Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

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51 views

Is the basis vector of a rotated vector in $E^3$ transformed differently than the components of the vector?

Do the basis vectors of a rotated vector in $E^3$ transform differently than the components of the vector? I've recently come across a blog where someone rotated the i,j,k basis vector using the ...
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63 views

Looking for a reflection of 30° at a line

I'm trying to find a matrix-expression of a 30° reflection at the line $g(x)=2x+4$ Somebody can give a hint? Greetings
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114 views

What is this inner product called?

While trying to define an inner product, I realized that what I ended up with was a special case of the following. It seems like it should be a standard way to define an inner product, but I'm not ...
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56 views

Witt Cancellation over $\mathbb{Z}/{p^e \mathbb{Z}}$?

I wonder whether someone knows if the Witt cancellation theorem also holds for the rings $\mathbb{Z}/{p^e \mathbb{Z}}$ where $p$ is an odd prime and $e \in \mathbb{N}$, i.e. for example, let $G = ...
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44 views

Linear Algebra Order

I normally start with Gaussian Elimination, vectors and then matrices. I know a number of books which start with matrices and then go on to Gaussian Elimination and vectors. What is the most ...
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28 views

the asymptotic cone of matrices

I am just curious about something. The asymptotic cone of $\mathbb{Z}^n$ minus one point is $\mathbb{R}^n$. In fact, I think the asymptotic cone of $\mathbb{Z}^n$ minus a finite number of points is ...
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83 views

Classifying A Matrix - matrix representation of an operator with linear operators as entries

Say that one has a matrix representation of an operator A with differential operators as entries in the matrix A. Is this a ...
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125 views

Map preserving one-dimensional affine subspaces is affine

Let $f: V \rightarrow V$ be a bijective map of a vector space to itself that preserves one-dimensional affine subspaces. Is $f$ already the composition of some invertible matrix and a translation? My ...
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274 views

eigenvalues of a product with a diagonal random matrix

I am interested in the eigenvalue spectrum of the following matrix A A=(id+diag random) M Where M is a given matrix with a known eigenvalue spectrum, lets say for simplicity that it is Hermitian. diag ...
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140 views

Number of solutions (excluding permutations of variables' values) and solving in distinct positive integers the following system of equations

Questions and important info in italics, very important ones in bold. Here we have the system; $V_{1}+V_{2}\cdots+V_{k}=A$ and $V_{1}^{2}+V_{2}^{2}+\cdots +V_{k}^{2}=B$ where $V_{1}$, $V_{2}$, etc. ...
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154 views

Calculation of stopping condition for Conjugate Gradient

I am a person with programming background and need some math help. I am looking at the source code for an implementation of the Conjugate Gradient iterative solver ...
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151 views

Pair of equations with any equal number of variables with unique solution?

$(a+b+c\cdots)\neq(a^{2}+b^{2}+c^{2}\cdots)$ given all distinct values for the variables? When I came across this topic, it made me curious as to explore other possibilities, as here, what other two ...
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93 views

Linear Least Squares: how to weight

I have a system of the form $Ax = b$, and I want to obtain the best $x$ in the least squares sense. $A$ is $M \times N$, where $M \approx 2N$ (or $4N$ for another variation). Rows in $A$ have 2 ...
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104 views

When do the invariant factors of a direct sum of diagonal matrices correspond to those of its summands?

I am trying to prove something about matroids, which I have reduced to the following question: Suppose I have a matrix $M$ which is a direct sum of submatrices $M_1,M_2,\ldots,M_k$. When do the ...
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138 views

Projecting onto subspace

Question Find the orthogonal projection of $$x = \begin{bmatrix}7 \\ 0 \\ -4 \\ -4 \end{bmatrix}$$ onto the subspace of $\mathbb R^4$ spanned by $$v_1 = \begin{bmatrix}-4 \\ 2 \\ 2 \\ -4 ...
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231 views

(Improved; Kindly assist )CSI Forensic Maths

In a square room with 7 m-long sides, five people have shot at random people. They all had two bullets and killed two people in the crowd. Other people in the room were not hurt. CCTV recordings show ...
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643 views

Linear algebra - linear transformation, basis, matrix representation

I have no idea how to solve it. Please help me out! Thank you! Let $\alpha$ be a plane in $\mathbb{R}^3$ passing through the origin, suppose $\alpha$ is given by the equation $$ax + by + cz = 0.$$ ...
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269 views

Wolfram alpha wrong on linear independence for 4d vectors?

The first three vectors in the following statement are linearly independent. I put this statement into Wolfram Alpha and it tells the 4 vector set is linearly independent linear independence of ...
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87 views

How to solve Rayleigh Quotient type problem?

How to solve Rayleigh Quotient type problem? $$\max (w+w_0)^tC(w+w_0) \text{ s.t. } w'w=1,$$ where $w_0$ is given. Thank you!
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60 views

Alternate Characterization of the Matrix of a Linear Transformation

Let $V$ and $W$ be finite dimensional vector spaces with respective ordered bases given by $v_1, \dots, v_n$ and $w_1, \dots, w_m$. Then, the matrix of a linear transformation $T:V \rightarrow W$ is ...
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34 views

greatest common divisor of the slots of $\operatorname{adj}(tI_n-A)$ and characteristic polynomial

I'm trying to show: Let $A\in \mathcal{M}_n(\mathbb{F})$ a matrix with characteristic polynomial $p_A(t)$ and minimal poliynomial $q_A(t)$. Let $d_{n-1}(t)$ 'greatest common divisor' of the slots of ...
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57 views

How to construct a matrix satisfying two semidefinite constraints

You are given matrices $A$, $B$ and $C$. $C$ is symmetric and positive semidefinite. How would you go about constructing a matrix $X \succeq 0$ such that $X \succeq AXA^T$ and $C \succeq BXB^T$? ...
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178 views

conjugate function for matrix mixed norm

I am familiar with the conjugate function of the vector norm, which uses the concept of dual norm and is defined as follows: ...
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70 views

Affine image of the convex hull is its subset

For a set of points $v_1,v_2,\dots, v_r\in \mathbb R^n$ let's use $\mathcal P(v_1,\dots,v_r)$ to denote the convex hull of these points. Let us consider $A\in\mathbb R^{n\times n}$ and $b\in \mathbb ...
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reducing the dimension of a tensor

I have a tensor $C$ of size $a \times b \times c$, all with real values. I need to compute for a vector $u$ of length $a$ and a vector $w$ of length $c$ the product: $(C \times_{3} w) \times_{1} u$ ...
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157 views

Lanczos Algorithm - Searching for Multiple Eigenvalues w/ Seeding Strategy

The question Assuming one has obtained an eigenpair of a matrix via Lanczos, is it viable to simply orthogonalize a random vector against any known eigenvectors and use that as the Krylov seed in ...
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113 views

Mathematical model for my experiment?

I'm running a benchmark to find the efficiency of a my computer. There are $p$ control variables say, $x_1,x_2,...,x_p$ and one output variable $Y$. For example, every time I run an experiment I ...
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1k views

if two homogeneous systems of linear equations in two unknows have the same solutions, then they are equivalent

I am self-studying Linear Algebra from Hoffman and Kunze. The Exercise $6$ on page $5$ asks to show that if two homogeneous systems of linear equations in two unknows have the same solutions, then ...
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310 views

Fast Walsh–Hadamard transform generalization for non-power-of-two orders?

I have to process vectors through a Hadamard matrix of order N. If N is a power of 2, I can use the Fast Walsh–Hadamard transform; but if N is not a power of two (for instance, N=12), it is not ...
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39 views

can one identify $U'A$ from the following decomposition?

Assume we have the matrix equation: $(U^{\top} A) (Q \Lambda Q^{\top}) (A^{\top} U) = \Sigma$ such that all matrices in this equation have real values and ($d > n$): $U$ is $d \times d$ and is ...
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109 views

Generate a 'natural' path through a set of 3D points

I have a set $P=\{p_0,\ldots,p_n\}$ of 3D points, such that a curve (spline) passes through each point in order. The curve is well-defined (in this case a cubic spline) and for each point $p_i$ I have ...
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144 views

is there a connection between singular vectors of $AB$ and $BA$?

Let $A$ and $B$ matrices of dimensions $d \times n$ and $n \times d$ respectively. We know that the non-zero eigenvalues of $AB$ and $BA$ are the same. Is there any connection between the top $m$ ...
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124 views

Democratic central planning model

I want to model following situation: 1) There is a number of representatives of social groups (e. g. political parties). 2) Each of them devises an economic plan for the next year (N+1, N being ...
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102 views

Constructing a unit vector in the direction of largest component

I am dealing with $\ell_\infty$ norms, and I need a "mathematical" way to take a vector of $p$ length (We'll call it $x$), and construct a vector pointing in the direction of $\|x\|_\infty$. I know ...
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56 views

formulas for exact values of singular values in low dimension?

Are there formulas for the singular values of a real matrix in low dimension, i.e. for a $2 \times 2$ matrix or a $2 \times 3$ matrix? Any comment is welcome.
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158 views

Rotate a line segment with an angular constraint to fit exactly between two rays in 3d space

Hi my Mathematics are somewhat rusty and I am trying to solve a problem where I take a 3d line segment described by the vector r*(cos[t], h, sin[t]) where t is unknown and describes a constraint that ...
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52 views

Alternating forms tangential to a subspace.

Let $V$ be a finite-dimensional vector space with euclidean product, and let $U$ be a subspace. Now let $P$ be the projection of $V$ onto $U$, and let $\omega$ be any alternating multilinear $k$-form. ...
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147 views

The fastest algorithm of computing Principal eigenvector of a non-negative-entries matrix

I am studying the QR algorithm, is it the fastest one in this situation?
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77 views

Polynomial function from $S^3$ to $S^3$ and quaternions

I am searching the polynomial functions from $S^3$ to $S^3$. We say $g$ is a polynomial function from $S^3$ to $S^3$, if there exists $f_1,f_2,f_3,f_4 \in \mathbb{R}[X_1,X_2,X_3,X_4]$ and ...
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60 views

Positive forms and dot products in some basis

For a real $n \times n$ matrix $A$ the following are equivalent: $A$ is symmetric and positive definite There is an invertible matrix $P$ such that $A = P^{t} P$ This makes geometric sense when ...
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380 views

Moore–Penrose pseudoinverse reference

Given the eigendecompositions $AA^{\top}=Q \Lambda Q^{\top}$ and $A^{\top}A=P \Lambda P^{\top}$, where $\Lambda$ is a diagonal matrix (of eigenvalues) and $P$ and $Q$ are unitary eigenvectors matrices ...
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131 views

Stable and efficient projection onto subspace along another subspace

Suppose we are given the euclidean space $\mathbb R^{n+m}$ with the decompositin $\mathbb R^n = V \oplus W$, which we however do not expect to be orthogonal. Let us describe the matrix $P$ that ...
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1k views

Understanding Power Method/Inverse Iteration in Linear Algebra

For a linear algebra class, we are currently learning about finding the largest/smallest eigenvalues of a matrix using the power method and inverse iteration methods. I just want to make sure that I ...
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43 views

Does the locality or non-locality of operators imply matrix structure?

I understand that an operator, $\hat{O}$, is said to be non-local if $$b(x)=\hat{O}a(x)=\int dx'O(x,x')a(x')$$ that is, to find $b(x)$ at aparticular value of $x$, we need to know ...
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71 views

Some question concerning curve of second order

Let $$F(x,y)=ax^2+2bxy+cy^2+2dx+2ey+f,$$ $$\phi(x,y)=ax^2+2bxy+cy^2,$$ $x,y \in \mathbb{R}$. Assume that for some $x_0, y_0 \in \mathbb{R}$ and for some $\alpha, \beta \in \mathbb{R}$ such that ...
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63 views

Slowly varying vectors and coefficients of a sine transform

Let $u_k$ be the vector in $\mathbb{R}^n$ whose $i$'th entry is $\sin(\pi ki/n)$. The vectors $u_1,\ldots, u_n$ are orthogonal and correspondingly every vector in $\mathbb{R}^n$ can be decomposed as a ...
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79 views

Sets of independent vectors

So we're working in Z2k, the group of bit-vectors of length k and componentwise addition modulo 2. Now I'm trying to make a function yj=1..?(vi) assigning elements of Z2k to n vertices, such that ...
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757 views

Best-fitting plane

I need to implement the algorithm described below. Everything is fine until the eigenvalues computation. I'm completely new to them and I found a lot of very complicated paper on the net. Is it ...
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159 views

Condition number of DA and D'A: row equilibrated matrix

I am with an exercise that first asks me to show that for any regular matrix $A$, there exists a diagonal matrix $D$ such that $A$ is transformed into a row equilibrated matrix by a left ...
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133 views

What does 'the subspace W = ( < \{> v_1 \})^o' mean?

Can anyone help me understand the notation used at the end of this sentence? 'Now we proceed to the dual space $V'$, and the subspace W = ( < {> v_1 })^o'