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Title: Singular Value Decomposition (Overview)
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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 49 | Singular Value Decomposition (Overview)
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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: ala49_sub_eng.srt missing
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Download bright video: Link on Vimeo
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Timestamps (n/a)
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Subtitle in English (n/a)
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Quiz Content
Q1: How does the general singular value decomposition for $A \in \mathbb{C}^{n \times n}$ look like?
A1: $ A = U \Sigma V^\ast $ with $\Sigma$ diagonal and $U,V$ unitary.
A2: $ A = U \Sigma U^\ast $ with $\Sigma$ diagonal and $U$ unitary.
A3: $ A = U \Sigma U^{-1} $ with $\Sigma$ diagonal and $U$ diagonal.
A4: $ A = U \Sigma V^{-1} $ with $\Sigma$ diagonal and $U,V$ invertible.
Q2: Let $ A = U \Sigma V^\ast $ be a singular value decomposition. What is always correct?
A1: $A \sim \Sigma$
A2: $A \approx \Sigma$
A3: $A \sim U$
A4: $A \sim V$
A5: $A \approx U$
A6: $A \approx V$
Q3: For which matrix is a singular value decomposition not possible?
A1: All shown matrices have singular value decompositions.
A2: $\begin{pmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1\end{pmatrix}$
A3: $\begin{pmatrix} 1 & 6 \ -5 & 1 \ 3 & 1 \end{pmatrix}$
A4: $\begin{pmatrix} 4 & -1 & 7 \ 0 & 3 & 2\end{pmatrix}$
A5: $\begin{pmatrix} 1 & 5 \ -5 & 4 \end{pmatrix}$
A6: $\begin{pmatrix} 1 & 1 & 1 \ 1 & 1 & 1 \ 1 & 1 & 1\end{pmatrix}$
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Date of video: 2025-04-25
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Last update: 2025-10
