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Title: Proof of Spectral Theorem
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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 48 | Proof of Spectral Theorem
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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: ala48_sub_eng.srt missing
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Download bright video: Link on Vimeo
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Timestamps
00:00 Introduction
00:40 Statement: Spectral Theorem for Normal Matrices
01:37 Proof of first implication
04:15 Proof of second implication
06:15 How do normal triangular matrices look like?
08:31 Induction proof that normal triangular matrices are diagonal
12:06 Correction: here we already use $w = 0$ for the matrix product of $R^{\ast}R$!
12:27 Using the induction hypothesis
13:45 Outro and credits
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Subtitle in English (n/a)
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Quiz Content
Q1: Let $A \in \mathbb{C}^{n \times n}$ be unitary. What can we conclude by the spectral theorem?
A1: $A$ can be unitarily diagonalized.
A2: $A$ is also unitary.
A3: $A$ has $n$ different eigenvalues.
A4: $A$ has eigenspaces of dimension $2$
A5: $A$ has no eigenvectors.
A6: $A$ is diagonal.
Q2: Let $R \in \mathbb{C}^{n \times n}$ be an upper triangular matrix. What is a correct implication?
A1: If $R$ is normal, then $R$ is diagonal.
A2: If $R$ is diagonal, then $R$ is selfadjoint.
A3: If $R$ is normal, then $R$ is unitary.
A4: If $R$ is selfadjoint, then $R$ is unitary.
A5: If $R$ is diagonalizable, then $R$ is normal.
Q3: Is the matrix $$ \begin{pmatrix} 2 & 3 & 1 \ 0 & 4 & 6 \ 0 & 0 & 1 \end{pmatrix} $$ a normal matrix?
A1: No!
A2: Yes!
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Date of video: 2025-04-14
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Last update: 2025-10
