Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc. For questions specifically ...

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Symplectic Eigenvalues of Wishart Matrix

We work over the reals. Fix a dimension $n$ and a symplectic form $\Omega$. This is a $2n \times 2n$ matrix s.t. $\Omega^2 = -1$. A symplectic matrix is a matrix $S$ such that $ S^T \Omega S = ...
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Intuitive Explanation For Why Dependent Equations Contain No Added Information?

I've always been taught that because dependent equations contain no added information they can be deleted without effecting the solution set. Now this makes sense to me if an equation is a constant ...
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How to estimate the world position of origin of a coordinate system based on its members global position?

I have n points of interests (POIs). I have a local coordinate system. For all these n POIs I know there position in this system, that is there x, y, z translation (local position) from origin of ...
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23 views

Skew-symmetric matrix property

This page (https://shiyuzhao.wordpress.com/2011/06/08/rotation-matrix-angle-axis-angular-velocity/), gives the following relation: ...
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30 views

Can anyone help out with this proof, certain steps are unclear. Norm of linear operator.

I have the following norm defined as follows (in $R^n$, $x=(x^1,x^2,\ldots,x^n)\ $)$\| x\|_1= \sum_{i=1}^{n}|x^i|$ Let $A:R^m \to R^n$ a linear map of the spaces $(R^m ,\| \cdot \|_1 )$ and $((R^n ...
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Reduction of product of quadratic forms into a single quadratic form

Let us assume we have the products $$f(x_1,\dotsc,x_K)=\prod^K_{i=1} \boldsymbol{\beta}\exp\left(\mathbf{C}_ix_i\right) \mathbf{C}_i,$$ and $$f(x_1,\dotsc,x_K)=\prod^K_{i=1} ...
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21 views

Arguing a stationary distribution exists

I am trying to show that there exists a stationary distribution when $q>p$ for the Markov process with one-step transition matrix $$ \begin{bmatrix} q & p & 0 & 0 & ...
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1answer
26 views

Transposing matrix when differentiating it

Hi so I am trying to understand the solution of linear regression with matrices (found at the following link) and an confused about how on page 10 he says the derivative of $2Y'XB$ with respect to $B$ ...
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30 views

(Nonunique) Solvability of Sylvester Equation

I am interested in stating existence of solution of a Sylvester equation $$ AX - XB = C, $$ where $A$, $B$, $C$, and $X$ are $(n,n)$ matrices. Existence of a unique solution $X$ is given, if $A$ ...
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Maximal determinant of a $\{1,−1\}$ matrix of size $n$ is $2n−1$ times the maximal determinant of a $ \{0,1\}$ matrix of size $n−1$.

Maximal determinant of a $\{1,−1\}$ matrix of size $n$ is $2n−1$ times the maximal determinant of a $ \{0,1\}$ matrix of size $n−1$. How to prove this result? (I found this statement while reading ...
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Is the direct sum of two orthogonal subspaces well defined in infinite-dimensional vector spaces?

Let's say that $V$ is an inner product space on some field $\Bbb{K}$ and $M$ is a subspace of $V$. If $M^{\perp}$ is the orthogonal complement of $M$ with respect to the inner product, can I make the ...
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18 views

Poritive orthant and positive functional

Let $A$ be a hyperplane of $\mathbb{R}^n$, and denote by $\mathbb{R}_+^n$ the positive orthant, i.e. $$ \mathbb{R}_+^n = \{ v \in \mathbb{R}^n \;\mid\; v_i\geq0 \quad \forall i = 1 \dots n \} $$ ...
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Product of a Householder transformation and reflection through the origin in 3 dimensions

This came up doing some research in quantum information. Let us consider two orthogonal three-dimensional unit vectors $v$ and $w$ $v^T\cdot w=0$, and the Householder transformation ...
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30 views

Prove that multiplying an elementary matrix to a matrix can produce the same effect as an elementary row operation.

Elementary row operations: 1) Interchange any two rows of the matrix 2) Multiply every entry of some row of the matrix by the same nonzero scalar 3) Add a multiple of one row of the matrix to ...
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Stochastic process, Fourier transform and $L^2$

Consider a time domain signal received at a sensor $x(t)$ over some time $t$, and we have performed Fourier transform on $x(t)$ to obtain $X(w)$. While performing Fourier Transform to find the ...
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Find the matrix representation in the standard basis for either rotation by an angle $\theta$ in the plane perpendicular to the subspace

Find the matrix representation in the standard basis for either rotation by an angle $\theta$ in the plane perpendicular to the subspace spanned by vectors $(1,~1,~1,~1)~and~(1,~1,~1,0)$ in ...
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21 views

What is the nature of the graph of number (y) of gallons of fuel left in the tank of a car moving at constant speed, plotted against time (x)? [on hold]

This is an interpreting linear functions question. I don't understand what it means to find "the nature of the graph". Please help me with this question. Thank you.
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Find a basis of a subspace defined by a linear equation

Let $B=\{v_1,v_2,v_3,v_4\}$ be a basis of $V$. Let $$V \supset S= \left \{v:v=\sum\limits_{i=1}^4 \alpha_iv_i, \alpha_1+2\alpha_2+\alpha_3-\alpha_4=0 \right \}$$ Find a basis of $S$. I don't ...
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24 views

Describe the kernel, and determine whether the given vector space linear transformation is invertible:

Let $F$ be the vector space of all functions mapping $\Bbb R$ into $\Bbb R$ $T:F\rightarrow\Bbb R$ defined by $T(f)=f(-4)$ $ker(T)=$ {$f\in F|f(-4)=0$}, by definition of kernel. To prove that $T$ ...
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11 views

Volume of the symmetric difference between a parallelotope and its translated.

Let $A$ be a n-dimensional parallelotope and $v \in \mathbb{R}^n$ a vector. Is there a formula giving the volume of the symmetric difference $A \Delta (v+A)$?
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1answer
25 views

Subspace of symmetric commuting matrices

I am given $W$ a subspace of real $n$-dimensional matrices which are symmetric and pairwise commuting. I have to prove that $dim(W) \leq n$. I have read some facts about commuting matrices over an ...
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48 views

How to prove there exists a unique linear map such that $T(e_i) = w_i$ in an infinite-dimensional vector space?

Problem: (a) Let $V$ and $W$ be two finite dimensional vectorspaces over a field $F$, and let $\left\{e_1, e_2, \ldots, e_n\right\}$ be a basis for $V$. Then there exists for each $w_i \in W$ an ...
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44 views

Prove that an $n$-dimensional non-unital algebra over a field $k$ is isomorphic to a subalgebra $\mathfrak{M}_n(k)$

Assume we have already proved this for unital algebras. Here's my book's solution: Construct the unital algebra $A^1$ [with unit $(1,0)$] as an algebra on the vector space $k\oplus A$ with the ...
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1answer
22 views

Strange “form” of the set of vertices $C(x,y,z)$ such that $ABC$ is a right triangle with hypotenuse $AB$

Let $A(1,-3,4)$ and $B(3,-2,-1)$ and find the set of all $C(x,y,z)$ such that $ABC$ is a right triangle with hypotenuse $AB$ What I did $$AB=(2,1,5)$$ $$BC=(x-3,y+2,z+1)$$ $$AC=(x-1,y+3,z-4)$$ ...
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1answer
50 views

General Linear Group over the quaternions is a a topological group

How to show that General Linear Group over the quaternions is a a topological group?
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29 views

What am I doing wrong? - Change of basis matrix

Problem: Let $\alpha$ be the standard basis of $\mathbb{R}^3$ and let $\beta = \left\{(1,0,0), (1,1,0), (1,1,1)\right\}$ be another basis. Consider the linear map $T: \mathbb{R}^3 \rightarrow ...
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1answer
27 views

proof of the singular-values of orthogonal matrix

What is a simple and intuitive proof that the singular-values of orthogonal matrix $A$ is $1$?
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37 views

Condition number of $A^TA$

if $n \times n$ full rank matrix $A$ has condition number $\kappa$, what would be the condition number of $A^TA$? Preferably If the derivation includes the following definition of $\kappa$: $$ \kappa ...
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Nonlinear-Variation of Helmholtz Equation

I was wondering on the solution of the equation $$\nabla^2P(\vec r)=v(\vec r)P(\vec r)^2\phantom{.......}(1)$$ Or more simply, if there exists a coordinate system where: $$\nabla^2P(\vec r)=P^2(\vec ...
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1answer
17 views

Solving Lyapunov equation for unknown A matrix and known P matrix

I need to solve the Lyapunov equation $A'P+PA+Q+PBR^{-1}B'P=0$ for matrix A. Note that usually the equation is solved to get unknown P matrix. But instead of the usual problem which can be solved ...
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Kernel and image of this linear transformations

I dont know how i can solve this problem, if you know, please, explain me... help...: Consider in $\mathbb{R}^2$ the following linear transformations, $f_{\alpha}:\mathbb{R}^2 ...
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2014 IMC first problem first day (eigenvalues of a product of symmetric matrices).

This was the first problem of the IMC 2014. Let $A$ and $B$ be two $n\times n$ symmetric matrices with real entries which have all their eigenvalues strictly larger than $1$. Prove all the ...
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1answer
77 views

$A^tA-AA^t$ in Mathematical Physics

In very different contexts of mathematical physics (rigid body mechanics, fluidodynamics, general relativity, quantum field theory,...) I have come across the following expression: $$ A^tA-AA^t, $$ ...
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Efficient Test For Commuting Matrices

I know that if $A$ and $B$ are two Hermitian matrices, then $A B= B A$ if and only if their eigenspaces coincide [1]. In order to apply this test one need to compute eigenvectors of both $A$ and $B$ ...
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Can the difference between a number and a set of numbers be calculated as follows?

I have a number $X$ (lets say $45$) and I have five other numbers, lets say $(3, 4, ,5, 6, 7)$ I want to know the difference between my number and the other numbers like this: ...
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If then as matrix calculation

Having simple script code a like to bring a if-then-condition into linear algebra form. How is it made? Example 1: Having $T=25$ (where T is temperature current in room). If $T>30$ the equ. ...
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1answer
12 views

Bound on maximum of product of matrix and vector

I need to bound the absolute maximum of each entry of a matrix-vector product: $\max_{|x|_{1}=1} |Ax|_{\infty}$ I tried to pose this in terms of the induced infinity norm of $A$, as in ...
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1answer
22 views

Find basis corresponding to dual basis

In the finite dimensional vector space $V$, suppose $\{f_1,f_2,\cdots,f_m\}$ are the dual basis, how can find the basis $\{e_1,e_2,\cdots,e_m\}$ s.t. $f_i(e_j)=\delta_{ij}$
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283 views

How to find the limit of this matrix function

Let $A$ be $n\times n$ real symmetric matrix that is positive definite. Let $x\in\mathbb{R^n}, \space x\ne 0$. Prove that the following limit $$ \lim_{m\to\infty}\dfrac{x^TA^{m+1}x}{x^TA^{m}x} $$ ...
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What are the main kinds of mathematics? [on hold]

I stumble upon as much on math I don't know (trascendal math, number theory) and math I know on the internet and elsewhere. I have a pretty good idea about differential and integral calculus, and I'd ...
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How to solve an equation involving euclidean norm operation?

On page 3 of Scalable, Versatile and Simple Constrained Graph Layout it describes the equation: $$|(\mathbf p-\mathbf r)-(\mathbf q+\mathbf r)|=d$$ Where $\mathbf p$ and $\mathbf q$ are known ...
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Semplify $\det\left(D+M+A\right)$.

Let $D$, $M$, $A$, $n\times n$ matrices, with $n\in\mathbb N$. $D$ is a diagonal matrix, $M$ with elements all equal to $k\in\mathbb R$, $A$ is an antisymmetric matrix. Is possible to calculate ...
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162 views

Symmetric matrices and eigenvalues

If the eigenvalues of a symmetric matrix $A$ are greater than 0, show that $v^{\top}Av > 0$ for every $v \ne 0$ I am trying to prove this as follows: If $v$ is an eigenvector of $A$, then $Av ...
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Find the population [on hold]

Every year, the emigration rate from country A to B is 𝛼 (0 < 𝛼 < 1), whereas the emigration rate from country B to A is 𝛽 (0 < 𝛽 < 1). Note that the fluctuation in population of both ...
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What does it mean to evaluate a polynomial at a linear map, e.g. $p(T)$ where $p(x)$ is a polynomial?

I'm learning about the Cayley-Hamilton Theorem and the Primary Decomposition Theorem, and have trouble understanding what the notations mean. What does $\chi_T(T)$ mean? ($\chi_T(x)$ is the ...
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1answer
20 views

What can I assume, when given a matrix with information about its eigenvalues but not its action?

Basically, I've had to use linearity a couple of times yesterday and today, in order to write up a few proofs. But I notice that I am only given information such as positivity conditions and ...
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39 views

T-invariant subspaces

I'm studying for a qualifying exam, and I'd really appreciate some help on the following questions. Let p be a prime integer, $F=\mathbb{Z/pZ}$, V a vector space over F, and $T\in\mathscr{L}(V)$. ...
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102 views

If det $A = 0$ and $\det B \neq 0$ then show that $abc = -1$

This has been hurting my head for a while now.... If $$ \det\begin{bmatrix}a&a^2&1+a^3\\b&b^2&1+b^3\\c&c^2&1+c^3\end{bmatrix}=0 $$ And $$ ...
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36 views

Asymptotic behavior of the minimum eigenvalue of a certain Gram matrix with linear independence

Consider the density matrices with the following spectral decompositions: $$\rho=\lambda_1|\nu_1\rangle+\lambda_{2}|\nu_2\rangle$$ and $$\sigma=\gamma_1|\omega_1\rangle+\gamma_2|\omega_2\rangle$$ such ...
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A subgroup of special linear group

Does anybody know if the subgroup of diagonal and antidiagonal matrices of $SL(n,F)$ has been given a particular name? By $SL(n,F)$ I mean $n \times n$ matrices over a field $F$ with determinant 1. ...