# 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.

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### Prove that $V = \ker(\phi^n) \oplus \text{image}(\phi^n)$

Let $V$ be a $n$-dimensional complex vector space and $\phi:V\to V$ a linear mapping. Prove that $$V = \ker(\phi^n) \oplus \text{image}(\phi^n)$$ Here is my attempt: Since $\phi^n$ is also a linear ...
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### Finding Jordan form

Find Jordan form of the following matrix: $$\left(\begin{matrix}4&-5&2 \\ 5&-7&3\\ 6&-9&4 \end{matrix}\right)$$ So I got stuck pretty much trying to find the eigenvalues. ...
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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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### Non-integral powers of a matrix

Question Given a square complex matrix $A$, what ways are there to define and compute $A^p$ for non-integral scalar exponents $p\in\mathbb R$, and for what matrices do they work? My thoughts ...
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### Finding the Jordan canonical form of this upper triangular $3\times3$ matrix

I am supposed to find the Jordan canonical form of a couple of matrices, but I was absent for a few lectures. \begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 2 \\ 0 & 0 & 3 \end{...
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### Finding Jordan Canonical form given the minimal and characteristic polynomial.

I have the following information: the characteristic polynomial of $A$ is $p_A(t)=(t-4)^3(t+6)^2$ and the minimal polynomial is $q_A(t)=(t-4)^2(t+6).$ I'm having problems seeing how one would work ...
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### Possible Jordan Canonical Forms Given Minimal Polynomial

I was supposed to find all possible Jordan canonical forms of a $5\times 5$ complex matrix with minimal polynomial $(x-2)^2(x-1)$ on a qualifying exam last semester. I took the polynomial to mean ...
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### $2\times2$ matrix $A$ such that $A$ has one independent eigenvector while $A^{2}$ has two independent eigenvectors

Give an example of $2\times2$ matrix $A$ such that $A$ has one independent eigenvector while $A^{2}$ has two independent eigenvectors. I would like to know a systematic answer of how to get this. My ...
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### Computation of transformation matrix for jordan normal form: how to choose eigenvectors

During this semester at university we we're introduced to the jordan normal form of a matrix. While we never wrote down an explicit algorithm of how to find the matrix $B$, such that $B^{-1}AB$ is a ...
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### Superdiagonal for the Jordan form of a Jordan block power

The question is an extension of the Prove that $A$ is similar to $A^n$ based on A's Jordan form. Let $J$ be Jordan block of any form. In what circumstances Jordan form of power $J^n$ has the ...
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### An intuitive approach to the Jordan Normal form

I want to understand the meaning behind the Jordan Normal form, as I think this is crucial for a mathematician. As far as I understand this, the idea is to get the closest representation of an ...
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### Jordan-Chevalley vs Jordan normal decomposition

I am confused about a proof of the Jordan-Chevalley decomposition I was reading in Peterson's linear algebra book. Let $T : V \to V$ be an $n$-dimensional operator on a complex vector space. The ...
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### Find all possible Jordan Canonical forms for a nilpotent matrix

$A$ is a $10 \times 10$ nilpotent matrix of order $4$ ($A^4=0$) over $\mathbb C$ with $\operatorname{rank} (A)=6$. Find all possible Jordan Canonical forms The nullity of $A$ is $4$ so there are $4$ ...
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### Prove Why $B^2 = A$ exists?

Define A = \begin{pmatrix} 8 & −4 & 3/2 & 2 & −11/4 & −4 & −4 & 1 \\ 2 & 2 & 1 & 0 & 1 & 0 & 0 & 0 \\ −9 & 8 & 1/2 & −4 & 31/...
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### $n$-th root of $3 \times 3$ invertible matrix

Yo, I couldn't solve this exercise after thinking for a while. For every $A \in GL_{3} (\mathbb{C})$ and $n$, there's a $B \in Mat_{3, 3}(\mathbb{C})$ such that $B^n = A$ The previous exercise was ...
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Let $A$ be a $(p\times p$)-Jordan block of generalized eigenvalue $\lambda$. Let $B$ be a $(q\times q$)-Jordan block of generalized eigenvalue $\mu$. Then what is the Jordan canonical form for $A\... 1answer 283 views ### Jordan normal form over$\mathbb{C}$Let there be$T:\mathbb{C}^8\rightarrow \mathbb{C}^8$Such that$ T\left(\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \\ x_{5} \\ x_{6} \\ x_{7} \\ x_{8} \end{array}\right)=\left(\begin{...
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 ...
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 ...