Articles of svd

interpretation of SVD for text mining topic analysis

Background I’m learning about text mining by building my own text mining toolkit from scratch – the best way to learn! SVD The Singular Value Decomposition is often cited as a good way to: Visualise high dimensional data (word-document matrix) in 2d/3d Extract key topics by reducing dimensions I’ve spent about a month learning about […]

SVD -obligation of normalization

let us suppose we have following matrix $ A= \left[ {\begin{array}{cc} 2 & 2 \\ -1 & 1 \\ \end{array} } \right] $ and i want to compute SVD of this matrix, i have calculated first of all $A*A’$ , which is equal to $ \left[ {\begin{array}{cc} 8 & 0 \\ 0 & 2 \\ […]

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 to this singular value? Usually I found the columns of U with this relation, so let’s […]

Do similar matrices have equal singular values?

Is it true that if $A$ and $B$ are similar matrices, $B=S^{-1}AS$, then $A$ and $B$ have the same singular values?

Intuitive understanding of SVD for a matrix

Could any give an intuitive understanding of SVD decomposition of a matrix? I know it can be used for image compress. But how to understand the decomposition within linear transform?

SVD: proof of existence

I’m reading “Numerical Linear Algebra” by Lloyd Thefethen. For Singular Value Decomposition proof of existence it starts like this: “Set $\sigma_1=||A||_2$. By a compactness argument, there must be vectors $ v_1 \in C^n$ and $u_1 \in C^m$ with $||v_1||_2=||u_1||_2=1$ and $Av_1=\sigma_1u_1$.” What exactly “by a compactness argument” means? I understand it should have something to […]

Singular Value Decomposition of Rank 1 matrix

I am trying to understand singular value decomposition. I get the general definition and how to solve for the singular values of form the SVD of a given matrix however, I came across the following problem and realized that I did not fully understand how SVD works: Let $0\ne u\in \mathbb{R}^{m}$. Determine an SVD for […]

Singular vector of random Gaussian matrix

Suppose $\Omega$ is a Gaussian matrix with entries distributed i.i.d. according to normal distribution $\mathcal{N}(0,1)$. Let $U \Sigma V^{\mathsf T}$ be its singular value decomposition. What would be the distribution of the column (or row) vectors of $U$ and $V$? Would it be a Gaussian or anything closely related?

$B – A \in S^n_{++}$ 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 positive definite matrices. How to prove that $$B – A \in S^n_{++}$$ if and only if $$I – A^{1/2}B^{-1}A^{1/2} \in S^n_{++}?$$

Why SVD on $X$ is preferred to eigendecomposition of $XX^\top$ in PCA

In this post J.M. has mentioned that … In fact, using the SVD to perform PCA makes much better sense numerically than forming the covariance matrix to begin with, since the formation of $XX^\top$ can cause loss of precision. This is detailed in books on numerical linear algebra, but I’ll leave you with an example […]