# Questions tagged [linear-algebra]

A field of mathematics concerned with the study of finite dimensional vector spaces, including matrices and their manipulation, which are important in statistics.

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### Multi-class LDA (pairwise classification)

From this post: $w=S_{W}^{-1}(μ1−μ2),$ is used to estimate $w_{0}=\frac{1}{2}(μ_{1}−μ_{2})^{T}S_{W}^{-1}(μ_{1}−μ_{2})−log(\frac{P1}{P2}),$ However, this is for a situation where there are only 2 ...
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### Laplace transform of sum of $N$ IID random variables where $N$ is itself random

Let $\{Y_i\}$ be a sequence of IID random variables so that $Y_i \sim Exp(\lambda)$ or equivalently $Y_i \sim CPH_1(1, -\lambda)$ (continuous phase-type distribution). Let $N$ be a discrete random ...
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### What are some beginner's references on algebraically structured models, and what's their connection with group actions and Fourier transform?

Since I didn't get an answer, I asked it on mathoverflow. I'm looking at a short-term position, whose project is on estimation in algebraically structured models. It also mentions in the required ...
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### Why does the smallest eigenvalue of the covariance matrix of a GMM equal the common variance?

I've been reading a paper "Introduction to Tensor Decompositions and Their Applications in Machine Learning". In it, the author describes an algorithm for estimating the means of the ...
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### Show that ABA=0 if and only if B is of the form C-PCP for some symmetric matrix C

If B is a symmetric matrix such that ABA is well-defined, then show that ABA=0 if and only if B is of the form C-PACPA for some symmetric matrix C. PA here is the projection matrix of A.
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### Probabilities concerning n-spheres

Let $S_{n-1}(R)=\{ {\bf x}\in {\mathbb{R}}^n : ||{\bf x}||^2=R^2\}$ be the sphere in ${\mathbb{R}}^n$ with radious $R>0.$ Let the projection map $\tau_m({\bf x})=(x_1,...,x_m)$ with $m\leq n.$ Now, ...
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### Visualizing data using vectors

Say there are 10 houses and we have three pieces of information for each of them, area, nbedrooms, price I can view this as 10 different vectors in space where there are 3 axes. Basically 10 arrows ...
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### Determine if orthogonal projection and full rank

Let $X=(X_1,X_2)$ be a $n \times p$ matrix of rank $p$, where $X_1$ is $n\times p_1$ and $X_2$ is $n\times p_2$. Let $P_1$ be orthogonal projection onto $C(X_1)$ and $P_2$ be orthogonal projection ...
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### Using matrices to find estimates in GLMs

How do we use matrices to find estimates in Linear Models? A 4 × 400 relay race is run as follows. There are four runners, each of whom runs 400 meters. The first runner carries a baton, which she ...
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### how to normalize data such that an estimated OLS regression vector has pre-specified length (= L_2 norm)

I have the following data: $n$ observations on $d$ variables $X$ and one outcome variable $Y$; i.e. $X$ is a $n \times d$ matrix and $Y$ an $n \times 1$ vector. I consider the following Ordinary Least ...
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### How vector projection works behind SVD?

I was reading a blog on mathematical intuition behind SVD. Here, author pointed out three information we get after vector decomposition. The directions of projection — the unit vectors (v₁ and v₂) ...
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### Condition numbers, invertibility and multicollinearity

The following is an excerpt from Greene's Regression Analysis (Seventh Edition): a) What does it mean to be "difficult" to invert a matrix accurately? Shouldn't all matrixes be either ...
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### Good concise, big picture, linear algebra book?

I have looked at this answer and am not satisfied with the results. Reference book for linear algebra applied to statistics? I have briefly looked at two of the books suggested by the answer, the one ...
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### Algebra for Intervention Effect in Interrupted Time Series (with delayed effect)

I have run an interrupted time series analysis using GLS regression model in R. My data consists of 48 observations [time 1:48], with the intervention implemented at time 20, but it's effect not ...
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### PCA dimension reduction on correlation matrix for invertability

I have a non-singular (correlation) matrix $C$ of dimension $N{\times}N$, this is a modified version of another correlation matrix, and therefore I don't think I am able to apply any calculations on ...
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### Published source for D-dimensional behaviour of Dot-Product

I am currently studying the behaviour of the dot product between two random vectors in $R^d$. Specifically I wanted to start with the case of uniform random vectors on $\mathcal{S}^{d-1}$. I found ...
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### Additional Property of Singular Value Decomposition

I am new to SVD so forgive me if the question is trivial. Following is my question. If I have two sets of linear equations, Y1 = T1.X Y2 = T2.X where T1 and T2 are mxn rectangular matrices. Now let'...
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