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 […]

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 \\ […]

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 […]

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

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?

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 […]

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 […]

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?

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 $$I – A^{1/2}B^{-1}A^{1/2} \in S^n_{++}?$$

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 […]

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