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
0answers
40 views

Can I clamp singular values of $3\times3$ matrix without effectively computing SVD?

I have a $3\times3$ matrix $A$, and compute its SVD $U \Sigma V^\star = A$. I clamp the singular values in $\Sigma$ to some small range (e.g. $[0.5, 1.5]$ ) and reconstruct matrix $\widetilde{A}=U ...
0
votes
0answers
10 views

the SVD (singular value decomposition) of an augmented matrix

Suppose we have a $4\times 3$ dimensional matrix $A$. Denote the SVD of $A$ by $USV^T$, where $U\in R^{4\times 3}, S\in R^{3\times 3}, V\in R^{3\times 3}$. Then, we construct a new matrix $B=[A;0]\in ...
1
vote
1answer
25 views

Using SVD to approximate matrix-vector multiplication?

Given some matrix A, is it possible to use Singular Value Decomposition to approximate Ax for some vector x within some error bound? According to Efficient low rank matrix-vector multiplication, it ...
0
votes
0answers
13 views

advantage and disadvantage of using SVD to solve least square problems

I usually just use $AA^T$ or QR decomposition of A to solve least square problems. But SVD seems to be the popular way to solve the problem. what is the advantage and disadvantage of SVD? thanks!
2
votes
0answers
36 views

Largest Singular Value / Singular Value

I was wondering, what if the eigenvalues of a matrix A are all negative. So does that simply mean there is no singular value for this particular matrix?, hence I can't calculate the conditional number ...
0
votes
0answers
34 views

Fundamental Theorem of Linear Algebra and SVD

Here is the pdf I am using to learn SVD Decomposition. On page $2$, it states how the eigenvectors from non-zero eigenvalues of $A^{T}A$ are in the rowspace of $A$ and how eigenvectors from non-zero ...
0
votes
0answers
21 views

a simple question about singular value decomposition(SVD)

This question have made my mind busy for a long time. Assume $X=X_1+X_2$ where $X$ is a $m \times n$ matrix and $X_1 \perp X_2$. Then how can I connect $SVD_X$ to $SVD_{X_1}$ and $SVD_{X_2}$. As I ...
1
vote
0answers
9 views

Simplify product (Hadamar / Kronecker) of matrices for singular value decomposition in Demmler-Reinsch Orthogonalization

Let $1_b = (1, \ldots, 1)^\prime$ be the $b$-dimensional vector of ones, let $\circ$ and $\otimes$ denote the Hadamar respectively the Kronecker product and let $n = m \cdot c$, where $n,m,c \in ...
0
votes
0answers
44 views

Finding $Q$ for any $A$ s.t. $QAQ^\top = I$

Given an invertible and PSD matrix $A$, I am looking to find $Q$ such that: $$ QAQ^\top = I $$ What is a/the right/efficient way to do this? Here is what I did: SVD gives $$ A \approx U S V^\top ...
0
votes
0answers
27 views

How close is Cartesian product of unit orthogonal bases of SVD to identity matrix?

If I have N unit orthogonal vectors of length N $\phi_{i,N\times 1}$ obtained from SVD of a $N\times M$ matrix $U$ : $$ U_{N\times M} = \sum_i^N \sigma_i\phi_{i,N\times 1}\times\psi_{i,1\times M}\\ ...
1
vote
0answers
13 views

SVD of a block partitioned matrix

Given a block partitioned matrix $\boldsymbol{A}$ $$ \boldsymbol{A} = \begin{bmatrix} \boldsymbol{A}_{1,1} & \boldsymbol{A}_{1,2} & \cdots \\ \boldsymbol{A}_{2,1} & ...
0
votes
0answers
10 views

General formula for $\hat{b}$ (least squares) using SVD and pseudoinverse

In a situation with an SVD for A given by $A=U\Sigma V^T$ I know about the relation $ x=(A'A)^{-1}A'b=A^+b $ Given b and matrix A, which general formula can one use to find $\hat{b}$?
0
votes
1answer
19 views

Solve Ax=0 using Single Value Decomposition

Trying to solve Ax=o when $A=\begin{bmatrix}2&1&-1\\1&2&1\\ \end{bmatrix}$ using single value decomposition. I have the s,v,u and was thinking that x was as simple as $x=s*s^t$ but ...
0
votes
0answers
26 views

Finding the largest singular value “easily”

Im only interested in finding the largest singular value. I don't need the singular vectors. Is there a way to do so without performing full SVD? Is there an analytical expression? If not, is ...
1
vote
0answers
15 views

SVD: How to find the columnvector of U corresponding to a singular value equal to zero

The question is if you have a situation where one of the singular values is equal to 0 in a singular value decomposition of a matrix, how to do you procede to find the column vector of U corresponding ...
0
votes
0answers
17 views

SVD and least squares solution: orthogonal projection $ \vec{\widehat b} $ of $ \vec{b} $ onto $Col(A^T)$

Given the following: $ A = \left(\begin{array}{rrr} -2 & 3 & 2 \\ 2 & 2 & 3 \end{array}\right).$ A has the SVD: $A = USV^T$ $ b = \left(\begin{array}{rrr} -6\\ 1 \\ 4 ...
0
votes
1answer
18 views

Simplifying the inverse of the sum of 2 matrices

I would like to simplify the following inverse computation : $$(D+A)^{−1}$$ where $A=UΣU^T$ (eigenvalue decomposition). And D is a diagonal matrix such that $D = \lambda \boldsymbol{I}$ I know the ...
0
votes
1answer
23 views

inverse of sum of diagonal matrix and eigendecomposition

I would like to simplify the following inverse computation : $$(D + A)^{-1}$$ where $A=U\Sigma U^T$ (eigenvalue decomposition). And D is a diagonal matrix I know the inverse of A is ...
2
votes
0answers
57 views

Restoring Bidiagonality to a Matrix in SVD Algorithms

Good Afternoon, I am implementing the Golub-Reinsch SVD algorithm and am having difficulty with a boundary case Given a bidiagonal matrix of the form: $$ \begin{bmatrix} b11 & ...
1
vote
1answer
46 views

Find unit singular vectors for two known singular values.

$$A=\begin{pmatrix}-3 & -1 \\ -3& 1\end{pmatrix}$$ Find the singular values of $σ_1$ and $σ_2$, Find unit vectors $v_1$ and $v_2$ such that $||Av_1|| =σ_1$ and $||Av_2|| =σ_2$ I figured ...
2
votes
1answer
49 views

SVD and transpose of a skinny matrix

Show: If $\mathbf{A}\in\mathbb{R}^{M\times N}$ with $M\geq N$, then there exists a matrix $\mathbf{G}$ with orthonormal rows so that $\mathbf{A}^T=\mathbf{G}\mathbf{A}\mathbf{G}$. I'm pretty lost on ...
0
votes
0answers
16 views

how to solve this low rank approximation problem

$$L = \sum_{i=1}^n (N_i-C_i)^T\Sigma^{-1}(N_i-C_i) =Tr((N-C)\Sigma^{-1}(N-C)^T)$$ $$=\sum_{i=1}^n \sum_{j=1}^d (N_{ij}-C_{ij})^2\sigma_j^{-2}$$ $$rank(C)=k<rank(N)$$ Basically i need to find the ...
1
vote
2answers
27 views

Why $||U\Sigma V^\intercal - X_k||_F = ||\Sigma - U^\intercal X_k V ||_F$

Here $U\Sigma V^\intercal $ is the svd decomposition of M.I know that U and V are othogonal but i dont know which property it is using. This in reference to the post Proof of Eckart-Young-Mirsky ...
0
votes
0answers
23 views

minimization of weighted frobenius norm for pca

So my problem is i like to derive pca solution as the maximum likelihood estimate for the true data.So basically i am assuming that my measured data has two component one is low rank component and ...
1
vote
1answer
84 views

vector as linear combination of other vectors with one more perpendicular vector

I am reading about Singular Value Decomposition (SVD) from book SVD CSTheory Infoage. At page 6, the chapter says: A matrix $A$ can be described fully by how it transforms the vectors $v_i$. Every ...
1
vote
2answers
17 views

Compactness argument in SVD existence proof

The classical proof of the existence of the SVD factorization by Trefethen and Bau reports Set $\sigma_1 = \mid\mid A \mid\mid_2$. By a compactness argument, there must be a vector $v_1 \in ...
0
votes
0answers
25 views

Singular value decomposition positive components

I am using Singular Value Decomposition (SVD) applied to Singular Spectrum Analysis (SSA) of a timeseries. ...
0
votes
2answers
25 views

If rows of matrix M are switched, do the singular values of M change?

I have looked for a source on the relationship between elementary matrix row operations and singular values, but I can't find a good, compact set of information. I'm really only interested in ...
2
votes
1answer
29 views

Efficiently compute the eigenvectors of the Laplacian of a symmetric positive matrix

I am working with a matrix A relatively large (200k x 200k), and I want to compute the eigenvectors of the Laplacian: $L = D - A^2$, where $A$ is symmetric. I don't need all eigenvectors, just a few ...
0
votes
2answers
124 views

Changes in singular values of matrix when rows are added

I know that if a column is added to a matrix then the matrix largest signular value increases and the smallest singular value decreases. That is: Given matrix $A \in R^{m \text{x} n}$, $m>n$, and ...
0
votes
0answers
19 views

Significance of minimum singular value

I came across this statement from Wikipedia on singular value decomposition. A total least squares problem refers to determining the vector $x$ which minimizes the 2-norm of a vector $Ax$ under ...
1
vote
0answers
47 views

Singular value decomposition for matrices that are not square?

I understand that the Singular Value Decomposition is defined as SVD = $U\Sigma V^T$ , but I am slightly confused about the calculations when the matrix is not square. For example, I have the matrix: ...
0
votes
1answer
32 views

how to calculate variance on SVD parameter estimation?

How do i Calculate the variance of a estimated parameter by SVD? I know that there is an uncertainty on the dataset, but how can that be used to calculate variance of an parameter?
0
votes
1answer
24 views

When PSD, singular value is equal to eigenvalue

It is known that If a matrix is PSD (symmetric), then its eigenvalues are equal to its singular value. How to prove it? Hope for a hint. thanks,
0
votes
0answers
30 views

Known the SVD of A, given a diagonal matrix C. How to find $SVD(CA)$?

I have an $M$x$N$ matrix $A$, given a diagonal matrix $C$ is$M$x$M $. What relationship can I find between the SVD decomposition of $A$ and $CA$? I suppose $C$ has no zero entry in the diagonal, so ...
0
votes
0answers
52 views

Generalized SVD and weighted SVD

I've the following question: How should I select the $A$,$B$ matrices in the generalized singular value decomposition (GSVD) such that it solves the weighted version of the generalized singular value ...
1
vote
1answer
38 views

If an upper bidiagonal matrix has a repeated singular value, it must have a zero on its diagonal or superdiagonal

I have a question that mentioned in the book "Matrix Computations" by Golub and van Loan. "Show that if $A\in \mathbb{R}^{n\times n}$ is an upper bidiagonal matrix having a repeated singular value, ...
1
vote
0answers
32 views

Is this matrix with SVD diagonalizable

Let $X=U\Sigma V^T$ is an (economical) SVD decompoisition of a square $n \times n$ stochastic matrix $X$, where $U$ and $V$ are two $n \times r$ matrices, and $\Sigma$ is a $r \times r$ matrix. Now ...
1
vote
0answers
59 views

Do I have to be a mathematician to understand the following papers?

I come from a CS&Machine Learning discipline. I have been looking to understand the core idea of Non-Negative Matrix Factorization. While most of the ML based work is understandable, mostly the ...
0
votes
0answers
47 views

Multiplying Matrices (using SVD)

I'm an online course and it has sections on it regarding SVD. I understand the concept, however, the maths eludes me. For example, if you view this: you can see the Matrix q * Matrix V = the output ...
1
vote
0answers
19 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
24 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
20 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
39 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
51 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
27 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
25 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
38 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
1answer
41 views

Left and right null vectors

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
64 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 ...