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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Finding $A^{\frac{1}{2}}$ where A is a symmetric positive definite matrix

Given the spectral decomposition $A = PDP^T$ where $D$ is the diagonal matrix of eigenvalues, I can define the following: $$ A^{\frac{1}{2}} = Q D^{\frac{1}{2}}Q^T $$ Is the following also true: if $...
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227 views

Calculate Homography with and without SVD

I've rendered an example for this question. With the trick of https://math.stackexchange.com/a/2619023/741822 I was able to calculate what seems to be the homography matrix (it passed 2 tests on $p_5$...
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46 views

Structure of an ill-conditioned matrix

I have the matrix $$A = \begin{bmatrix} 0.501 &1 & & &\\ &0.502& 1& &\\ & & \ddots & \ddots& \\ & & & 0.509& 1\\ &...
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How to show the relationship between the eigenvector of $A^TA$ and $AA^T$?

I was looking though this answer and had a question regarding the following statement: Then he shows that if $v$ is a unit eigenvector of $A^TA$ with eigenvalue $\sigma^2$, then $u = \frac{1}{\...
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262 views

Fitting a plane to points using SVD

I am trying to find a plane in 3D space that best fits a number of points. I want to do this using SVD. To calculate the SVD: Subtract the centroid of the points from each point. Put the points in ...
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20 views

SVD Calculation

Given a matrix $X\in \mathbb{R}^{2 \times 2}$ as follows $$X=\begin{bmatrix} a &b\\ c&d \end{bmatrix}.$$ Assume the dominant singular vector of $X$ is $u\in \mathbb{R}^{2\times1}$. Let another ...
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41 views

SVD starting with U instead of V

Most examples that I have seen online starts with by calculating the right hand singular vectors, and then from then use that $AV$ basically gives you U. If I instead wanted to start from by ...
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“Square-normal” matrices are normal

This is from a practice exam for my quals. Let $A$ be an $n \times n$ complex matrix. Suppose $A$ satisfies the following property: $(AA^\dagger)^2 = (A^\dagger A)^2$ Prove that $A$ is normal, ...
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Problem in calculation of SVD for a matrix

SVD Decomposition for the matrix: $$A=\begin{bmatrix} 3&-1&1\\3&0&2\\0&2&2\\-1&0&-2\end{bmatrix}$$ We want: $A=U\sum V^T$ We know that : $V=eigenvectors(A^T*A)$ $A^...
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SVD of matrix with complex coefficients

This is a question from an old exam: Perform the svd decomposition of the following matrix $$A=\begin{bmatrix} 1 & i\\ i & 1\end{bmatrix}$$ What I have done: I planned to use the svd ...
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49 views

Same SVD for $U M V^{\dagger}$ and $M$ where $U$ and $V$ isometries: why?

In the following document https://arxiv.org/pdf/1106.1445.pdf, on page 252 is stated the following property: Let $U$ and $V$ be isometries. I emphasize the fact that they are not necesserally ...
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how can I calculate the inverse of matrix?

Matrix has 16 rows and 1166 columns so I did not inverse directly. I have to calculate the SVD of matrix. I used the Matlab and I try to calculate pseudo inverse of matrix. [S V D]= svd(A) A= SVD' ...
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Convergence of singular vectors

I know that with Weyl's inequality or Hoffman-Wielandt inequality we can obtain the convergence of singular value from the convergence of matrix. I wonder if there is any lemma or theorem about the ...
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Rank of product of two orthogonal matrices

For $U,V\in\mathbb{R}^{n\times r}$, where each column $u_i$ satisfies that $u_i^Tu_j=0,j\ne i, u_i^Tu_i=1,\forall i=1,...,r$. So is $v_i$. Suppose we have $$\Vert UU^T-VV^T \Vert<1$$ My question is ...
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The smallest non-zero singular value of AB

For a matrix $A$, let $σ_{min}(A)$ denote the smallest $\textbf{non-zero}$ singular value of A. I saw some materials (e.g. https://pdfs.semanticscholar.org/e74f/89ac85239811b295787619e73c99fc867724....
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Statements about the SVD of a matrix.

Given a matrix $A \in \mathbb{R^{m \times n}}$ is some matrix with the single value decomposition of $A = U \Sigma V^T$, where $U$ contains the left orthogonal singular vectors and $V$ contains the ...
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60 views

How to compute efficiently and robustly if a given set of vectors is linearly independent?

Say we're given a set of $d$ vectors $S=\{\mathbf{v}_1,\dots,\mathbf{v}_d\}$ in $\mathbb{R}^n$, with $d\leq n$ (obviously). We want to test in an efficient way if S is linearly independent. Now, write ...
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How to prove a singular value expansion of a second derivative kernel

I am trying to prove a singular value expansion, when given \begin{equation} K(s,t) = \begin{cases} t(s-1) & \text{0} {\le}{t}{\le}s \\ s(t-1) & \text{s} {<}{t}{\le}1 ...
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Singular Values for $A$ and $A^2$

$\forall A \in R^{n \times n}$, if $\sigma$ is a singular value of $A$, then $\sigma^2$ is a singular value of $A^2$ Intuitively, this seems to be false, but how can I prove this?
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Singular value decomposition of identity matrix

If you run a SVD on a identity matrix, $I = U S V^T$, will the matrix $S$ also be an identity matrix?
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Kronecker product SVD

Suppose $a \in \mathbb R^{n_1n_2n_3}$. Show how to compute $f \in \mathbb R^{n_1}$ and $g \in \mathbb R^{n_2}$ so that $$\| a - h \otimes g \otimes f \|_2$$ is minimized where $h \in \mathbb R^{n_3}$ ...
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Singular vectors of random Gaussian matrix with non-isotropic rows

Suppose $G \in \mathbb{R}^{m \times n}$ has i.i.d. rows $g_i \sim \mathcal{N}(0, \Sigma)$ for some diagonal matrix $\Sigma = \text{diag}(\lambda_1,\dots,\lambda_n)$ where the diagonal entries satisfy $...
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90 views

The trace of the product of diagonal matrix and another arbitrary matrix

Let $D$ be a non-negative diagonal matrix with decreasing order in the diagonal, i.e. $D_{11}\geq D_{22}\geq\dots\geq 0$, And $X$ be an arbitrary square matrix with SVD decomposition, $X=U\Sigma V^{T}$...
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Changing basis of singular vectors in an SVD

If we have a SVD of $A$ such as $U \Sigma V^*$ then $U$ and $V$ are transformation matrices with columns made from eigenvectors of $AA^*$ and $A^*A$ respectively. Now if I multiply a column of $U$ by $...
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Does there exist a real normal matrix that's not symmetric, antisymmetric, orthogonal, and has distinct singular eigenvalues?

This question points out a matrix on Wikipedia that is real, normal, and neither symmetric, antisymmetric, or orthogonal. However, its singular values are 2, 1, 1. Is there one that has distinct ...
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If $v \in Ker(A)$ does that mean $v \in Ker(\Sigma)$?

Using SVD, if $v \in Ker(A)$ does that mean $v \in Ker(\Sigma)$? How can I proof this, if so?
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Prove that the Least Squares solution is Orthogonal to the Kernel of A

The least squares problem tries to minimize $||Ax - y ||^{2}$ . I'm trying to prove that $x^{LS} \perp Ker(A)$ where $x^{LS} = A^{\dagger}y$ and that $x^{LS}$ is the solution with the smallest $L2$ ...
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48 views

How to prove: if singular values equal the absolute values of eigenvalues then A is normal

Another post shows that if $A$ is normal that $|\lambda_i|=\sigma_i$ but how does one show the converse?
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Matrix column permutation with optimal low rank approximation

Say we are allowed to shuffle the elements in the columns of a matrix $M$. In other words, we can switch any $M_{i,j}$ with $M_{k,j}$. What would be an approach to identify a set of swaps that will ...
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Finding $3$D Lines passing through $3$D Points, all sharing a single Common Point

I'm writing a track finding algorithm, which finds the $3$d line going through a set of $3$d points such that the square distance for all points to that line is minimized. I found this fantastic ...
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How do you prove $\|A-B\|_F^2 \geq \|\Sigma_A - \Sigma_B\|_F^2$?

Given the following matrices $A,B\in \Bbb{R}^{m\times n}$ with $A=U \Sigma_A V^t$ and $B=Q \Sigma_B R^t$ (full SVD), how do you prove $$\|A-B\|_F^2 \geq \|\Sigma_A - \Sigma_B\|_F^2$$
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Is singular value matrix uniquely determined up to permuting rows and columns.

Given the SVD $A = U\Sigma V^T$, is $\Sigma$ uniquely determined up to permuting the rows and columns? My take is that singular value matrix is uniquely determined. Its diagonal elements are square ...
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SVD: $\mbox{im}(A)$ and $\ker(A)$ as orthonormal basis

In Gilbert Strang's book in the SVD chapter, he states that $u$ and $v$ give the bases for the image and kernel of a matrix. Say we have a matrix $A$ that has rank $r$. Why is it that the ...
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35 views

why must $|A\vec v_1|^2$ is the highest value ?if the $\vec v_1$ is the vector which is corresponding to the biggest singular value of A

If A is a $3$ by $2$ matrix,if we do the singular value decomposition (SVD) to A,that is $A= \begin{bmatrix} \vec u_1 & \vec u_2 &\vec u_3 \end{bmatrix} \begin{bmatrix} ...
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How to find reflection matrix in PCA?

I have bunch of 3d objects of different shapes (point clouds) and I am finding their principal components (three orthogonal vectors) using PCA. When I am checking the determinant of PC vectors then 60%...
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SVD of a matrix when its columns are shuffled and when certain columns are removed

as far as I can tell, if the columns of a matrix are shuffled, an SVD would still give the same left singular vectors and eigenvalues, and a permutation (based on which column is now in which place) ...
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SVD: why are the results different: hand-calculation and Matlab computation?

I am solving a SVD computatiob question. The following matrix is given: A=: $$\begin{bmatrix} 1.5 & 0.5 & 0&0 \\ 0.5 & 1.5 & 0 & 0\\ 0 &...
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97 views

Eigenvalues of $UV^T$ in SVD decomposition

After performing a singular value decomposition (SVD) of a real square matrix $A$, $$A=USV^T$$ How to prove that the absolute value of all eigenvalues of $UV^T$ are one? Is there any relation ...
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Does Shilov's Linear Algebra cover SVD?

I'm teaching myself Georgi E. Shilov's Linear Algebra, and have just finished chapter 6 except the final two starred sections. So far, I haven't seen Singular Value Decomposition (SVD). I skimmed the ...
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How to prove that the singular value of product of two orthonormal matrix is related to the principal angles between their columns space?

Assume that $A$, $B$ $\in R^{p\times d}$ both have orthonormal columns, then the vector of $d$ principal angles between their column spaces is give by $(\cos^{-1}\sigma_1,\cos^{-1}\sigma_2, \dots, \...
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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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91 views

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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31 views

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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67 views

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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199 views

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 ...
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22 views

Maximize in Closed form SVD

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 ...
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89 views

Condition number of principle submatrix

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 ...

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