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Title: Block Diagonalization
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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 40 | Block Diagonalization
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Subtitle on GitHub: ala40_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 $V$ be complex vector space and $U_1$ and $U_2$ be two subspaces with $V = U_1 \oplus U_2$. Can any map $\ell: V \rightarrow V$ be put into a block diagonal form $\ell_{\mathcal{B} \rightarrow \mathcal{B}}$ for a suitable basis $\mathcal{B}$?
A1: Only in special cases.
A2: Yes, always.
A3: No, never!
Q2: Let $V$ be complex vector space and $\ell: V \rightarrow V$ be a linear map such that $\ell_{\mathcal{B} \rightarrow \mathcal{B}}$ is in a block diagonal form, where the first block is of size $2\times2$. Does $\ell$ have non-trivial invariant subspaces?
A1: Only in special cases.
A2: Yes, always.
A3: No, never!
Q3: Let $A \in \mathbb{C}^{n \times n}$ be diagonalizable with $n\neq2$. What is not correct in general?
A1: $A$ is similar to a block triangular matrix written as $ \begin{pmatrix} 0 & \mathbb{1} \ 0 & 0\end{pmatrix}$.
A2: $\mathbb{C}^n = U_1 \oplus U_2 \oplus \cdots \oplus U_n$ for $A$-invariant subspaces $U_j$.
A3: $A$ has at least $n$ invariant subspaces.
A4: If $A$ has two different eigenvalues, it’s similar to a block diagonal matrix.
A5: $A$ is similar to a block diagonal matrix.
Q4: Let $T \in \mathbb{C}^{4 \times 4}$ be a square matrix and $U_1$, $U_2$ invariant subspaces under $T$, where both have dimension $2$, and we have $\mathbb{C}^4 = U_1 \oplus U_2$. What is the correct block diagonalization in this case?
A1: $\left(\begin{array}{cc|cc} a_{11} & a_{12} & 0 & 0 \ a_{21} & a_{22} & 0 & 0 \ \hline 0 & 0 & b_{11} & b_{12} \ 0 & 0 & b_{21} & b_{22} \end{array}\right)$
A2: $\left(\begin{array}{c|ccc} a_{11} & 0 & 0 & 0 \ \hline 0 & b_{11} & b_{12} & b_{13} \ 0 & b_{21} & b_{22} & b_{23} \ 0 & b_{31} & b_{32} & b_{33} \end{array}\right)$
A3: $\left(\begin{array}{cc|cc} a_{11} & 0 & 0 & 0 \ 0 & a_{22} & 0 & 0 \ \hline 0 & 0 & b_{11} & 0 \ 0 & 0 & 0 & b_{22} \end{array}\right)$
Q5: Let $A \in \mathbb{C}^{n \times n}$ where $1$ and $5$ are eigenvalues. Is there a block diagonalization like $$ A \approx \begin{pmatrix} B_1 & 0 \ 0 & B_2 \end{pmatrix} \text { ? }$$
A1: Only in special cases.
A2: Yes, always.
A3: No, never!
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Date of video: 2025-02-03
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Last update: 2025-10
