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Title: Schur Decomposition
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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 45 | Schur Decomposition
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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: ala45_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 $A \in \mathbb{C}^{n \times n}$. Can one find a unitary matrix $U \in \mathbb{C}^{n \times n}$ such that $U^\ast A U$ is an upper triangular matrix?
A1: Yes, it’s always possible!
A2: No, it’s only possible if $A$ is a normal matrix.
A3: No, it’s only possible if $A$ is a unitary matrix.
A4: No, it’s never possible!
Q2: Let $U \in \mathbb{C}^{n \times n}$ be a unitary matrix. What is not always correct?
A1: The columns of $U$ form an ONB of $\mathbb{C}^n$.
A2: $U$ only has $1$ and $0$ as entries.
A3: The rows of $U$ form an ONB of $\mathbb{C}^n$.
A4: $U$ is invertible.
A5: $|\det(U)| = 1$.
Q3: Let $A \in \mathbb{C}^{4 \times 4}$ be a square matrix with characteristic polynomial $p_A(\lambda) = (5 - \lambda)^4$. What do we find on the diagonal of the Schur normal form?
A1: Only the number $0$.
A2: Only the number $5$.
A3: Only the number $4$.
A4: $0,1,2,3,4$
A5: $1,2,3,4$
A6: $0,0,4,4$
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Date of video: 2025-04-09
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Last update: 2025-10
