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Title: Definition of Jordan Normal Form
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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 35 | Definition of Jordan Normal Form
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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: ala35_sub_eng.srt missing
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Timestamps (n/a)
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Subtitle in English (n/a)
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Quiz Content
Q1: Let $A \in \mathbb{C}^{n \times n}$. What is correct?
A1: There is at least one eigenvalue and an eigenvector for $A$.
A2: There are examples where $A$ has no eigenvalue.
A3: There are examples where $A$ has no eigenvectors.
A4: There are exactly $n$ different eigenvalues for $A$.
Q2: Let $A \in \mathbb{C}^{n \times n}$. Is $A$ diagonalizable?
A1: No, there are non-diagonalizable matrices.
A2: Yes, each square matrix is diagonalizable.
Q3: Let $A \in \mathbb{C}^{n \times n}$. Is $A$ similar to a Jordan normal form?
A1: No, there are counterexamples.
A2: Yes, for every square matrix there is a Jordan normal form $J$ and an invertible matrix $X$ such that $A = X J X^{-1}$.
A3: One needs more information.
Q4: Which of the following matrix is a Jordan normal form?
A1: $\begin{pmatrix} 1 & 2 \ 0 & 3 \end{pmatrix}$
A2: $\begin{pmatrix} 1 & 1 \ 0 & 1 \end{pmatrix}$
A3: $\begin{pmatrix} 1 & 1\ 0 & 3 \end{pmatrix}$
A4: $\begin{pmatrix} 2 & 2 \ 0 & 2 \end{pmatrix}$
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Date of video: 2024-11-27
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Last update: 2025-10
