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.

learn more… | top users | synonyms

1
vote
1answer
11 views

If A and A' are approximately the same, are their principal components/SVD very close?

If we have that two matrices $A\approx A'$ within some guaranteed error bound for each term, and $A=U\Sigma V$ is the singular value decomposition for $A$, and $A'=U'\Sigma' V'$ is the SVD for $A'$, ...
0
votes
1answer
8 views

Replacing Singular Values of a Matrix with Complex Ones

Is there a procedure to replace singular values of a real valued matrix according to: s1 -> i*s1 s2 -> i*s2 ... without going through any singular value decomposition (change singular values and ...
0
votes
0answers
14 views

A question in paper “Fitting helices to data by total least squares” writen by Yves Nievergelt in 1996

does anyone read this paper before? I got a problem in this paper. Specifically, I do not understand the Step 2.3 to 2.5. Two variables, r and s, are involved in the calculation but I have no idea ...
0
votes
1answer
39 views

Can a matrix have more than one inverse (Singular Value Decomposition)

Assume there's a matrix $A$ with SVD as below $$ A = U \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} ...
0
votes
3answers
43 views

Approximate matrix by a rank 2 matrix using singular values

I only understand the singular value decomposition process. Do I apply it to this matrix? \begin{bmatrix} 0 & 0 & \pi \\ 0 & e & 0 \\ 1&0&0 \end{bmatrix} What is the idea ...
1
vote
0answers
20 views

SVD, ALSWR tutorial

Please advise good introductory books for neophyte on the following subjects: LSA (Latent Semantic Analysis) SVD (Single Value Decomposition). SGD(Stochastic Gradient Descent) ...
0
votes
0answers
35 views

Algorithm to efficiently compute$ A^k$

If a symmetric matrix $A$ has SVD $A=U\Sigma U^{\top}$, then $A^k=U\Sigma^kU^{\top}$. What would be the most efficient algorithm to compute $A^k$ such that the worst case time complexity is as low as ...
0
votes
1answer
25 views

If a symmetric matrix $A$ has SVD $A=U\Sigma U^T$, then $A^k=U\Sigma^k U^T$ [closed]

How can I prove that for symmetric matrix for SVD, the following condition is true $$A^k=UΣ^kU^T$$
0
votes
1answer
17 views

The less than full rank case, can be done with SVD decomposition?

The matrix is $$ A = \left( \begin{matrix} 1 & 2 \\ 2 & 4 \\ 3 & 6 \end{matrix} \right), $$ The rank is 1, there only one nonzero eigenvalue, and when I was doing the svd decomposition, ...
0
votes
0answers
35 views

Why are singular values of a positive

I read this in my textbook but couldn't understand why this is true: For a real positive semi-definite matrix A, the singular values are the same as the eigenvalues. Could someone please explain ...
1
vote
1answer
13 views

why can SVD handle rank deficient matrices?

I am currently reading a book on data analysis (Nathan Kutz, Data-Driven Modeling & Scientific Computation) and a bit stuck in the chapter about SVD. It states that SVDs can be used to handle ...
0
votes
0answers
17 views

Best fitting circle to points in 3D

I have a set of n ≥ 3 points in 3D that are measurements of a possible circle. The measured points are "noisy" so best-fitting algorithms are involved. I'm programming in C# and have put together some ...
1
vote
1answer
83 views

Incorrect angle detected between two planes

I want to calculate the angle between 2 planes, Reference plane and Plane1. When I feed the X,Y,Z co-ordinates of pointCloud to the function plane_fit.m (by Kevin Mattheus Moerman), I get the output ...
0
votes
1answer
23 views

principal eigenvectors of an unknown matrix

Do you have any idea about how we can find the principle eigenvectors of an unknown matrix ${H}$. The only information that we have is that $H$ has only a few (up to 3) dominant eigen modes regardless ...
1
vote
1answer
32 views

Analytical solution to the first PCA direction

It is known that the first PCA direction for a dataset of $n$ points is the unit vector with max variance after projecting the points onto this vector. I wonder whether there are some analytical ...
0
votes
0answers
23 views

Non-orthogonal basis

I have a set of complex vectors (maybe 10,000 vectors, each of which has maybe 200 elements). I know that each of the complex vectors is a linear combination of a small (maybe 10) collection of ...
1
vote
2answers
63 views

Singular Value Decomposition in Axler's book

In Axler's "Linear Algebra Done Right", he gave the singular-value decomposition as: $Tv = s_1\langle v,e_1\rangle f_1 + \cdots + s_n\langle v,e_n\rangle f_n$, where T is an operator; ...
0
votes
2answers
19 views

What does it mean by one matrix is **unitarily similar** to another?

I am reading a tutorial about the Lanczos method for eigen problem / SVD. It mentioned "Then the tridiagonal matrix $B^∗B$ is unitarily similar to $A^∗A$. " What does it mean? I can derive this: ...
0
votes
0answers
20 views

Understand singular vectors and unit-phase factor

Wikipedia says "Non-degenerate singular values always have unique left- and right-singular vectors, up to multiplication by a unit-phase factor $e^{i\theta}$". I don't understad it. Can you explain it ...
2
votes
1answer
40 views

Is the product eigenvalues less than or equal to the product of singular values?

On a numerical math course I recently saw the following statement without a proof. How would one prove it? Let $A$ be an $n$ x $n$ real matrix with singular values $s_1 \geq s_2 \geq \ldots \geq ...
1
vote
1answer
54 views

Connection between results of two SVDs

Consider SVD of $M$: $$ M = U \Sigma V^\top $$ And SVD of $N= \ln M$: $$ N = U^\prime \Sigma^\prime V^{^\prime\top} $$ Anyone knows/has seen/can think of any interesting connection/relation between ...
0
votes
0answers
19 views

Different SVD results in Matlab

my question relates to calculating SVD in Matlab. I have been reading a lot and somehow I have jumbled up all the facts. It would be great if you experts could get me to the right track. My task is ...
0
votes
2answers
30 views

Prove that $\text{rank}(A) = \text{rank}(A^T)$ using SVD

The title pretty much says it all, I need to prove that $\text{rank}(A) = \text{rank}(A^T)$ using SVD. It seems quite trivial, but I'd like to hear a second opinion. My thoughts are exposed below. ...
0
votes
1answer
34 views

Converting between SVD and Eigenvector-based expressions

For the well known ordinary least squares there are the well known solution $$\beta=(X'X)^{-1}X'Y$$ This can be expressed in "canonical form" using eigenvector decomposition. ...
1
vote
0answers
17 views

A matrix being “diagonal with a $c$-column border”, what does it mean?

The matrix in the middle of the beginning part, i.e., $$ Q=\begin{bmatrix} \operatorname{diag}(s) & L \\ 0 & K \end{bmatrix} $$ In the context, $L$ is a $r$ by $c$ matrix. What is the ...
0
votes
0answers
14 views

Relations between SVD and QR decomposition, and matrix projection onto orthogonal basis

For a $rank-r$ matrix $M_{p\times q}$, we can calculate a SVD: Equation 1: In the following, the authors of a paper want to update the SVD of matrix $M$ when new data (as new column) come, i.e., ...
1
vote
1answer
29 views

Find the singular value decomposition for the following matrix and try to use the decomposition to create a sketch of the range in R3?

Let $$D=\begin{pmatrix}1&2\\1&0\\1&0\end{pmatrix}.$$ I found the SVD to be ...
0
votes
2answers
23 views

Creating a random square matrix with known singular values

The first step in one question has me creating a random square matrix A with singular values given as $2^{-1}, 2^{-2}\dots 2^{-n}$. There is no other information about what assumptions can be made ...
0
votes
1answer
15 views

How to prove $V*V^T=I$ in SVD? [duplicate]

How to prove $V*V^T=I$ in SVD: $M=U*S*V^T$? It's easy to understand $V^T*V=I$. It seems $V*V^T=I$, but how to prove it?
0
votes
0answers
8 views

Matrix with highly correlated adjacency entries

I am learning about SVD from this book. One of the exercise questions asks me to create matrix with highly correlated adjacency entries and then conduct some experiments to discover the nature of the ...
0
votes
1answer
57 views

Why does SVD provide the least squares solution to $Ax=b$?

I am studying the Singular Value Decomposition and its properties. It is widely used in order to solve equations of the form $Ax=b$. I have seen the following: When we have the equation system $Ax=b$, ...
0
votes
0answers
26 views

Prove this relation between truncated SVD and eigen decomposition?

For a real matrix $M$, we have a full SVD $M=USV^T$ and a truncated SVD $M_{k}=U_kS_kV_k^T$. The truncated SVD means (matlab grammar): $U_k=U(:,1:k), S_k=S(1:k,1:k), V_k=V(:,1:k)$. Based on the ...
0
votes
0answers
27 views

Perpendicular distances versus vertical distances

Why is it better to use perpendicular distance rather than vertical distance along a particular coordinate axis when finding the best fit subspace? This is an exercise question in a chapter related to ...
1
vote
0answers
22 views

Bound on Signal Amplitude for subspace methods (MUSIC, ESPRIT)

MUSIC and ESPRIT are methods that use subspace decomposition to identify signal Parameters. Subspace decomposition is achieved either by SVD or Eigen Value Decomposition. Subspace decomposition ...
2
votes
1answer
47 views

How do I find 2x2 orthonormal diagonalizing matrices using only trigonometry?

I have a matrix $A=\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ (where all values are known), and I eventually want to diagonalize it into: $$ A=UDV^T $$ for orthonormal U and V. If I ...
0
votes
0answers
18 views

Issues with connecting the SVD and Eigenvalues for block matrix

In class, we have talked about the singular value decomposition and its connection to Eigenvalues. Specifically, for a matrix A, if the columns of a matrix contain linearly independent eigenvectors, ...
0
votes
1answer
141 views

Plane fitting using svd

I am trying to get a best fit plane in a 3d space of points. I am using an svd as described in http://stackoverflow.com/questions/10900141/fast-plane-fitting-to-many-points. If I use the data provided ...
1
vote
0answers
32 views

Double Normalization Before SVD

It is usually suggested that we normalize a matrix before SVD. For a certain recommender application we found out normalizing both columns and rows is necessary otherwise SVD based recommendations do ...
0
votes
0answers
35 views

Modifying U=mxn SVD Algorithm to U=mxm Algorithm

I have painstakingly ported this Python source "svd.py" to C++. I confirm it works for the example it comes with. While testing another example (this one, from Wikipedia), the assert statement trips ...
1
vote
0answers
9 views

What is pseudodiagonality in matrix/tensor?

What is the difference between diagonality and pseudodiagonality? Does this apply to tensor too? https://www.math.uzh.ch/fileadmin/math/preprints/06_11.pdf
0
votes
1answer
63 views

What is the Singular Value Decomposition for the Zero Matrix?

I am interested in the singular value decomposition of a matrix: $\mathbf{M} = \mathbf{U} \mathbf{S} \mathbf{V}^T$. Suppose $\mathbf{M} = \mathbf{0}$ (zero matrix) and square. Clearly, $\mathbf{S} = ...
5
votes
1answer
107 views

Can I turn $Ax=b$ into $Ax=0$?

For a system of equations $$ \begin{bmatrix}d_1 & d_2 & \dots & d_n \end{bmatrix} \begin{bmatrix}u_1\\u_2\\ \vdots \\ u_n \end{bmatrix} = d_{n+1} $$ where each $d$ is a column of ...
0
votes
1answer
24 views

Idea of singular vectors in matrix completion

I am trying to understand the concept of matrix completion. I came across the following line (M is a low rank matrix, and we only have few samples of it. The task is to recover the entire matrix from ...
1
vote
1answer
77 views

Do these two rearranged matrices have the same singular values (or the same rank)?

This is the origin of my problem: I have a set of data which expresses which user ($U$ set) applies what tag ($T$ set) to which item ($I$ set). So it is actually a $U×I×T$ tensor $A$ (or 3-dimensional ...
0
votes
1answer
30 views

How to find parameters that minimize the sum of squares, using Matlab?

I have a system of linear equations in the following form. How can I solve it in Matlab? $$\operatorname*{argmin}_{a,b} \sum_{i,j} [X(i,j)-a\times Y(i,j)-b]^2$$ Where X and Y are known. I need to ...
0
votes
0answers
25 views

Weighted Rigid Body Transformation

Usually if one talks about rigid body transformation between 2 sets of points, it means: Performing rigid body transformation upon 1 set of points so that the least square error between the 2 sets of ...
0
votes
0answers
39 views

Is the eigenvalue decomposition equal to the singular value decomposition for real symmetric matrices?

Question is as the title states. I've read something similar for hermitian matrices, but am unsure if this is correct as well for real symmetric matrices.
0
votes
1answer
30 views

Finding SVD of a Matrix

If $a_{1},a_{2} \in \mathcal{R}^{2},$ $\ \|a_{1}\|_{2} = \|a_{2}\|_{2} = K$, and the angle $\theta$ between $a_{1}$ and $a_{2}$ is between $0$ and $\pi/2$, we want to compute the SVD of the matrix $A ...
1
vote
1answer
50 views

Find rigid transform from coordinates of the same points in different reference frames

Given two* 3-dimensional points $p_1$ and $p_2$ expressed in different reference frames $A$ and $B$, find the rigid transform (rotation and translation) between frames $A$ and $B$. The answer to this ...
1
vote
1answer
24 views

Cross-product is a left singular vector?

Assume A is a 3x2 matrix with rank(A)=2. u1 and u2 are already left singular vectors... How would I go about proving that the cross-product of the two is also a left singular vector? Hints would be ...