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

0
votes
0answers
6 views

Proof of Optimality for Approximation of Probability Spaces by PCA

I have come across a theorem that states, that the $d$-dimensional subspace found by PCA is the optimal approximation of a probability space with such a plane, in the sense that it minimises the ...
0
votes
0answers
14 views

Gaussian variance estimation via spectral decomposition

I was given a dataset (a mat file) of 100,000 observations, each with 50 dimensions (coordinates). Denote matrix $X$ is a 100,000x50 matrix in which each column was generated according to: ...
2
votes
1answer
19 views

$B - A \in S^n_{++}$ and $I - A^{1/2}B^{-1}A^{1/2} \in S^n_{++}$ equivalent?

Define $S^n_{++}$ to be the set that contains all the positive definite matrices. That is, if $A \in S^n_{++}$, then $A$ is a positive definite matrix. Now suppose that $A,B \in S^n_{++}$ are two ...
1
vote
1answer
36 views

the relation between positive definite matrix and its ellipsoid

Define $S^n_{++}$ to be the set that contains all the positive definite matrices. That is, if $A \in S^n_{++}$, then $A$ is a positive definite matrix. Then, we associate with each $A \in S^n_{++}$ ...
1
vote
0answers
24 views

In the SVD of $A = U \Sigma V^T$, how does one know that V actually spans the row space $C(A^T)$ of A and U the column space $C(A)$?

In the SVD of $A = U \Sigma V^T$, how does one know that V and U actually span the column and row space of A (respectively for each one)? I do know how to find such a U and V and $\Sigma$ by just ...
1
vote
1answer
18 views

Computing right null vector with smallest eigenvalue

I know that the null space can be calculated from the equation $Ax=0$ by computing $svd(A).$ If we compute svd, $svd(A)=USV,$ $S$ are the singular values, $V$ are the eigenvectors of $A^TA$ and U are ...
1
vote
1answer
13 views

A and P*A*Q have same singular values being P and Q orthonormal matrixes:

Let P and Q be two orthogonal matrices such that it makes sense to calculate PAQ. Show that A and PAQ have the same singular values. So far, I've come to the fact that the SVD of an orthogonal matrix ...
3
votes
1answer
27 views

How does additive noise change the SVD

For matrix $M$ with SVD $M=U\Sigma V^*$ and random matrix $A$, what is the SVD of $M+A$? That is, how will $A$ change the singular values and vectors of $M$? Let's even say that the entries of $A$ ...
0
votes
0answers
23 views

Left and right Null vector

Can somebody please explain what is the meaning of left null vector and right null vector? I know that null space of $Ax=0$ (where $x$ is the null vector) can be found out using row echelon form or by ...
0
votes
3answers
62 views

$\text{det}(A+E) = 0 \implies \Vert E\Vert_{2} \geq \sigma_n$?

Suppose $A,E$ are $n\times n$ matrices and $A$ has singular values $\sigma_1\geq \sigma_2 \geq \cdots \geq \sigma_n >0$. Please help me to prove that $\Vert E \Vert_2 \geq \sigma_n$ if $A+E$ is ...
0
votes
0answers
16 views

Matching degrees of freedom of SVD and original matrix

Every real $M\times N$ matrix has at least one singular value decomposition where $U$ is $M\times M$, $V$ is $N\times N$, and $S$ is a $M\times N$ diagonal matrix with at most $\min(M,N)$ non-zero ...
0
votes
1answer
24 views

uniqueness of svd decomposition and its role in statistical analysis

let us consider following model according to following link http://www4.ncsu.edu/~ipsen/REU09/chapter4.pdf it says that : The singular values are unique, but the singular vector matrices are ...
0
votes
0answers
103 views

Singular value decomposition of a matrix multiplied with a semi-unitary matrix

Say I have a $m \times n$ matrix A and I do a singular value decomposition (SVD), $\quad A = U \Sigma V $ where $U$ is $m \times m$ unitary, $\Sigma$ is $m \times n$ with non-zero elements only on ...
0
votes
2answers
44 views

Does the following type of “SVD” exist?

SVD of $A$ gives $U$, $\Sigma$ and $V$ such that $A = U \Sigma V$. I am interested in a different problem. Given an $A$ and $\Sigma_1,\ldots,\Sigma_{n-1}$ diagonal matrices, such that we know that ...
3
votes
0answers
49 views

Tikhonov regularization vs truncated SVD

To find $\mathbf{x}$ such that $$A\mathbf{x}=\mathbf{b}$$ we can use least squares when the problem is not well posed. Further, we can use Tikhonov regularization when $A$ is ill-conditioned. In ...
2
votes
0answers
32 views

If we have SVD of a submatrix, is that useful for SVD of full matrix?

Say we have a matrix that has a singular value decomposition $M_1=U_1\Sigma_1V_1$, and we have another matrix $M_2$, which has the same number of rows. Can we say anything about the SVD of their ...
0
votes
1answer
26 views

Why the largest singular value of a megic matrix is its magic constant?

A magic matrix is a square matrix such that the sums of the elements of each row, each column and diagonal equal to a same number, the magic constant. As reported ...
2
votes
1answer
34 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
10 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
17 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
43 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
51 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
47 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
36 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
20 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
38 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
18 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
36 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
104 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
25 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
34 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
33 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
66 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
21 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
24 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
44 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
56 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
30 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
33 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
37 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
18 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
26 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
30 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
29 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
18 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
10 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
68 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
37 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
32 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
27 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 ...