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Title: Generalized Eigenspaces
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Series: Abstract Linear Algebra
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Chapter: Some matrix decompositions
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YouTube-Title: Abstract Linear Algebra 36 | Generalized Eigenspaces
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Forum: Ask a question in Mattermost
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Quiz: Test your knowledge
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Subtitle on GitHub: ala36_sub_eng.srt missing
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Definitions in the video: generalized eigenspace, generalized eigenvector
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Timestamps (n/a)
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Subtitle in English (n/a)
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Quiz Content
Q1: Let $A \in \mathbb{C}^{n \times n}$ for $n\geq 2$ and $\lambda = 0$ be an eigenvalue. What is always correct?
A1: There is a generalized eigenvector of rank $1$.
A2: There is a generalized eigenvector of rank $2$.
A3: There is a generalized eigenvector of rank $n$.
A4: There are no generalized eigenvectors of rank $n$.
Q2: Let $A \in \mathbb{C}^{n \times n}$ and $\lambda = 0$ be an eigenvalue. Assume that $x$ is a generalized eigenvector of rank $2$. What is correct?
A1: $Ax$ is an ordinary eigenvector for the eigenvalue $\lambda = 0$.
A2: $x$ is also an ordinary eigenvector for the eigenvalue $\lambda = 0$.
A3: $Ax = 0$.
A4: $\mathrm{Ker}(A^2) = \{ 0 \}$.
Q3: Let $A = \begin{pmatrix} 0 & 1 \\ 0 & 0\end{pmatrix}$. What is correct?
A1: $\mathrm{Ker}(A^2) = \{ 0 \}$.
A2: $\mathrm{Ker}(A^2) = \mathbb{C}^2 $.
A3: $\mathrm{Ker}(A^2) = \mathbb{C}$.
A4: $\mathrm{Ker}(A)$ is two-dimensional.
A5: $\mathrm{Ker}(A) = \{ 0 \}$.
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Date of video: 2024-11-28
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Last update: 2026-03
