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Title: Example 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 43 | Example for Jordan Normal Form
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Subtitle on GitHub: ala43_sub_eng.srt missing
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Quiz Content
Q1: Let $N \in \mathbb{C}^{6 \times 6}$ be a square matrix with only one eigenvalue $\lambda = 0$. Furthermore, we know $\mathrm{dim}(\mathrm{Ker}(N^3))= 6$, $\mathrm{dim}(\mathrm{Ker}(N^2))= 4$, $\mathrm{dim}(\mathrm{Ker}(N^2))= 2$. What is correct?
A1: The algebraic multiplicity of $\lambda$ is $6$.
A2: The geometric multiplicity of $\lambda$ is $6$.
A3: The geometric multiplicity of $\lambda$ is $1$.
A4: The algebraic multiplicity of $\lambda$ is $1$.
A5: We have $6$ Jordan chains of length $2$.
A6: We have $2$ Jordan chains of length $6$.
Q2: Let $N \in \mathbb{C}^{5 \times 5}$ be a square matrix with only one eigenvalue $\lambda = 0$. Furthermore, we know $\mathrm{dim}(\mathrm{Ker}(N^3))= 5$, $\mathrm{dim}(\mathrm{Ker}(N^2))= 4$, $\mathrm{dim}(\mathrm{Ker}(N^2))= 2$. What is correct?
A1: We have $2$ Jordan chains of length $2$.
A2: We have $5$ Jordan chains of length $2$.
A3: We have $2$ Jordan chains, one with length $3$ and another one with length $2$.
A4: We have $2$ Jordan chains, one with length $5$ and another one with length $1$.
A5: We have $3$ Jordan chains, two with length $1$ and another one with length $2$.
A6: We have $1$ Jordan chain of length $5$.
Q3: Let $N \in \mathbb{C}^{5 \times 5}$ be a square matrix with only one eigenvalue $\lambda = 0$. Furthermore, we know $\mathrm{dim}(\mathrm{Ker}(N^4))= 4$. What can we say?
A1: $N$ is diagonalizable.
A2: The geometric multiplicity of $\lambda$ is greater or equal than 2.
A3: The algebraic multiplicity of $\lambda$ is at most $4$.
A4: We have $2$ Jordan chains, one with length $5$ and another one with length $1$.
A5: We have $3$ Jordan chains, two with length $1$ and another one with length $2$.
A6: We have $1$ Jordan chain of length $5$.
A7: There are different possibilities for the Jordan normal form because we don’t have enough information given.
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Date of video: 2025-02-25
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Last update: 2025-10
