Review Note
Last Update: 09/23/2024 04:12 PM
Current Deck: LinAl_SuperDecks
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Front
Spectral Theorem deals with the ability to diagonalize a normal matrix. The special cases that spectral theorem applies to involve a matrix \(\mathbf{A}\) that is a (Hermitian) symmetric. What is guaranteed about \(\mathbf{V}\) in:
\[ \mathbf{A} = \mathbf{V} \mathbf{\Lambda} \mathbf{V}' = \sum_{n=1}^{N} \lambda_n \mathbf{v}_n \mathbf{v}_n' \]
\[ \mathbf{A} = \mathbf{V} \mathbf{\Lambda} \mathbf{V}' = \sum_{n=1}^{N} \lambda_n \mathbf{v}_n \mathbf{v}_n' \]
Back
- \(\mathbf{V}\) is orthogonal/unitary. Hence, eigendecomposition also known as unitary decomposition.
- If \(\mathbf{A}\) is real, \(\mathbf{V}\) is also real
Click for Hint
\( \mathbf{V}' \mathbf{V} = \mathbf{V} \mathbf{V}' = \mathbf{I} \), so \( \mathbf{V}^{-1} = \mathbf{V}' \)
Index
pg. 3.7
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