# 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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### Jordan Normal Form of $A_{\pi}: \mathbb{C}^n \to \mathbb{C}^n$ given by $A_{\pi}(v) = A_{\pi}(v_1,…,v_n) = (v_{\pi(1)},…,v_{\pi(n)})$.

Let $\pi \in S_n$ be a permutation. Prove that $A_{\pi}: \mathbb{C}^n \to \mathbb{C}^n$ given by $A_{\pi}(v) = A_{\pi}(v_1,...,v_n) = (v_{\pi(1)},...,v_{\pi(n)})$. Show that $A_{\pi}$ is linear and ...
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### Is this a family of similar matrices $\left(\begin{smallmatrix} 0&x\\ 0&0 \end{smallmatrix}\right)$?

Is matrix $A = \begin{pmatrix} 0&1\\ 0&0 \end{pmatrix}$ similar to matrix $B=\begin{pmatrix} 0&2\\ 0&0 \end{pmatrix}$? If so, how do I prove this? I came here from following the ...
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### If $\text{tr}(A) = \text{rank}(A) = 1$, find the Jordan Canonical Form of $A$.

Let $A \in M_{n \times n}(\mathbb{C})$ with $n > 1$. If $\text{tr}(A) = \text{rank}(A) = 1$, find the Jordan Canonical Form of $A$. Since $A$ is a complex matrix, it must have a Jordan Form. ...
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### If $B$ is nilpotent and $AB=BA$ then $\det(A+B)=\det(A)$ (Asking for other method) [duplicate]

Let $K$ be some field and $A, B \in M_n(K)$. Prove that: If $B$ is nilpotent and $AB=BA$ then $\det(A+B)=\det(A)$. I believe there is a nice solution here. However, it seems that this problem could ...
### Let $A, B, C$ be some complex matrices. If $AB - BA = C$ and $AC = CA$, then $C^k = 0$ for some $k$. [duplicate]
Let $A, B, C$ be some complex matrices. Suppose that $AB - BA = C$ and $AC = CA$. Prove that: $C^k = 0$ for some $k$. It is an exercise in the section of "Jordan Canonical Form of Nilpotent ...