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.

Filter by
Sorted by
Tagged with
0
votes
0answers
14 views

Does Sharing same singular vectors lead to some properties?

What properties do a pair of matrices have if they share same ordered right singular vectors? Or for a specical case the matrices are square, what properties do they have if their ordered eigen-...
5
votes
2answers
196 views

Gradient of $A \mapsto \sigma_i (A)$

Let $ A $ be an $m \times n$ matrix of rank $ k \le \min(m,n) $. Then we decompose $ A = USV^T $, where: $U$ is $m \times k$ is a semi-orthogonal matrix. $S$ is $k \times k$ diagonal matrix , of ...
3
votes
1answer
73 views

Stability of eigenvalues/singular values on altering the matrix

From Strang's Introduction to Linear Algebra (p375), there is a paragraph on the instability of eigenvalues in relation to the stability of singular values when $A$ is altered slightly. Moreover, it ...
0
votes
1answer
15 views

Simple estimator for biggest singular value

I was reading a paper where they used $\sqrt{\|A\|_\infty \|A\|_1 }$ as an estimate of the biggest singular value $\sigma_{max}(A)$ of a non-singular matrix $A \in \mathbb{R}^{n \times n}$. However, ...
1
vote
0answers
74 views

SVD in multilinear algebra

I'm completely new in this field of algebra and I would like to better understand the extension of the SVD in the multilinear case. In particular, I'm interested in understanding what the so-called "...
0
votes
0answers
29 views

SVD uniquely determined for a matrix

I'm trying to prove the following: Given a matrix, the singular values of that matrix are uniquely determined. If that matrix is square, and the singular values are distinct, the left and right ...
0
votes
1answer
76 views

How does the outer product work in matrix approximation using SVD?

SVD factorizes a matrix $A \in \mathbb{R}^{mxn}$ such that $A = U\Sigma V^T$, quoting from Deisenroth et al: Instead of doing the full SVD factorization, we will now investigate how the SVD allows ...
1
vote
0answers
39 views

Every matrix is diagonal

Using SVD, prove the following: Let $A \in \mathbb{R}^{m\times n}$ and $T(x)=Ax$, a linear transformation for which A is the representative matrix in the canonical coordinates. The existence of SVD ...
1
vote
0answers
35 views

Advantage of singular value decomposition of $X$ over eigenvalue analysis of $XX^T$

Suppose I have a real $n \times N$ matrix $X$ where $n$ is a small number like $<20$ and $N$ is a big number like $>100000$. For example $X$ might be a time series of $n$ measurment quantities ...
2
votes
0answers
39 views

Functional calculus for compact operators using singular value decomposition and regularisation filters

Notation: Let $X,Y$ be Hilbert spaces, $\mathcal{K}(X,Y)$ denote linear compact operators $X \to Y$, $L(X)$ the linear continuous operators $X \to X$, $\mathcal{N}(K)$ is the kernel of $K$, $\mathcal{...
0
votes
1answer
37 views

Iterated matrix multiplication when matrix is defective

Is there any easy method of computing $$A^{n}x$$ when $A$ is not diagonalizeable? That is, we cannot find $P,D$ such that $A=PDP^{-1}$. If there is no easy way of computing, can we say anything ...
2
votes
0answers
62 views

If the sum of the singular values of $df$ is constant $1$, must $f$ have a constant Jacobian?

Let $D \subseteq \mathbb{R}^2$ be the closed unit disk. Let $f:D \to \mathbb{R}^2$ be a smooth map, such that the singular values of $df$, $\sigma_1(df),\sigma_2(df)$ satisfy $$ \sigma_1(df)+\...
1
vote
1answer
26 views

Help with understanding a paper: Solving Fredholm Integrals of the first kind with Tensor Product structure in 2 and 2.5 dimensions

I would very much like to implement the algorithm present in this paper, however, I think I am way over my head. My interest is to implement the algorithm contained in it to get 2D maps of two ...
2
votes
0answers
31 views

Possible application of Laws of Large Numbers and Central Limit Theorem to SVD of dual covariance matrices

Let $X:=[x_1\dots x_n] \in \mathbb{R}^{d\times n}, x_i \in \mathbb{R}^{d\times 1}$ be a data matrix where $x_i \in \mathbb{R}^{d\times 1}$ are iid random vectors with mean $\mu$ and covariance $\...
1
vote
1answer
59 views

What is the best way to solve square integer matrices of 8-bit?

Assume that we want to solve this linear system: $$A^TA x = b$$ Matrix $A$ is square and random integer of 8-bit, e.g numbers between 0 and 255. Vector $b$ is known as well and also integer of 8-bit....
0
votes
0answers
13 views

Can I use Singular value Decomposition to do Fisherfaces?

Just as Eigenfaces. It's possible to do eigenfaces with Singular Value Decomposition. But is it the same for Fisherfaces? Doing fisherfaces with Singular Value Decomposition?
1
vote
0answers
40 views

Is this matrix field contractible inside the space of non-conformal matrices?

Set $\mathcal{NC}:=\{ A \in M_2(\mathbb{R}) \, | \det A \ge 0 \, \,\text{ and } \, A \text{ is not conformal} \,\}$, where by a non-conformal matrix, I refer to a matrix whose singular values are ...
0
votes
0answers
19 views

Show that $A^+=V_rS^{-1}U_r^T$

Let $A^+:=V\sum U^T$ and $A:=USV^T$ Show that $$A^+=V_rS^{-1}U_r^T$$ Where the columns of $V_r, U_r$ are the first $r$ singular vectors. My try: $A^+=\sum_{i=1}^r \frac{1}{\sigma_i}v_iu_i^T,\...
0
votes
0answers
18 views

Distribution of Gaussian projected on orthogonal direction

Let $X$ be a matrix-valued random variable with each entries are independent Gaussian. Let $\mathbf{u}$ be the first left singular vector of $X$. What is the distribution if we project $X$ on $\mathbf{...
0
votes
0answers
25 views

How to I compute Jacobi rotation for Singular Value Decomposition

I want to compute $A \in \Re^{mxn}$ so that $$A = U\Sigma V^T$$ Singular Value Decomposition in other words. There is lot of algorithms to do that. One is One-Sided Jacobi that can handle general ...
0
votes
1answer
34 views

Is there any way to prove stability for a discrete state space model by using Singular Value Decomposition?

Assume that we have this state model representation $$x(k+1) = Ax(k) + Bu(k)$$ To check stability of this model, we need to check the eigenvalues. In discrete mode, $\lambda$ will result both a ...
1
vote
0answers
34 views

Computational stability of calculating EV-decomposition vs SVD with the Läuchli matrix.

In a lecture on PCA the lecturer claimed that (and I've also found it in this answer but it had 17 comments and I didn't want to make them 17 more) instead of calculating the eigenvalue ...
0
votes
1answer
60 views

prove that sum of quadric form less than the frobenius norm

I was reading this famous paper (Clustering Large Graphs via the Singular Value DecompositionClustering Large Graphs via the Singular Value Decomposition) and find difficulty in understanding a step ...
0
votes
0answers
51 views

How do I find $\min_{z,z'~s.t. \|z\|=\|z'\|=1} \|A^Tz-B^Tz'\|$ where $A$ and $B$ have length 1 columns?

For $n\times m$ matrices $A$ and $B$ ($n>>m$), which also have length 1 columns $rank(A)=rank(B)=m$ $n$-dimensional vector $z$ The $l2$ norm for vectors and Frobenius norm for matrices I need ...
0
votes
1answer
59 views

how to make singular matrix becomes non-singular using pseudo inverse?

I have a problem of making a slight change on a singular matrix to make that matrix become full rank. My thought is to add some numbers to the diagonal to recover some 0 eigenvalues, so that the ...
0
votes
0answers
44 views

Correct way to represent eigenvalue spectra from varying eigendecomposition techniques (SVD, etc)?

I've noticed a discrepancy between the eigenvalue spectrum retrieved via different eigendecomposition methods. As an example, here's code showing PCA of the Iris dataset, whereby both (Fig A) ...
0
votes
1answer
33 views

Smallest singular value of product of 2 random matrices

Let $A\in\mathbb{R}^{n\times m}$ and $B\in\mathbb{R}^{m\times k}$ be two random matrices (each element is drawn iid from a normal distribution). Also $n<m<k$. Let $\sigma_{min}(A)$ be the ...
0
votes
1answer
32 views

Recovering the singular values of a matrix from partial information

This is a self-answered question, which I post since it wasn't immediately trivial for me. Of course, I would be happy to find easier or quicker solutions. Let $A:\mathbb{R}^2 \to \mathbb{R}^2$ be an ...
0
votes
1answer
69 views

Spectral Norm of block diagonal matrix

I was reading a proof utilizing some property of the spectral norm, but fail to understand some steps. The part of the proof goes like, \begin{equation*} \begin{split} \left\|\left[\begin{array}{cc}{\...
0
votes
0answers
7 views

Row space, column space and SVD

In the lecture on the SVD by Strang around 3:14 he talks about vectors in the row space such as v being mapped into the column space by transformations like Av. I understand why Av is in the column ...
0
votes
1answer
19 views

Confusion over formula for finding V during SVD

I understand any rectangular mxn matrix A is a factorization of the form: $$ A = U\Sigma V^{\top}$$ where U and V are orthogonal matrices and $\Sigma$ is diagonal. Say, I have found U and $\Sigma$ and ...
0
votes
0answers
22 views

Confusion about finding V given U and S during SVD

I understand any rectangular mxn matrix A is a factorization of the form: $$ A = U\Sigma V^{\top}$$ where U and V are orthogonal matrices and $\Sigma$ is diagonal. Say, I have found U and $\Sigma$ and ...
0
votes
1answer
20 views

When the singular values of a convex sum are preserved?

Let $A,B$ be two real $2 \times 2$ matrices with identical singular values $0<\sigma_1<\sigma_2$ and with a positive determinant (which is $\sigma_1 \sigma_2$). Is there a known ...
0
votes
1answer
29 views

Can the existence of a QR decomposition PROVE the existence of a SVD decomposition?

I have written a demonstration of the existence of SVD based on the existence of QR, but I am not sure if it's correct or if I am missing something. QR and L'Q' We know that B = QR , with Q ...
1
vote
1answer
19 views

Problem with the SVD of a large complex matrix

I have a fat and complex matrix $H$ on which I would like to perform an SVD. Because of its size, I'm forced to using a trick based on the computation of a covariance matrix. $H H^\dagger = U S V^\...
0
votes
0answers
24 views

Find transformation matrix based on other variable

I want to find a mapping from a space X that contains 100 dimensions to a space Y that contains 100 dimensions. The problem here is that i want that the mapping will consider big information just ...
0
votes
1answer
52 views

How to perform SVD of a matrix with symbols?

Let's assume that we have the following matrix: \begin{pmatrix} 0& 2& 3& 4a& 5a\\ 6& 7& 1& 8a& 9a\end{pmatrix} Can we perform SVD on this matrix and get the output in ...
1
vote
1answer
43 views

SVD to transform regularization problem

Can anyone explain this transformation to me : $$ ||Ax-b||_2^2 + \delta ||x||_2^2 : A \in R^{m,n}, b \in R^m \rightarrow \\ \tilde{x} = (V^Tx, V_2^Tx), \tilde{b} = (U^Tb, U_2^Tb)\\ V_2 \in R^{n x (n-...
0
votes
1answer
46 views

regularization of $\hat{x} = \arg\min_x|Ax - b|^2 + \lambda|x|^2$ using SVD of $A$

Suppose that the following energy is provided $$ \hat{x} = \arg\min_x|Ax - b|^2 + \lambda|x|^2 $$ with a given matrix $A$, a vector $b$ and regularization paramter $\lambda$. Analyze how the solution ...
12
votes
0answers
210 views

Is there a volume-preserving diffeomorphism of the disk with prescribed singular values?

This question has now been cross-posted at mathoverflow. While working on a variational problem, I have reached to the following question. Let $0<\sigma_1<\sigma_2$ satisfy $\sigma_1\sigma_2=...
0
votes
1answer
42 views

summation notation of singular value decomposition

In Boyd's optimization book, he explains the singular value decomposition of a matrix of rank $r$ can be written like this: $$A \in R^{mxn} = U\text{diag}(\sigma)V^T = \sum_{i=1}^r \sigma_i u_i v_i^T\\...
0
votes
0answers
37 views

Equality for matrix norms [duplicate]

Let $A \in \mathbb{C}^{n,n}$ invertible and we use a norm $\|\cdot\|$. Then it holds that $$\|A^{-1}\|= \max_{||x||=1} \|A^{-1}x\|=\max_{\|x\|=1}\frac{1}{\|Ax\|}. $$ I can see, why that holds for ...
0
votes
1answer
40 views

Piece of advice for a SVD

For my machine learning course, I have to find the SVD of the following matrix $\begin{pmatrix} 2 & 1 \\ 2 & 1 \\ \frac{2}{5} & \frac{11}{5} \\ \frac{2}{5} & \frac{11}{5} \\ \end{...
2
votes
1answer
66 views

Special Eigenvalues of a Matrix in Strang p.368

This question arises from Strang's Linear Algebra p.368. It concerns the matrix $$A = \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & ...
1
vote
1answer
46 views

Diagonal matrices are not necessarily square?

I have seen it implied, in the context of singular value decomposition, that diagonal matrices are not necessarily square. Is this true? How can it be true? Can someone please explain this in more ...
0
votes
0answers
45 views

Pseudoinverse of $\mathbf{A} \in \mathbb{R}^{m \times n}$ multiplied by $\mathbf{A}$

The Moore-Penrose pseudoinverse of a matrix $\mathbf{A} \in \mathbb{R}^{m \times n}$ is \begin{equation} \mathbf{A}^+ = (\mathbf{A}^T\mathbf{A})^{-1}\mathbf{A}^T. \end{equation} Now, using this we ...
0
votes
1answer
27 views

Independence of singular values

If $X$ is a matrix valued random variables, are its singular values independent as random variables?
0
votes
0answers
23 views

Classification with SVD - Eigenfaces

I just saw a play list on Youtube where Professor Brunton teach how SVD works and its applications. He mention that with SVD, classification can be done. https://www.youtube.com/watch?v=gXbThCXjZFM I ...
0
votes
0answers
15 views

Calculation of Principal component variables

Consider the SVD decomposition of a data matrix $$X = UDV^T$$ where rows of $X$ are samples. In Elements of Statistical Learning it is said that Then $Z = UD$ is the matrix of principal ...
0
votes
1answer
85 views

Optimization problem which requires differentiation of Kabsch algorithm

I want to solve some complicated optimization problem which involves differentiation through Kabsch algorithm. So i need to obtain jacobian vector product and vector jacobian product of Kabsch ...

1 2
3
4 5
21