# 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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### Is every symmetric matrix in $\mathbb R^n$ a (non-uniform) stretch?

I conjecture that in $\mathbb R^n$, every symmetric matrix is a non-uniform stretch. Am I correct? By non-uniform stretch, I mean that if $T$ is a non-uniform stretch, there exists an orthonormal ...
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### Does SVD work for non-square matrices?

I'm trying to decompose a matrix of keypoints for Tomasi-Kanada factorization I am not doing this by hand right now simply because there are libraries available for this and I know I will get the ...
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### SVD: intuition of why linear transformation takes some orthogonal basis to orthogonal vectors

The singular value decomposition (SVD) was introduced to us as $MV = U\Sigma$, i.e., finding some mutually orthogonal normalized vectors (columns of $V$) that map to orthogonal vectors $U$ (then we ...
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### SVD of matrix of eigenvectors

Let $A$ be a (complex) matrix with eigendecomposition $A = U S U^{-1}$, where $U$ is the invertible matrix of eigenvectors. What is known about the singular value decomposition (SVD) of $U$?
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### A proof of permutation matrix.

For a square matrix $A \in \Bbb R^{n\times n}$, if its singular values are identity matrix $I$, moreover, the summation of each row is equal to $1$, and all elements are non-negative, can we prove ...
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### How does the usual algorithm for SVD matrix factorization work?

I get that SVD is a way of representing an arbitrary linear transformation on ${R}^n$ as a composition of a scaling, a rotation, and a reflection transformation, and I've seen plenty of insightful ...
1 vote
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### Why are there two solutions to this SVD decomposition problem?

I was faced with the task of decomposing the following matrix via SVD, i.e. decomposing the matrix into the form $USV^T$. \begin{align*} A = \begin{bmatrix}3&2&2\\2&3&-2\end{bmatrix} ...
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### Working with change of coordinates in singular value decomposition (SVD)

I am given a matrix A which I need to take the SVD of. Currently I have found the vectors U, Σ and V such that A=UΣV. As the next part of my task I need to find the matrix A in V coordinates. To solve ...
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### Working with singular value decomposition (SVD)

I am given a matrix $A$ which I need to take the SVD of. Currently I have found the vectors $U$, $\Sigma$ and $V$ such that $A = U \Sigma V$. As the next part of my task I need to find the matrix $A$ ...
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### SVD of $A$ has rank $r$. How to write $A$ as a sum of $r$ rank-$1$ matrices? [duplicate]

Suppose $A = U \Sigma 𝑉^*$ is the SVD of $A$, and $A$ has rank $r$. Using this decomposition, how do I write $A$ as a sum of $r$ rank-$1$ matrices? I am very confused about how to approach this. I ...
1 vote
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### Uniqueness of $UV^T$ of non-singular matrix $A$

Let $A$ be a $n \times n$ non-singular matrix. Then we can take many alternative SVD from from $A$. Like $A = U_1 \Sigma V_1^\top = U_2 \Sigma V_2^\top = \cdots$ where $\Sigma$ is an ordered diagonal (...
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### What is the singular value of such a simple permutation matrix?

The matrix $\left[\begin{array} &0 &1\\1& 0\end{array}\right]$ The following two solutions give different answers: The first |A-λI|=0 (-λ) 1 1 (-λ) = 0 ∴(-λ)×(-λ)-1×1=0 ∴(λ2)-1=0 ∴(λ2-...
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### Why does subtracting a rank-$1$ matrix shift the singular vectors?

To compute SVD of a matrix: Say you start with the matrix $A$ and you compute $v_1$. You can use $v_1$ to compute $u_1$ and $\sigma_1(A)$. Then you want to ensure you're ignoring all vectors in the ...