# 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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### Proof Help : $A \in \mathbb{R}^{n \times n}, \sigma > 0$ if and only if the next matrix is singular.

I've been stuck with the next proof. I've tried to approach the first implication using that if $\sigma$ is singular value of A, then is solution of $P(\lambda) = \det(A^t A - \lambda I)$, but i don'...
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### SVD with rectangular matrix U

I have a matrix $A_{3\times2}$ , $U_{3\times2}$ , $V_{2\times2}$. I was asked to find the singular values, which I did with by multiplying $U'AV$. I got the $U'$ by following this rule: Since $U$ is ...
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### How is SVD computing unitary square matrices for rank-1 matrices (Matlab)

Consider matrix $\mathbf X=[\mathbf x ~\mathbf x] \in \mathbb R^{D \times 2}$. Of course, $\mathbf X$ has rank-1. Background: $\bullet$The full Singular Value Decomposition (SVD) of $\mathbf X$ is ...
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### If $(U,\Sigma,V)$ is a singular value decomposition of $A$, do the first $\text{rank}A$ columns of $V$ and $U$ form orthonormal bases?

Let $m,n\in\mathbb N$ $A\in\mathbb R^{m\times n}$ and $|A|:=\sqrt{A^TA}$ $r:=\operatorname{rank}A$ $\sigma_1>\cdots>\sigma_r>\sigma_{r+1}=\cdots=\sigma_n=0$ denote the singular values of $A$ ...
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### Solving for $X$ using the SVD of $QX$ when $Q$ is orthogonal

I inherited some code (see below), and I am not quite sure what it does. It is part of a factor analysis-type model that learns a latent variable $X \in \mathbb{R}^{N \times K}$ with $N > K$ that ...
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### If the singular value decomposition $U\Sigma V^T$ has rank $r$ and $Q$ is semi-orthogonal, then $Q\Sigma V^T$ has rank $r$ as well

Let $m,n\in\mathbb N$ and $A\in\mathbb R^{m\times n}$ with reduced singular value decomposition $U\Sigma V^T$, \begin{align}U&\in\mathbb R^{m\times r},\\\Sigma&\in\mathbb R^{r\times r},\\V&...
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### SVD of a 3x3 symmetric matrix shortcuts

Say there is a 3x3 symmetric matrix such as the following: $$A=\begin{bmatrix} a & b & a \\b & b & b \\ a & b & a \end{bmatrix}$$ By being symmetric:$$AA^T=A^TA=A\cdot A =A^2$$...
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### Why does SVD solve $\underset{U,V}{\min}\| A - UV^T\|_F^2$

I read here the following: You can solve the quadratic problem below through Singular Value Decomposition (SVD) of the matrix $A$. \begin{align} \underset{U,V}{\min} \| A - UV^T\|_F^2 \end{...
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### The norm $\|S-Q\|_F$ where $Q$ is orthogonal is minimised by $Q=I$

Problem: Suppose that $S$ is symmetric and semi-positive-definite. Let $\|\cdot \|_F$ be the Frobenius norm. Show that $$\|S-I \|_F \leq \|S-Q\|_F$$ for all orthogonal matrices $Q$, where $I$ is ...
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### Relation between eigenvectors and singular vectors of complex skew-symmetric matrices

As shown in this answer, if $A$ is a real skew-symmetric matrix, and $v,w$ are a pair of orthogonal singular vectors with $$Av=sw \qquad\text{ and }\qquad Aw=-sv,$$ for some $s>0$, then the ...
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### In SVD why is $\Sigma$ the square root of $V$'s Eigen values?

Following a problem, it was not explained why the $\Sigma$ matrix is the square root of $V$'s Eigen values rather than the values themselves.
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### How do you compute the reduced SVD?

I know how to compute the full SVD by hand, but surprisingly, I couldn't find much information on how to compute the reduced SVD by hand. What is the easiest way to do this?
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### What are the most relevant applications of polar decomposition?

Assume there exists a new and very efficient algorithm for calculating the polar decomposition of a matrix $A=UP$, where $U$ is a unitary matrix and $P$ is a positive-semidefinite Hermitian matrix. ...
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### SVD - why do we complete matrices $U$ and $V$ with the vectors that form a basis for the nullspace?

SVD picture Normally, in theoretical demonstrations, $U$ has size $(m, n)$ instead of $(m, r)$. Why is that so? Is there a reason why we include a basis for the nullspace as well in the matrices $U$ ...
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### SVD for discovering patterns?

I have a matrix of 1000 rows as the instances or observations of some kind, values are between 0-1. Every row has 10 positions as columns, 1000 rows X 10 columns. The data is outliers-free. Every row ...
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### Distribution of the entries of a particular matrix product

Let us assume a Complex Gaussian i.i.d. matrix $A$ which can be decomposed using the SVD into $A=UDV^*$, where $U$ and $V$ contain the left and right singular vectors, respectively, and $D$ is a ...
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### Proof that transposed matrix can be used for ZCA

I am using Zero-phase Component Analysis (ZCA) for the processing of images. The images are provided as a matrix $X$ with $m=$ number of rows = number of images and $n=$ number of features (pixels) = ...
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### SVD convergence on eigenvectors

I’m trying to understand why QR iteration results in convergence of the SVD matrices on eigenvectors and eigenvalues rather than arbitrary similar matrices?
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### diagonal form of a complex matrix

Let us consider a complex symmetric matrix \begin{equation} A = \begin{pmatrix} a+ib & c \\ c & -a+ib \end{pmatrix} \end{equation} where the coefficients $a,b,c$ are real. I am interested in ...
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### Singular value decomposition.

I would like to solve the following problem with singular value decomposition. We have $x_{1},...,x_{m}$ vectors of $\mathbb{R}^{d}$. But $d$ is too high for your computer. So Let's find a linear ...
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### Is gaining enough understanding of SVD/QR factorization until you have their formulas memorized useful?

Is knowing how to compute SVD/QRfactorization,Power iteration/power method by hand without notes useful in applications such as Statistics? Now adays, we can do everytime by Mathematica/computer ...
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### SVD for image compression

I want to make sure I understood the concept behind SVD for image compression. So, we start off with a rectangular $m \times n$ matrix that stores all the pixel values of the image. We then compute ...
Given a set of points $a_i(x,y)$ in the 2-Dimensional plane how can I determine the unit vector $u$ which produces the biggest standard-deviation in the $<a_i,u>$ (scalar product) values? I ...