# Questions tagged [svd]

In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix, with many useful applications in signal processing and statistics.

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### Prove that $\|{X}\|_{*}=\min _{{A B}={X}}\|{A}\|_{F}\|{B}\|_{F}=\min _{{A B}={X}} \frac{1}{2}\left(\|{A}\|_{F}^{2}+\|{B}\|_{F}^{2}\right)$.

For any matrix $X\in\mathbb{R}^{m\times n}$, I am confused with $$\|{X}\|_{*}=\min _{{A B}={X}}\|{A}\|_{F}\|{B}\|_{F}=\min _{{A B}={X}} \frac{1}{2}\left(\|{A}\|_{F}^{2}+\|{B}\|_{F}^{2}\right),$$ ...
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### System of differential equations with a NOT positive definite matrix (SVD?)

After solving the dynamics of a system, I arrived at an apparently simple system of differential equations of the form: $$\tau \frac{d}{dt}\vec{w}=Q\vec{w}+\vec{c}$$ where $\vec{c}\neq\vec{c}(t)$...
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### Is the orthogonal polar factor of matrices of nullity 1 smooth?

Let $M_n$ be the space of real $n \times n$ matrices, and let $\text{GL}_n^-=\{A \, | \, \det A < 0 \}$. Set $$S=\text{GL}_n^- \cup \{A \in M_n \, | \, \text{rank} (A) = n-1 \}.$$ Claim: There ...
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### Trace of SVD low rank in Frobenius norm

I'm trying to understand the low rank approximation matrices using SVD and Frobenius norm, and one line I keep encountering and cannot understand is the following : \operatorname{Tr}((A-M)^*(A-M)) ...
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### Describe in a sentence to perform Singular Value Decomposition (SVD) for all $t$.

This question might be more of a linguistic thing than a pure mathematical question. However, I am struggling with writing something down in a proper manner. I'm trying to describe that I want to ...
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### Given $W=ULV^T$ and a vector $\mathbf{x}$, can we compute $UL^kV^T\mathbf{x}$ without doing the SVD, for any integer k?

Consider a matrix $W \in \mathbb{R}^{n\times m}$ with corresponding singular value decomposition, $W = ULV^T$, and a vector $\mathbf{x} \in \mathbb{R}^{m}$. Is it possible to compute matrix-vector-...
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### Angle between the singular vectors of a matrix A and the singular vectors of Transpose(A)

There is a clear relation between the eigenvectors of $A$ and $A^T$. They are mutually orthogonal. But I cannot find a similar relation between the singular vectors of $A$ and $A^T$. I am looking ...
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### Restrictions to simplify computation of matrix powers with SVD

For a given $2 \times 2$ matrix $A$ ($\in \mathbb{R}^{2 \times 2})$ I want to compute $A^n$ using singular value decomposition, where the intermediate terms are "nice". Assuming we have a singular ...
I am looking to maximize, where $R$ is a 3x3 rotational matrix, $q'_i$ and $q_i$ are known data points. $\mbox{maximize}_{R} \ \sum_i \mathbf{q'^{t}_i} \mathbf{R} \mathbf{q_i}$ The solution to ...
Suppose we have a $m\times n$ matrix. Consider its principle submatrices (removing the $i$th row and col). How do I find the principle submatrix with the smallest condition number? Currently I loop ...