Mathematics Stack Exchange
Mathematics Stack Exchange

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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Why does this matrix give the derivative of a function?

I happened to stumble upon the following matrix: $$ A = \begin{bmatrix} a & 1 \\ 0 & a \end{bmatrix} $$ And after trying a bunch of different examples, I noticed the following remarkable pattern. If $P$ is a polynomial,…
ASKASK
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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 arbitrary endomorphism towards the diagonal form. As…
user66906
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When can two linear operators on a finite-dimensional space be simultaneously Jordanized?

IN a comment to Qiaochu's answer here it is mentioned that two commuting matrices can be simultaneously Jordanized (sorry that this sounds less appealing then "diagonalized" :P ), i.e. can be brought to a Jordan normal form by the same similarity…
Mark
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Why does the $n$-th power of a Jordan matrix involve the binomial coefficient?

I've searched a lot for a simple explanation of this. Given a Jordan block $J_k(\lambda)$, its $n$-th power is: $$ J_k(\lambda)^n = \begin{bmatrix} \lambda^n & \binom{n}{1}\lambda^{n-1} & \binom{n}{2}\lambda^{n-2} & \cdots & \cdots &…
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Why do we need a Jordan normal form?

My professor said that the main idea of finding a Jordan normal form is to find the closest 'diagonal' matrix that is similar to a given matrix that does not have a similar matrix that is diagonal. I know that using a diagonal matrix is good for…
atefsawaed
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Motivation for Jordan Canonical Form

I took linear algebra and understood the proof that linear operators on a vector space over an algebraically closed field have a Jordan Canonical Form. Why should I care about this theorem? I understand that it can be useful in doing some…
nigel
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What is the purpose of Jordan Canonical Form?

I don't claim at all to be an expert on this topic. In many (advanced) linear algebra textbooks for undergraduates, I usually find something about the "Jordan Canonical Form" of a matrix. What is the purpose of such a form? I have taken a usual…
Clarinetist
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How to calculate the matrix exponential explicitly for a matrix which isn't diagonalizable?

How can I compute an expression for $(\exp(Qt))_{i,j}$ for some fixed $i, j$ and matrix $Q$? When $Q$ is diagonalizable, we can diagonalize, but what can be done otherwise? Thanks.
Ben Derrett
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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 Integral exponents Defining $A^k$ for $k\in\mathbb N$ is…
MvG
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Why are there multiple Jordan Blocks corresponding to the same eigenvalue?

Though the title seems clear enough, I'd like to start with a discussion of how I personally came to derive the Jordan Normal Form, because my question is very specific to the details of my derivation. Notation To start, let $X$ be a finite…
user3002473
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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 Jordan-Chevalley decomposition tells us that $T = S +…
Ethan Alwaise
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Why does the largest Jordan block determine the degree for that factor in the minimal polynomial?

Let $A$ be a square matrix, so $A$ has some Jordan Normal form. Then $A$ has a minimal polynomial, say $m(X)=\prod_{i=1}^k (t-\lambda_i)^{m_i}$. Wikipedia says The factors of the minimal polynomial $m$ are the elementary divisors of the largest…
Boma
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“Geometric” problems on the Jordan normal form of a particular operator

Assume you have a class of students more or less familiar with the notion of the matrix of a linear operator. They have seen and calculated lots of examples in various context: geometric transformations (rotations, reflections, scaling along axes,…
Andrei Smolensky
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Find the Jordan normal form of a nilpotent matrix $N$ given the dimensions of the kernels of $N, N^2, N^3$

Let $N\in \text{Mat}(10 \times 10,\mathbb{C})$ be nilpotent. Furthermore let $\text{dim} \ker N =3 $, $\text{dim} \ker N^2=6$ and $\text{dim} \ker N^3=7$. What is the Jordan Normal Form? The only thing I know is that there have to be three blocks,…
Hans
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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 mapping of $V$ into $V$, we have that $$\dim V =…
rehband
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