typos in reg slides
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@@ -1465,25 +1465,9 @@ As an example, the above defective matrix can be decomposed as
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with eigenvalues $\sigma_1=2$ and $\sigma_2=0$.
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The SVD exits always!
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!split
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===== Another Example =====
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Consider the following matrix which can be SVD decomposed as
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!bt
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\[
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\bm{X} = \frac{1}{15}\begin{bmatrix} 14 & 2\\ 4 & 22\\ 16 & 13\end{bmatrix}=\frac{1}{3}\begin{bmatrix} 1& 2 & 2 \\ 2& -1 & 1\\ 2 & 1& -2\end{bmatrix} \begin{bmatrix} 2& 0 \\ 0& 1\\ 0 & 0\end{bmatrix}\frac{1}{5}\begin{bmatrix} 3& 4 \\ 4& -3\end{bmatrix}=\bm{U}\bm{\Sigma}\bm{V}^T.
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\]
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!et
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This is a $3\times 2$ matrix which is decomposed in terms of a
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$3\times 3$ matrix $\bm{U}$, and a $2\times 2$ matrix $\bm{V}$. It is easy to see
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that $\bm{U}$ and $\bm{V}$ are orthogonal (how?).
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And the SVD
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The SVD
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decomposition (singular values) gives eigenvalues
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$\sigma_i\geq\sigma_{i+1}$ for all $i$ and for dimensions larger than $i=2$, the
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$\sigma_i\geq\sigma_{i+1}$ for all $i$ and for dimensions larger than $i=p$, the
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eigenvalues (singular values) are zero.
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In the general case, where our design matrix $\bm{X}$ has dimension
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