Review Note
Last Update: 09/23/2024 04:12 PM
Current Deck: LinAl_SuperDecks
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Front
Spectral Theorem Special Case:
\( \mathbf{A} = \mathbf{V} \mathbf{\Lambda} \mathbf{V}' \)
What is the "sum of outer products" form?
- \( \mathbf{A} \in \mathbb{F}^{N \times N} \) is (Hermitian) symmetric, i.e., \( \mathbf{A} = \mathbf{A}' \)
- \(\mathbf{V} = [\mathbf{v}_1, \dots, \mathbf{v}_N]\) are the matrix contains eigenvectors of \(\mathbf{A}\)
- \( \mathbf{\Lambda} = \text{Diag}\{\lambda_1, \dots, \lambda_N\} \) where \(\lambda_i\)s are the eigenvalues of \(\mathbf{A}\)
\( \mathbf{A} = \mathbf{V} \mathbf{\Lambda} \mathbf{V}' \)
What is the "sum of outer products" form?
Back
\( \mathbf{A} = \mathbf{V} \mathbf{\Lambda} \mathbf{V}' = \sum_{n=1}^{N} \lambda_n \mathbf{v}_n \mathbf{v}_n' \)
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Index
pg 3.7
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