-
Title: Jordan Chains
-
Series: Abstract Linear Algebra
-
Chapter: Some matrix decompositions
-
YouTube-Title: Abstract Linear Algebra 42 | Jordan Chains
-
Bright video: Watch on YouTube
-
Dark video: Watch on YouTube
-
Ad-free video: Watch Vimeo video
-
Original video for YT-Members (bright): Watch on YouTube
-
Original video for YT-Members (dark): Watch on YouTube
-
Forum: Ask a question in Mattermost
-
Quiz: Test your knowledge
-
Dark-PDF: Download PDF version of the dark video
-
Print-PDF: Download printable PDF version
-
Thumbnail (bright): Download PNG
-
Thumbnail (dark): Download PNG
-
Subtitle on GitHub: ala42_sub_eng.srt missing
-
Download bright video: Link on Vimeo
-
Download dark video: Link on Vimeo
-
Timestamps (n/a)
-
Subtitle in English (n/a)
-
Quiz Content
Q1: Let $A \in \mathbb{C}^{n \times n}$ be a square matrix where all eigenvalues are given by $\lambda_1, \ldots, \lambda_r$. Is $A$ similar to a block diagonal matrix?
A1: Yes, the block sizes are exactly given by the algebraic multiplicities.
A2: Yes, the block sizes are exactly the geometric multiplicities.
A3: No, in general it’s not possible.
Q2: Let $A \in \mathbb{C}^{k \times k}$ be a square matrix with only one eigenvalue $\lambda$. What is always correct?
A1: $A$ is similar to a block diagonal matrix where the blocks are given by Jordan boxes.
A2: $A$ is diagonalizable.
A3: $A$ is a unitary matrix.
A4: $A$ is has two Jordan boxes of size $\alpha(\lambda)$.
Q3: Let $N \in \mathbb{C}^{k \times k}$ be a square matrix with only one eigenvalue $\lambda=0$. What is correct?
A1: If $x,y \in \mathrm{Ker}(N^2)$ are linearly independent and $\mathrm{Span}(x,y) \cap \mathrm{Ker}(N) = {0 }$, then $Nx, Ny$ are also linearly independent.
A2: If $x,y \in \mathrm{Ker}(N^2)\setminus \mathrm{Ker}(N)$ are linearly independent, then $Nx, Ny$ are also linearly independent.
A3: If $x,y \in \mathrm{Ker}(N)$ are linearly independent, then $Nx, Ny$ are also linearly independent.
A4: If $x,y \in \mathrm{Ker}(N^2)$ are linearly independent, then $Nx = Ny$.
-
Date of video: 2025-02-18
-
Last update: 2026-01
