typos in reg slides

This commit is contained in:
mhjensen
2020-09-11 11:19:35 +02:00
parent 243a68c942
commit a6f462e86f
128 changed files with 21520 additions and 21811 deletions
+2 -18
View File
@@ -1465,25 +1465,9 @@ As an example, the above defective matrix can be decomposed as
with eigenvalues $\sigma_1=2$ and $\sigma_2=0$.
The SVD exits always!
!split
===== Another Example =====
Consider the following matrix which can be SVD decomposed as
!bt
\[
\bm{X} = \frac{1}{15}\begin{bmatrix} 14 & 2\\ 4 & 22\\ 16 & 13\end{bmatrix}=\frac{1}{3}\begin{bmatrix} 1& 2 & 2 \\ 2& -1 & 1\\ 2 & 1& -2\end{bmatrix} \begin{bmatrix} 2& 0 \\ 0& 1\\ 0 & 0\end{bmatrix}\frac{1}{5}\begin{bmatrix} 3& 4 \\ 4& -3\end{bmatrix}=\bm{U}\bm{\Sigma}\bm{V}^T.
\]
!et
This is a $3\times 2$ matrix which is decomposed in terms of a
$3\times 3$ matrix $\bm{U}$, and a $2\times 2$ matrix $\bm{V}$. It is easy to see
that $\bm{U}$ and $\bm{V}$ are orthogonal (how?).
And the SVD
The SVD
decomposition (singular values) gives eigenvalues
$\sigma_i\geq\sigma_{i+1}$ for all $i$ and for dimensions larger than $i=2$, the
$\sigma_i\geq\sigma_{i+1}$ for all $i$ and for dimensions larger than $i=p$, the
eigenvalues (singular values) are zero.
In the general case, where our design matrix $\bm{X}$ has dimension