Under what conditions is it the case that for a matrix $M$ whose rows and columns are indexed by a countably infinite set $S$ one has a Hamel basis consisting of generalized eigenvectors (i.e. $v \in \ker(M - \lambda I)^n$) of $M$? Must $M$ be a…
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What commutes with a matrix in Jordan canonical form?

The question I would like answered is the following: Given a matrix $G$ and that $G$ commutes with another matrix $X$, that is $[G, X] = 0$, what is $X$? Or more generally, what properties of $X$ may we infer? I understand however that this question…
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How to express a matrix as a product of two symmetric matrices?

Let $A$ be a matrix and $J$ its Jordan canonical form. How can one express $A$ as a product of two symmetric matrices? I expressed $J$ as a product of two symmetric matrices: block by block in the following manner: $$\begin{bmatrix}\lambda&1 & \\…
Leo
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How does one obtain the Jordan normal form of a matrix $A$ by studying $XI-A$?

In our lecture notes, there's the following example problem. Find a Jordan normal form matrix that is similar to the following. $$A=\begin{bmatrix}2 & 0 & 0 & 0\\-1 & 1 & 0 & 0\\0 & -1 & 0 & -1\\1 & 1 & 1 & 2\end{bmatrix}$$ The solution begins by…
goblin GONE
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The Jordan Canonical Form of linear operator in two variables polynomial

I've been trying to solve the following exercise, In the space of bivariate polynomials of the form $f(x,y)=\sum_{n,m=0}^2a_{n,m}x^ny^m$, the lineal operator $T$ is defined by $Tf(x,y)=f(x+1,y+1)$. Obtain a Jordan Canonical Form of T I think i've…
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Find Jordan Decomposition of $\left(\begin{smallmatrix} 4 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 3 \end{smallmatrix}\right)$ 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 modulo 5. What I've done so far The characteristic polynomial is…
Ramanujan
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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 the characteristic polynomial $$p_A(x)=x^3-ax^2-bx-c$$ and I know from…
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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 $J_{A}$ for the matrix $A$ and a basis $X$ for the endomorphism $\varphi$ such…
MP3129
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A linear map $T: \mathbb{R^3 \to \mathbb{R^3}}$ has a two dimensional invariant subspace.

Let $T: \mathbb{R^3 \to \mathbb{R^3}}$ be an $\mathbb{R}$-linear map. Then I want to show that $T$ has a $2$ dimensional invariant subspace of $\mathbb{R^3}.$ I considered all possible minimal polynomial of $T$ and applying canonical forms I found…
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Center of a non-abelian subgroup of $GL(2, \mathbb{C})$

I'm trying to do the following exercise: Let be $G$ a non-abelian subgroup of $GL(2, \mathbb{C})$. Prove that the center of $G$ is contained in the center of $GL(2, \mathbb{C})$ My (very partial) attempt: Suppose that there is a matrix $A$ in the…
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How to find the Jordan canonical form of tensor products.

Let $k$ be a field, $ A \in M_{m\times m}(k)$ be a single Jordan block with eigenvalue $a$, and $B \in M_{n\times n}(k)$ be a single Jordan block with eigenvalue $b$. $A$ and $B$ together define a linear transformation $$A\otimes B : k^{m\times n}…
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