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Misleading and with no references "Derivation from calculus of variations" section removed instead added a reference to Ritz method where it is correctly described. Section "Applications in mechanical engineering" moved to Ritz method |
→Example: a few corrections and explanations added |
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Line 101:
0 & 1 & 0 & 0\\
1 & 0 & 0 & 0
\end{bmatrix}</math>,
where the columns of the first multiplier from the complete set of the left singular vectors of the matrix <math>A</math>, the diagonal entries of the middle term are the singular values, and the columns of the last multiplier transposed (although the transposition does not change it)
:<math>
\begin{bmatrix}
0 & 0 & 0 & 1\\
0 & 0 & 1 & 0\\
0 & 1 & 0 & 0\\
1 & 0 & 0 & 0
\end{bmatrix}^*
\quad = \quad
\begin{bmatrix}
0 & 0 & 0 & 1\\
0 & 0 & 1 & 0\\
0 & 1 & 0 & 0\\
1 & 0 & 0 & 0
\end{bmatrix}
</math>
are the corresponding right singular vectors.
Let us take
:<math>W = \begin{bmatrix}
Line 108 ⟶ 126:
0 & 0\\
0 & 0
\end{bmatrix}
with the column-space that is spanned by the two exact right singular vectors
:<math>
\begin{bmatrix}
0 & 1\\
1 & 0\\
0 & 0\\
0 & 0
\end{bmatrix}
</math>
corresponding to the singular values 1 and 2.
Following the algorithm step 1, we compute
Line 156 ⟶ 184:
</math>
recovering from its rows the two right singular vectors <math>v</math> as <math>[0, 1, 0, 0]^*</math> and <math>[1, 0, 0, 0]^*</math>.
We validate the first vector: <math>Mv=\sigma u</math>
: <math>
\begin{bmatrix}
Line 183 ⟶ 211:
\begin{bmatrix}0\\ 1\\ 0\\ 0\end{bmatrix}.
</math>
Thus, for the given matrix <math>W</math> with its column-space that is spanned by two exact
== See also ==
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