Lagmental Vicfred

Singular Value Decomposition Works for Every Matrix by Vicfred

tags: mathematics, matrix factorizations, least squares, positive matrices

Any rectangular matrix factors into two unitary matrices and a nonnegative diagonal singular-value matrix. The point is to make the formal expression readable enough to audit line by line.

Set-up

Matrix factorizations expose different geometry: \(A=QR\) separates an orthonormal frame, while \(A=U\Sigma V^\ast\) separates rotations from axis scaling.

$$ A\in\mathbf C^{m\times n} $$

The formulas should not be merged too early. The datum \(\mathsf D\), the conclusion \(\mathsf C\), and the bridge \(\Longrightarrow\) have three different logical jobs.

$$ A=U\Sigma V^\ast,\qquad\sigma_1\ge\cdots\ge\sigma_r>0 $$

The calculation

Now evaluate one representative case. The result should agree with the structural law above, but it is obtained without assuming the conclusion.

$$ A^\ast A=V\begin{pmatrix}\sigma_1^2&&\\&\ddots&\\&&\sigma_r^2\\&&&0\end{pmatrix}V^\ast $$

What survives abstraction

The formula is reusable precisely because it says which pieces are structural and which belong only to the worked example.

$$ \begin{aligned} \mathsf{D}\;&:\quad A\in\mathbf C^{m\times n},\\[5pt] \mathsf{C}\;&:\quad A=U\Sigma V^\ast,\qquad\sigma_1\ge\cdots\ge\sigma_r>0. \end{aligned} $$

The boundary

Conditioning matters numerically. An exact algebraic identity can be a poor computational method when it squares the condition number.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] A=U\Sigma V^\ast,\qquad\sigma_1\ge\cdots\ge\sigma_r>0 \end{gathered}} $$

A symbolic summary is trustworthy only because the example and limitation remain visible beside it. The box compresses the conclusion without hiding its origin.

This article was posted on Thu 20 November 2025. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.