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:<math>\mathbf{T}_L = \mathbf{U}_L\mathbf{\Sigma}_L = \mathbf{X} \mathbf{W}_L </math>
The truncation of a matrix '''M''' or '''T''' using a truncated singular value decomposition in this way produces a truncated matrix that is the nearest possible matrix of [[Rank (linear algebra)|rank]] ''L'' to the original matrix, in the sense of the difference between the two having the smallest possible [[Frobenius norm]], a result known as the [[Low-rank approximation#Proof of Eckart–Young–Mirsky theorem (for Frobenius norm)|Eckart–Young theorem]] [1936].
<blockquote>
'''Theorem (Optimal k‑dimensional fit).'''
Let P be an n×m data matrix whose columns have been mean‑centered and scaled, and let
<math>P = U \,\Sigma\, V^{T}</math>
be its singular value decomposition. Then the best rank‑k approximation to P in the least‑squares (Frobenius‑norm) sense is
<math>P_{k} = U_{k}\,\Sigma_{k}\,V_{k}^{T}</math>,
where V<sub>k</sub> consists of the first k columns of V. Moreover, the relative residual variance is
<math>R(k)=\frac{\sum_{j=k+1}^{m}\sigma_{j}^{2}}{\sum_{j=1}^{m}\sigma_{j}^{2}}</math>.
</blockquote>
<ref>{{cite book
| last = Holmes
| first = M.
| title = Introduction to Scientific Computing and Data Analysis
| edition = 2nd
| year = 2023
| publisher = Springer
| isbn = 978-3-031-22429-4
}}</ref>
== Further considerations ==
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