# 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.

1,006 questions
Filter by
Sorted by
Tagged with
13k views

### 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 ...
8k views

### 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 ...
6k views

### 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 ...
3k views

### Why do we need a Jordan normal form? [duplicate]

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 ...
10k views

1k views

### Jordan form step by step general algorithm

So I am trying to compile a summary of the procedure one should follow to find the Jordan basis and the Jordan form of a matrix, and I am on the lookout for free resources online where the algorithm ...
759 views

### 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 ...
1k views

### Conjugacy classes in $SU_2$

I'm trying to find all conjugacy classes in $SU_2$. Matrices in $SU_2$ are of the form: $M = \begin{bmatrix} \alpha & \beta \\ - \bar{\beta} & \bar{\alpha} \end{bmatrix}$...
665 views

965 views

88 views

### Misconception on Jordan Canonical Form

Say we have matrix $$M= \begin{pmatrix} 2 & 0 & 1 & -3\\ 0 & 2 & 4 & 8\\ 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 3 \end{pmatrix}\DeclareMathOperator{\Id}{Id}$$ ...
5k views

### 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 ...
3k views

### Proof for real Jordan canonical form

Let A $\epsilon$ Mat(nxn, $\mathbb R$) be a matrix that is diagonalizable in $\mathbb C$ with k real eigenvalues of algebraic multiplicity 1 and (n-k)/2 pairs of complex-conjugated eigenvalues of ...
If $N$ is a $k \times k$ elementary nilpotent matrix, i.e. $N^k = 0$ but $N^{k-1} \ne 0$, then show that $N^\top$ is similar to $N$. Now use the Jordan form to prove that every complex $n \times n$ ...
Show that $\exp(\mathrm{Tr}(X))=\det(\exp(X))$ where $X$ is a matrix using the concept of the Jordan normal form I realised this formula by considering that: $\det(\exp(X))=\exp(\lambda_1) \times\... 2answers 179 views ### Find Jordan form of a$3\times 3\$ matrix
$$\left( \begin{array}{ccc} 0 & 1 & 2 \\ -5 &-3 & -7 \\ 1 & 0 & 0 \end{array} \right)$$ I figured out the eigenvalues are all -1 from the characteristic polynomial, but I'm ...