• Title: Invariant Subspaces

  • Series: Abstract Linear Algebra

  • Chapter: Some matrix decompositions

  • YouTube-Title: Abstract Linear Algebra 38 | Invariant Subspaces

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  • Subtitle on GitHub: ala38_sub_eng.srt missing

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  • Definitions in the video: invariant subspace, invariant under linear map

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  • Quiz Content

    Q1: Let $V$ be complex vector space. Consider the map $\ell: V \rightarrow V$ given by $x \mapsto x$. What is always correct?

    A1: All subspaces of $V$ are invariant under $\ell$.

    A2: The subspace ${0}\subseteq V$ is not invariant under $\ell$.

    A3: The subspace $V$ is not invariant under $\ell$.

    A4: There are no invariant subspaces for $\ell$.

    A5: There are exactly two different invariant subspaces for $\ell$.

    Q2: Let $A \in \mathbb{C}^{2 \times 2}$ given by $$\begin{pmatrix} 2 & 1 \ 0 & 2 \end{pmatrix},.$$ Which of the following subspaces is not invariant under $A$?

    A1: $\bigg{ \begin{pmatrix} 0 \ 0 \end{pmatrix} \bigg}$

    A2: $\bigg{ \begin{pmatrix} x \ 0 \end{pmatrix} ~\bigg|~ x \in \mathbb{C} \bigg}$

    A3: $\bigg{ \begin{pmatrix} 0 \ y \end{pmatrix} ~\bigg|~ y \in \mathbb{C} \bigg}$

    A4: $\bigg{ \begin{pmatrix} x \ y \end{pmatrix} ~\bigg|~ x,y \in \mathbb{C} \bigg}$

    A5: All of the other ones are invariant subspaces.

    Q3: Let $A \in \mathbb{C}^{n \times n}$ for $n\geq 2$ and $\lambda$ be an eigenvalue. Are the subspaces $\mathrm{Ker}((A-\lambda \mathbf{1})^k)$ and $\mathrm{Ran}((A-\lambda \mathbf{1})^k)$ for every $k \in \mathbb{N}$ invariant under $A$?

    A1: Yes!

    A2: No, only if $k$ is the Fitting index

    A3: No, only the kernel is invariant for every $k \in \mathbb{N}$.

    A4: No, only the range is invariant for every $k \in \mathbb{N}$.

  • Date of video: 2025-01-13

  • Last update: 2025-10

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