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| author | 8oldr <8oldr@proton.me> | 2026-10-16 16:59:36 +0300 |
|---|---|---|
| committer | 8oldr <8oldr@proton.me> | 2026-10-16 16:59:36 +0300 |
| commit | 1773466fef0b6fd417f257c1f3806f9421abf6e2 (patch) | |
| tree | 2616bc98fc0e6c8e89ca9ced47c658c70fa9b96c | |
| parent | 5c444886997ea588c0594895cc06fa7bb16dcff1 (diff) | |
Corrected columns of V in SVD, and singular vectors definitions
| -rw-r--r-- | lecture_1.typ | 4 |
1 files changed, 2 insertions, 2 deletions
diff --git a/lecture_1.typ b/lecture_1.typ index b46df53..756691d 100644 --- a/lecture_1.typ +++ b/lecture_1.typ @@ -158,7 +158,7 @@ $ $ We denote $sigma_i = Sigma_(i,i) : 1 <= i <= k$, and call $sigma_1, dots, sigma_k$ the singular values of $M$. -Let $u_1, dots, u_n$ be the columns of $U$, and $v_1, dots, v_n$ be the columns of $V$. This allows the following: +Let $u_1, dots, u_n$ be the columns of $U$, and $v_1, dots, v_m$ be the columns of $V$. This allows the following: $ M dot v_i & = U dot Sigma dot overbrace(V^T dot v_i, e_i) \ & = U dot Sigma dot e_i \ @@ -174,7 +174,7 @@ $ M^T = (U Sigma V^T)^T = V Sigma^T U^T $ -*Definition*: $U = u_1, dots, u_n, V = v_1, dots, v_n$ are called the _singular vectors_ of $M$. +*Definition*: $U = u_1, dots, u_n, V = v_1, dots, v_m$ are called the _singular vectors_ of $M$. *Definition* (operator norm): $M in M_(n times m) (RR), quad norm(M)_"op" = max{norm(M dot v) : v in RR^m and norm(v) = 1}$ (may be considered the maximum increase that $M$ can do to a vector). |