Tagged Questions

Questions on the various algorithms used in linear algebra computations (matrix computations).

681 views

Fast computation/estimation of the nuclear norm of a matrix

The nuclear norm of a matrix is defined as the sum of its singular values, as given by the Singular Value Decomposition of the matrix itself. It is of central importance in Signal Processing and ...
94 views

How to get the SVD of $2AA^T-\operatorname{diag}(AA^T)$ given $A$ and its SVD $A=USV^T$?

Given a matrix $A\in R^{n\times d}$ with $n>d$, and we can have some fast ways to (approximately) calculate the SVD (Singular Value Decomposition) of $A$, saying $A=USV^T$ and $V\in R^{d\times d}$. ...
601 views

125 views

How to solve linear system of form $(A \otimes B + C^{T}C)x = b$ when $A \otimes B$ is too large to compute?

For the given linear system: $$(A \otimes B + C^{T}C)x = b$$ where $\otimes$ is the Kronecker product, $A$ and $B$ are dense and symmetric positive-definite, and $C^{T}C$ is a sparse symmetric block ...
17k views

Distance/Similarity between two matrices

I'm in the process of writing an application which identifies the closest matrix from a set of square matrices $M$ to a given square matrix $A$. The closest can be defined as the most similar. I ...
2k views

Trace minimization with constraints

For positive semi-definite matrices $A,B$, how can I find an $X$ that minimizes $\text{Trace}(AX^TBX$) under 'either' one of these constraints: a) Sum of squares of Euclidean-distances between pairs ...
735 views

Fast algorithm for approximating Eigenvalue distribution of large sparse matrix

I am interested in the eigenvalue distribution of a huge $2^{16}$x$2^{16}$ Hermitian sparse matrix with spectrum contained in $[-1,1]$. That is I don't need to know all eigenvalues exactly, but rather ...
327 views

First of all, I´m very sorry for my bad english, especially writing. Ok, for differents problems i´m studing a Bachelor degree in Mathematics. These degree is online. Now, the problem with my school ...
226 views

Generate arbitrary numerically invertable matrix

I'm designing a unit-test for a matrix inversion function. Currently I make a random matrix as a test case by generating its elements with random numbers uniformly distributed in $[0,1)$. If I ...
2k views

Is there a faster way to calculate a few diagonal elements of the inverse of a huge symmetric positive definite matrix?

I asked this on SO first, but decided to move the math part of my question here. Consider a $p \times p$ symmetric and positive definite matrix $\bf A$ (p=70000, i.e. $\bf A$ is roughly 40 GB using 8-...
2k views

3k views

Proof that Newton Raphson method has quadratic convergence

I've googled this and I've seen different types of proofs but they all use notations that I don't understand. But first of all, I need to understand what quadratic convergence means, I read that it ...
948 views

Uniform sampling of points on a simplex

I have this problem: I'm trying to sample the relation $$\sum_{i=1}^N x_i = 1$$ in the domain where $x_i>0\ \forall i$. Right now I'm just extracting $N$ random numbers $u_i$ from a uniform ...
1k views

2k views

A book for self-study of matrix decompositions

I am a third year math student and I noticed that there are many uses for decomposing a matrix (I mean decompositions like SVD, LU etc'). Is there a good book for self-study of the subject ? Note ...
1k views

What is the practical impact of a matrix's condition number?

Let's say I am trying to solve a square linear system $Ax = b$ for whatever reason. A perturbation $\delta b$ in $b$ will lead to a perturbation $\delta x$ in $x$, whose relative norm is bounded by ...
204 views

how can a matrix vector product reduce to a scalar?

I have an Excel spreadsheet with the following formula (paraphrased): ...
5k views

Properties of sum of real symmetric, positive semi-definite matrices

I have two correlation matrices A and B. They are: Real symmetric (with ones on the diagonal) Positive semi-definite (eigenvalues are $\ge 0$) I want to try to prove that the average of these two ...
761 views

When does an eigenvector of a matrix contain only a constant?

When I compute the eigenvectors of a certain matrix, the first of them is composed entirely of a single constant. What properties of a matrix lead to this result? Update By "a vector composed ...
4k views

When do two matrices have the same column space?

Recently I started learning about matrices and know for example that the pivot columns of a matrix form a basis for the column space of this matrix. I just can't seem to find out when two matrices ...
1k views

Least square principles with Lagrange multiplier

I have a function to minimize: $$f(a_1,a_2,a_3,a_4)=\sum_{i=1}^n\left(\sum_{k=1}^3 a_k\ p_i^k -a_4\right)^2$$ subjected to this constraint: $$a_1^2+a_2^2+a_3^2=1$$ and $$a_4\geq0$$ I am trying ...
1k views

Matrix Calculus in Least-Square method

In the proof of matrix solution of Least Square Method, I see some matrix calculus, which I have no clue. Can anyone explain to me or recommend me a good link to study this sort of matrix calculus? ...
990 views

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 $XX^\top$...
168 views

real eigenvalue

Let matrix $A$ be $$\begin{bmatrix} -5& 1& 0& 0\\ a &2& 1 &0\\ 0& 1 &1 &1\\ 0 &0&1& 0 \end{bmatrix}$$ where $a$ is a constant between 1 and ...
216 views

solution to $\min \|A-BXC \|$

I have the following problem. Let $A$, $B$ $C$ be real-valued matrices of size $m \times q$, $m \times n$, $p\times q$ respectively. I would like to find matrix $X$ of size $n\times p$ and maximum ...
2k views

What is the time complexity of conjugate gradient method

I have been trying to figure our the time complexity of conjugate gradient method I have to solve a system of linear equations given by $$Ax=b$$ where A is sparse and positive definite symmetrix ...
200 views

QR factorization of a special structured matrix

A friend asked me the following interesting question: Let $$A = \begin{bmatrix} R \\ \xi{\rm I} \end{bmatrix},$$ where $R \in \mathbb{R}^{n \times n}$ is an upper triangular and ${\rm I}$...
Check whether two subgroup of $GL(n,\mathbb Z)$ are conjugate
Suppose I have two finite subgroups of $GL(n,\mathbb Z)$. Is there an algorithm to find out whether these two belong to the same conjugacy class in $GL(n,\mathbb Z)$? I tried by using the Jordan ...
If $A,B$ are $2 \times 2$ matrices of real or complex numbers, then AB = \left[ \begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right]\cdot \left[ \begin{array}{cc} b_{11} ...