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Title: Application for 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 44 | Application for 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: ala44_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: What is correct for every matrix $A \in \mathbb{C}^{n \times n}$?
A1: $A$ is similar to Jordan normal form where the eigenvalues of $A$ are on the diagonal.
A2: $A$ is similar to a diagonal matrix where the eigenvalues of $A$ are on the diagonal.
A3: $A$ is similar to a self-adjoint matrix.
A4: $A$ is similar to a unitary matrix.
A5: $A$ is similar to the zero matrix.
A6: $A$ is similar to a triangular matrix that has the numbers $1,2, \ldots, n$ on the diagonal.
Q2: What is correct for every matrix $A \in \mathbb{C}^{n \times n}$?
A1: $\det(A) = \prod_{j=1}^n \lambda_j$ where $\lambda_j$ are the zeros of the characteristic polynomial, counted with multiplicities.
A2: $\det(A) = \prod_{j=1}^k \lambda_j$ where $\lambda_j$ are the eigenvalues of $A$, counted by the geometric multiplicities.
A3: $\det(A) = \sum_{j=1}^k \lambda_j$ where $\lambda_j$ are the eigenvalues of $A$, counted by the geometric multiplicities.
A4: $\det(A) = \sum_{j=1}^n \lambda_j$ where $\lambda_j$ are the zeros of the characteristic polynomial, counted with multiplicities.
Q3: A Jordan normal form $J$ can be split up into two parts $D+N$. What are these matrices?
A1: $D$ is a diagonal matrix and $N$ a nilpotent matrix.
A2: $D$ is a triangular matrix and $N$ a unitary matrix.
A3: $D$ is a unitary matrix and $N$ a nilpotent matrix.
A4: $D$ is a diagonal matrix and $N$ a non-triangular matrix.
A5: $D$ is a self-adjoint matrix and $N$ an invertible matrix.
Q4: Let $J \in \mathbb{C}^{2 \times 2}$ be the Jordan normal of a matrix that has only the eigenvalue $2$ but with geometric multiplicity $1$. What is $\exp(J)$?
A1: $ \begin{pmatrix} e^2 & 0 \ 0 & e^2 \end{pmatrix} \begin{pmatrix} 1 & 1 \ 0 & 1 \end{pmatrix} $
A2: $ \begin{pmatrix} e^2 & 0 \ 0 & e^2 \end{pmatrix} \begin{pmatrix} 0 & 1 \ 0 & 0 \end{pmatrix} $
A3: $ \begin{pmatrix} e^2 & 0 \ 0 & e^2 \end{pmatrix} \begin{pmatrix} 0 & e^1 \ 0 & 0 \end{pmatrix} $
A4: $ \begin{pmatrix} e^2 & 0 \ 0 & e^2 \end{pmatrix} + \begin{pmatrix} 0 & e^1 \ 0 & 0 \end{pmatrix} $
A5: $ \begin{pmatrix} e^2 & e^2 \ e^2 & e^2 \end{pmatrix} + \begin{pmatrix} 0 & e^1 \ 0 & 0 \end{pmatrix} $
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Date of video: 2025-03-07
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Last update: 2025-10
