Let \(f:V\to V\) be an endomorphism and let \(f^k = \underbrace{f\circ\cdots\circ f}_{k-\text{times}}.\) If \(\operatorname{Im}(f^k) = \operatorname{Im}(f^{k+1})\ne\{0_V\}\) for some \(k\in\mathbb N,\) then \(f|_{\operatorname{Im}(f^k)}:\operatorname{Im}(f^k)\to\operatorname{Im}(f^k)\) is invertible.

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The endomorphism \(\frac{\mathrm d}{\mathrm dx}:\mathsf P_n(\mathbb{R})\to\mathsf P_n(\mathbb{R})\) is diagonalisable.

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The Jordan normal form of \(\frac{\mathrm d}{\mathrm dx}+\mathrm{Id}_{\mathsf P_n(\mathbb{R})}:\mathsf P_n(\mathbb{R})\to\mathsf P_n(\mathbb{R})\) consists of one Jordan block of size \((n+1)\times (n+1).\)

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Every matrix \(\mathbf{A}\in M_{n,n}(\mathbb{R})\) admits a Jordan normal form \(\mathbf{J}\in M_{n,n}(\mathbb{R}).\)

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If \(s\ne 0,\) the Jordan normal form of \(\begin{psmallmatrix} 1 & s \\ 0 & 1\end{psmallmatrix}\) is given by \(\begin{psmallmatrix}1 & 1 \\ 0 & 1\end{psmallmatrix}.\)

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• False

Let \(\mathbf{A}\in M_{2,2}(\mathbb{C}).\) Then \(\mathbf{A}^2\) if and only if \(\mathbf{A}= \mathbf{0}_2\) or \(\mathbf{A}= \mathbf{C}\mathbf{J}_2(0)\mathbf{C}^{-1}\) for some invertible matrix \(\mathbf{C}\in M_{2,2}(\mathbb{C}).\)

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• False

If \(\mathbf{A}\in M_{n,n}(\mathbb{C})\) is invertible, its inverse can be written as a linear combination of powers \(\mathbf{A}^k,\) where \(k\in\{0,1,\ldots,n-1\}.\)

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• False

If \(\mathbf{A}\in M_{n,n}(\mathbb{C}),\) then there exists an invertible matrix \(\mathbf{C}\in M_{n,n}(\mathbb{C})\) such that \(\mathbf{A}=\mathbf{C}(\mathbf{D}+\mathbf N)\mathbf{C}^{-1},\) where \(\mathbf{D}\in M_{n,n}(\mathbb{C})\) is diagonal and \(\mathbf N\in M_{n,n}(\mathbb{C})\) is nilpotent.

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• False

It holds that \(\mathbf{J}_2(\lambda)^n = \begin{psmallmatrix}\lambda^n & n\lambda^{n-1} \\ 0 & \lambda^n\end{psmallmatrix}\) for all \(n\in\mathbb N.\)

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• False

If \(\mathbf{A},\mathbf{B}\in M_{n,n}(\mathbb{C})\) have the same eigenvalues, then they have the same Jordan normal form.

• True
• False