# 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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### How do we know V in the SVD is the eigenvectors of M*M?

if we have any real matrix M nXm, the SVD (singular value decomposition) allows us to decompose it into $U{\Sigma}V^T$, where V is an orthogonal real matrix composed of the eigenvectors of $M^TM$. ...
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### Proper Generalized Decomposition

I have researched some topics regarding Proper Generalized Decomposition. All the text more or less state the what is the algorithms to conduct PGD as Assume that the solution is separable in x(space)...
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### Trying to understand why the largest singular value of $M_n^{-1}$ is $1.4286$ no matter what $n$ I choose

Let $n \ge 3$ be an odd number. Let $I_n$ be the $n$ dimensional identity matrix and let $A_n$ be the $n\times n$ matrix where every element is zero except the central element which is, say, $0.3$. ...
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### How to find the SVD when eigenvalue is $0$? [duplicate]

After calculating the eigenvalues, I get $1040400$ and $0$. Since one of them is $0$, how do I calculate (orthogonal) matrix $U$? $$u_i = \frac{1}{\sqrt{\lambda_i}} A v_i$$ I know there are a similar ...
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### Proof that A+A projects onto R(A) and AA+ projects onto N(A)

I have been able to understand the proof that shows both are projections by proving P^2 = P for both of them. I don't understand how these projections project onto R(A) and C(A) though. Proof that ...
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### $\| Ax \|= \|x\| \forall x \in \mathbb{R}^n$ if and only if all singular values of $A = 1$? [closed]

it is true? I think it is true but I don't know why
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### Condensed SVD decomposition of an outer product

Let $A = uv^{T} \in \mathbb{R}^{m \times n}$. Find the (condensed) SVD decomposition of $A$. Theorem (Condensed SVD decomposition) Let $A \in \mathbb{R}^{n \times m}$ be a non-zero matrix of rank $r$....
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### Subgradients of the trace norm for a singular value decomposition

I'm following CMU convex optimization course. I am doing homework 2 but I'm stuck at this question:$\DeclareMathOperator{\tr}{tr}$ For $f(X)=\lvert\lvert X \rvert\rvert_{\tr}$, show that the ...
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### Finding Orthonormal Basis using SVD and comparing it with Gram-Schmidt shows different result

I was trying to find the orthonormal basis for the column space of the following matrix "A" \begin{pmatrix} -1 & -1 & 2 & 3 \\ -1 & 1 & -3 & -4 \\ 2 & -2 & 5 ...
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### Find SVD of the given matrix

Find a decomposition $X=U \Sigma V^{T}$ of the matrix $X=\begin{bmatrix} 2 & 1 & 2\\ -2 & -1 & -2\\ 4 & 2 & 4\\ 2 & 1 & 2 \end{bmatrix}$ where $\Sigma$ is a ...
I have encountered a task, which I do not really know how to approach. I need to find a decomposition $$A = U\Sigma V^T$$ of the matrix: A = \begin{pmatrix} 2&1&2&\\ -2&-1&-2&... 1answer 18 views ### How to show that each rank one decomposition of SVD is exactly rank one? by using the fact that rank(AB) \leq min{rank(A), rank(B)}, I can only figure out that rank(u\cdotv^{T}) \leq 1. How to show that it is exactly rank one? Appreciate any help! 0answers 20 views ### Conditions for equality for the nearest rank-k matrix my proof for the best rank approximation, using the Frobenius norm, is as follows: \begin{equation} \begin{alignedat}{1} \vert\vert{\mathbf{A-B}}\vert\vert_F^2 &= <\mathbf{A - B}, \mathbf{... 0answers 10 views ### Multiplying an eigenvector by -1 while constructing the V matrix in SVD decomposition While performing SVD I found eigenvectors, which will allow me to write down my V matrix. However, I need to multiply the 3rd eigenvector by -1 because it will satisfy some condition that my task ... 0answers 16 views ### Swapping the orthonormalized eigenvectors in the V matrix while performing SVD decomposition. While performing an SVD decomposition, I found eigenvalues and eigenvectors, I orthonormalized them and they look something like this:\mathbf{eig_{1}}=\left(\begin{array}{c} a\\ b\\ c \end{array}\...
Suppose there is a square matrix $A$ and a positive semi-definite matrix $X\in\Re^n$, such that \begin{equation} \mathrm{trace}(AX)\leq0 \end{equation} Is there any ways I could do the rank one ...