# Tagged Questions

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

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### 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 ...
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### 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 ...
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### Floating point arithmetic operations when row reducing matrices

A numerical note in my linear algebra text states the following: "In general, the forward phase of row reduction takes much longer than the backward phase. An algorithm for solving a system is usually ...
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### Product of positve definite matrix and seminegative definite matrix

Let $A$ a spd (symmetric positive definite) matrix and $B$ a symmetric seminegative definite matrix. Is tr $AB \leq 0$ and more general is $AB$ seminegative definite? I know that tr $AB \leq 0$ ...
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### To create a special matrix !!

How to create a $N \times N$ matrix with $1$ and $-1$ as its elements, such that when this matrix is multiplied with its transpose the resultant matrice is $N \times \mathbb{I}_N$. Where $N$ is a ...
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### Choose $\rho$ such that $\rho$-norm minimizes the matrix condition number

I'm solving questions from am exam that I failed miserably, so I would love it if someone can take a look at my proof and make sure I'm not making any gross mistakes. Question Let $A$ a symmetric ...
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### Augmented Reality Transformation Matrix Optimization

i am a software developer, i'm working on an Augmented Reality system. I'd like to receive some advice in order to optimize my math model. My program has to be slim and fast. Here's the situation: ...
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### 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? ...
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### If modulus of each one of eigenvalues of $B$ is less than $1$, then $B^k\rightarrow 0$

Let $B$ be a $n\times n$ matrix and let $X$ be the set of all eigenvalues of $B$. Prove that if $|m|<1$ then $\lim \limits_{k\rightarrow\infty}B^k=0$, where $m=\max X$. Thanks. Actually, there ...
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### How to substitute and matrix into a functions?

I have $f(x)=2*x_1 +x_2$ how to find $f(m*x)$ if m is a matrix $m=\begin{bmatrix} 1 & 2 \\ 3 & 1 \end{bmatrix}$
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[![enter image description here][1]][1]Please tell me how this formula for numerical differentiation derived. I think it has something to do with Vandermonde Matrices but I am not quite sure how to go ...
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### 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 ...
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### Factorize a Symmetric matrix as an 'Approximation' with an outer product.

(deprecated-taken back based on discussion(OLD)) What is a good way to factor a symmetric matrix $X$ as an outer product of two vectors $u$ and $v$. i.e, Find two vectors $u$ and $v$ such that $X=uv^T$...
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### Conditions for the equivalence of $\mathbf A^T \mathbf A \mathbf x = \mathbf A^T \mathbf b$ and $\mathbf A \mathbf x = \mathbf b$

I have an application where I have to minimize a cost function of the form: $J(\mathbf x) = \| \mathbf A \mathbf x - \mathbf b \|^2 \quad (1)$ By calculating the gradient, I derived that I have to ...
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I'm looking for a numerically stable method for computing the angle between two 3D vectors. Which of the following methods ought to be preferred? Method 1: $$u\times v = ||u||~||v|| \sin(\theta) \... 1answer 200 views ### QR factorization of a special structured matrix A friend asked me the following interesting question: Let$$A = \begin{bmatrix} R \\ \xi{\rm I} \end{bmatrix},$$where R \in \mathbb{R}^{n \times n} is an upper triangular and {\rm I}... 1answer 55 views ### Strictly diagonal matrix Suppose that matrix A is strictly diagonally dominant, show that$$\|A^{-1}\|_{\infty}\leq\left[\min_i\left(|a_{ii}|-\left|\sum_{\substack{j\neq i}} a_{ij}\right|\right)\right]^{-1}.$$0answers 45 views ### Wiedemann for solving sparse linear equation I am new member. I am researching in Wiedemann algorithm to find solution x of$$Ax=b$$Firstly, I will show a Wiedemann's deterministic algorithm (Algorithm 2 in paper Compute A^ib for i=0..... 0answers 157 views ### Algorithm to determine matrix equivalence I'm a physicist who's not particularly good at linear algebra so please accept my apologies if this is standard textbook stuff that I'm just unaware of. I have two real rectangular matrices A_{mxn} ... 0answers 427 views ### Really confused about LU decomposition and Doolittle algorithm I'm really confused about the Doolittle algorithm, so I need some help. At the end of the description at wikipedia, it says It is clear that in order for this algorithm to work, one needs to ... 0answers 46 views ### Compression of a matrix A by V I can't understand and even can't find any text on Compression of a matrix A by V. meaning if B=V^*AV then B is called the compression of A. What does it mean??? 2answers 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 ... 1answer 1k views ### 3-sigma Ellipse, why axis length scales with square root of eigenvalues of covariance-matrix This is my first post on math.stackexchange and i am not a mathematician, but i took some undergrad math courses and some grad mathematical modelling courses, so i come with a basic understanding of ... 3answers 2k views ### Approximate a convolution as a sum of separable convolutions I want to compute the discrete convolution of two 3D arrays: A(i, j, k) \ast B(i, j, k) Is there a general way to decompose the array A into a sum of a small number of separable arrays? That is: ... 1answer 73 views ### Numerically Solving a Poisson Equation with Neumann Boundary Conditions The Problem Suppose I have an equation of the form \nabla^2 \phi(x) = f(x) on the interval A \le x \le B, where f(x) is known and \phi(x) is unknown. I have Neumann-type boundary conditions: ... 0answers 144 views ### The norm of the matrix This problem is in Trefethen'book Numerical Linear Algebra Suppose the m\times n matrix A has the form A=\begin{pmatrix}A_1\\A_2 \end{pmatrix} where A_1 is a nonsingular matrix of dimension ... 1answer 563 views ### equations solved with Newton's method by finding the zeros of functions? I found this statement in one paper I read recently: This problem can be solved by finding the zero of functions: ... 3answers 189 views ### On integral of a function over a simplex Help w/the following general calculation and references would be appreciated. Let ABC be a triangle in the plane. Then for any linear function of two variables u.$$ \int_{\triangle}|\nabla u|^...
Let $F$ be a $100 \times 100$ DFT matrix, and $U$ be a diagonal matrix with its diagonal entries being all positive, denoted by $U=\mathrm{diag}(u_1, u_2,\cdots, u_{100})$. My question is: Under ...