# Tagged Questions

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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### Singular value decomposition of a matrix and its transpose

Consider a matrix A. The SVD of A will be : A =USV'. The SVD of A' will be: A' = U1*S1*V1' (say). Since, A' = V*S'*U', therefore the values of U1, S1, and V1 should be as follow: ...
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I am studing the SSA (Singular Spectrum Analysis) algorithm and after reading some papers about SSA, the ideia was not consolidated. I found a Matlab SSA algorithm here and I was trying to implement ...
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### Decompose a matrix into diagonal term and low-rank approximation

For a matrix $A$ the Singular Values Decomposition allows getting the closest low-rank approximation $$A_K=\sum_i^K\sigma_i \vec{v}_i \vec{u}_i^T$$ so that $\|A-A_k\|_F$ is minimal. I'd like to do ...
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### Relation between row-space and column-space vectors

Let $A$ be any $n$ by $m$ matrix. $V$ is an orthonormal vector in column-space of $A$. $U$ is an orthonormal vector in row-space of $A$. Now, why is the following relation True? $$AV=U\Sigma$$ , ...
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### Singular value decomposition: does the choice of eigenvectors matter?

I'm trying to calculate the SVD-decomposition of a certain matrix, i.e. $A = U \Sigma V^T$. My solution doesn't yield $A$ again; I just can't get the signs correct. I'm wondering if this is just a ...
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### SVD parallel for distributed memory

I want to build a code executing a parallel SVD via the one sided Jacobi method within a distributed memory approach (MPI). The only possibility therefore seems to be a block method. Sofar it looks ...
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### How to solve the following quadratic matrix equation?

Solve the following matrix equation in $D$ $$A=D^{T}(DVD^{T}+\alpha\lambda_{\max}(D^{T}D)I)^{-1}D$$ where $I$ is the identity matrix, $A$ and $V$ are known matrices, $\alpha$ is a known ...
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### Maximizing the trace of a complex matrix

Let's say I have the following maximization problem: $max_U{tr(AU)}$ where $A\in\mathbb{C}$ and $UU^\dagger=1$ I know that for $A\in\mathbb{R}$ and $UU^T=1$ the solution is: $U=XZ^T$ where $X$ ...
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### Singular values of the product of two (semi)orthogonal matrices

Let's assume we are given two (semi)orthogonal matrices $U_1$ and $U_2$ with dimension $m\times n$ such that $m>n$. The (semi)orthogonality means $$U_1^TU_1 = I_n$$ and $$U_2^TU_2 = I_n$$ but ...
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### Proof that left singular vectors in SVD are orthogonal, and proof of low-rank approximation

I've been reading about SVDs and have a couple questions. First, let $A\in \mathbb{R}^{n\times d}$ be a matrix with SVD $U\Sigma V^T$. Let $\sigma_i$ denote the $i$'th singular value, with ...
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### Compute the SVD of $AB$ from the SVDs of $A$ and $B$

Knowing the SVD of $\mathbb{C}^{m*n} \ni A = U_A\Sigma_AV_A$ and $\mathbb{C}^{n*s} \ni B = U_B\Sigma_BV_B$, is there any way to speed up the calculation of the SVD of $AB = U_{AB}\Sigma_{AB}V_{AB}$? ...
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### Given A $m\times n$, $\{u_1,…,u_n\}$ ON basis of $R^n$, prove eigenvalues aren't negative [duplicate]

Given $A$ $(m\times n)$, $\{u_1,...,u_n\}$ ON(orthonormal) basis of $R^n$ which are eigenvectors of $A^TA$ with $\lambda_1 , ... , \lambda_n$ eigenvalues accordingly. Prove: Eigenvalues are not ...
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### Minimizing $\|Ax\|_2$ subject to $\|x\|_2 = 1$

I have a Matlab program to estimate a vector $x$ from noisy measurements. I use the singular value decomposition (SVD) to solve the linear equation $Ax=0$ (where the number of equations is greater ...
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### Find SVD of $A$

How do I find the singular values? They somehow show that $\lambda_1 = 27, \lambda_2 = 6, \lambda_3 = 0$. I still can't see how they found them with the equations I made in my solution.
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### invertible subspaces and SVD

Suppose X is a $m\times d$ real matrix. I am trying to prove the following claim: $XX^T$ is invertible iff $span\{x_1,x_2,...,x_m\}=\mathbb{R}^d$ such that $x_1, x_2, ..., x_m$ are X's columns. ...
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### low rank approximations and diagonalization

I would like to discuss or hear an opinion about the following. Given is the (hermitian) $n\times n$ matrix $A = D+M V M^{\dagger}$ with D diagonal. I would like to calculate the eigenvalues (and ...
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### Independent components with a known relationship between sources

I'm performing an experiment, and I expect that my results of the form: $$\mathbf{M}=\mathbf{ce}, \mathbf{M}\in\Bbb{R^{m\times n}},\mathbf{c}\in\Bbb{R^{m\times 2}},\mathbf{e}\in\Bbb{R^{2\times n}}$$...
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### constrain general solution of ill-posed linear system to $\Re_+$?

I have a solution space of an under-determined linear system Ax = b with n x m matrix A: $$x= x0 + V2 * c (1)$$ with [U, S, V] = svd(A); V2 = V(:,r+1:end); $x0 = A^+ b;$ r = rank(A); I ...
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### square eigenvectors for Singular Value Decomposition?

This is from my textbook What I don't understand is, $V$ and $U$ are already square, why the textbook says "if we want to make them square"?
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### Rank of product of SVD vectors

Given a matrix $M$, we can compute its singular value decomposition $M=U\Sigma V^*$ where $^*$ is the complex conjugate transpose. $U$ and $V$ are unitary, so $UU^*=I$, $VV^*=I$. Let's take the $i$-th ...
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### Perturbations on SVD left singular matrix

Given a symmetric matrix $A\in \mathbb{R}^{n\times n}$, with all the entries greater than zero $A_{i,j}>0$ with rank $k<n$, we can calculate its SVD decomposition: $$A = USU'$$ Assuming now ...
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### Jacobi SVD algorithm implementation

Is this implementation of Jacobi SVD algorithm according to the standard algorithm? Please verify. Is this Hestenes Jacobi method? I have seen pseudo code of Jacobi algorithm like here which ...
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### null space from SVD

It is said that a matrix's null space can be derived from QR or SVD. I tried an example: $$A= \begin{bmatrix} 1&3\\ 1&2\\ 1&-1\\ 2&1\\ \end{bmatrix}$$ I'm convinced that QR (more ...
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### Possible quick solution of SVD of covariance matrix of Xv, where v may change, while X does not.

I am current trying to work on one algorithm, that for Iteration $t$, I need to calculate the SVD of $(X\text{diag}(v^t))^T(X\text{diag}(v^t))$. This could be very slow if $X$ is of high dimension. ...
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### SVD and homogeneous equation

Suppose a $m \times n$ matrix $A$, and column vector $h$. ($A$'s rank is equal or smaller then $n$(=$h$'s length).) If, $$Ah=0$$ then $h$ can be the last column of $V$ where $A = UDV^T$. ($UDV^T$...
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### Compute a similarity matrix of documents from SVD (LSI)

I'm computing a SVD from a Matrix $m$ (columns = Documents ($D$) and rows = Terms ($T$) and truncate this matrix to lower the dimension of $m$ to $k$. From my resulting matrix $A = U \Sigma V^T$ I ...
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### SVD - Decomposed Matrix Sizes

I had a question about SVD. Specifically about the size of matrices $U$, $\Sigma$ and $V$ decomposed from the $m\times n$ matrix $X$ using the formula $$X = U \Sigma V^T$$ Most of the the tutorial ...
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### Singular value decomposition of the real part of a complex matrix

An $N\times N$ complex matrix $K$, with SVD of $K = U\Sigma V^H$. I only have access to the real part of the elements of $K$, i.e. $K_R$. If I did SVD of $K_R$ to get $U_R \Sigma_R V_R^H$, my ...
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### Most efficient method for computing Singular Value Decomposition of a triangular matrix?

There are several methods available for computing SVD of a general matrix. I am interested in knowing about the best approach which could be used for computing SVD of an upper triangular matrix. ...