# Tag Info

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### How we can show this matrix is nilpotent?

In general, suppose that $I-XY=k(I-YX)$ for some complex number $k$ that is not a root of unity. (In your case, $k=-2$.) Since $I-XY$ and $I-YX$ have the same spectrum $S$, we have $S=kS$. As $k$ is ...
• 130k
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### Every matrix that commutes with $A \in \Bbb C^{n \times n}$ is a polynomial in $A$

Hint $:$ To disprove $(a)$ and $(b)$ just consider the non-diagonalizable nilpotent matrix (of index $2$) $$A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}.$$ It's easy to check that all the ...
• 1,929
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• 660
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### Polynomial maps of finite-dimensional vector spaces

$\operatorname{Hom}(V,W)$ refers to the $linear$ maps between vector spaces $V,W$. Although linear maps are always polynomial maps, polynomial maps needn't be linear maps. In the case that $W=k$ is ...
• 3,820
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• 214k
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### Find the linear transformation in terms of the canonical base

There are two ways of doing this: As others suggested in the comments --- which given the amount of structure this problem admits is probably the fastest --- one expresses each element of the new ...
• 3,564
1 vote

### Computing $(A\otimes I + I \otimes A)^{-1} \text{vec}B$

One thing to keep in mind: the axioms of a vector space provide a binary sum operation, and so you get linear combinations of the form $$a_1 v_1 + a_2 v_2 = \sum_{i=1}^2 a_i v_i$$ By applying the ...