# Questions tagged [jordan-normal-form]

The Jordan normal form of a matrix is a similar block matrix having diagonal blocks when the matrix is diagonalizable and diagonal + nilpotent blocks more generally.

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### Using power of matrix to find JCF

Given a $5 \times 5$ matrix $A$, find any Jordan canonical form for $A$. There is a hint, that you should calculate $A^3$ first (crucially, not $(A-\lambda E_5)^3$). What information does the power ...
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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 ...