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Title: Example of 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 46 | Example of Schur Decomposition
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Quiz: Test your knowledge
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Subtitle on GitHub: ala46_sub_eng.srt missing
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Quiz Content
Q1: For which of the following matrix a Schur decomposition is not needed?
A1: $\begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix}$
A2: $\begin{pmatrix} 1 & 1 \\ 2 & 3 \end{pmatrix}$
A3: $\begin{pmatrix} 1 & 5 \\ 2 & 3 \end{pmatrix}$
A4: $\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$
Q2: What is a Schur normal form of $\begin{pmatrix} 4 & 1 \\ -2 & 1 \end{pmatrix}$?
A1: $\begin{pmatrix} 3 & 2 \\ 0 & 2 \end{pmatrix}$
A2: $\begin{pmatrix} 3 & 3 \\ 0 & 3 \end{pmatrix}$
A3: $\begin{pmatrix} 3 & 5 \\ 0 & 1 \end{pmatrix}$
A4: $\begin{pmatrix} 3 & 3 \\ 3 & 3 \end{pmatrix}$
A5: $\begin{pmatrix} 1 & 1 \\ 2 & 3 \end{pmatrix}$
Q3: Is it correct to say that every matrix $A \in \mathbb{C}^{n \times n}$ is similar to a triangular matrix?
A1: Yes, the Schur normal form shows that.
A2: No, there are counterexamples.
A3: Yes, but the Schur normal form does not show it!
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Date of video: 2025-04-10
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Last update: 2025-10
